**Analytical Techniques**

dynamical systems. The chapter considers both the predictability of atmospheric and cli‐ mate processes with respect to the initial data errors (predictability of the first kind) and the predictability with respect to external perturbations (predictability of the second kind). Chapter 6 extends the dynamical systems theory to quantum systems. Time-like operators are derived by exploiting the properties of operators and quantum states that are conjugated to the Hamiltonian operator and eigenstates when the Hamiltonian spectrum is continuous. Chapter 7 introduces some recent fixed-point techniques for the study of fractional set-val‐ ued dynamical systems. A general class of cyclic operators that satisfy the implicit contrac‐ tivity condition is considered. A number of fixed-point-inclusion results for fractional set-

The second section of the book is composed of four chapters that center on computational techniques. Chapter 8 explores the relationships between linear interpolation and differential equations. A class of spectral collocation (pseudospectral) methods, which are derived by a linear interpolation process, is constructed by exploiting the close relationship between the Green's function and Peano's kernel. These methods are illustrated through numerical solu‐ tions of several initial value and boundary value problem examples. Chapter 9 presents a computational technique that employs accurate, efficient, and reliable solvers based on ap‐ propriate combinations of surface integral equations, discretizations, numerical integrations, fast algorithms, and iterative techniques. As a case study, nanowire transmission lines are investigated in wide frequency ranges, demonstrating the capabilities of the computational technique. Chapter 10 is devoted to the existence of a true solution near a numerical approxi‐ mate random periodic solution of stochastic differential equations. A general finite-time ran‐ dom periodic shadowing theorem is proved under some suitable conditions, and an estimate of shadowing distance via computable quantities is provided. The applicability of this theo‐ rem is demonstrated through numerical simulations of random periodic orbits of the stochas‐ tic Lorenz system for certain parameter values. Finally, Chapter 11 covers some aspects of the analytical and numerical analysis procedures in the study of dynamical systems. It provides a brief summary to basic solution techniques and classification of ordinary and partial differen‐ tial equations. The chapter focuses on the two classes of most commonly used numerical methods, namely finite difference methods and finite element methods. Only a very limited number of techniques for solving ordinary differential and partial differential equations are discussed, as it is impossible to cover all the available techniques in a single chapter. The ap‐

plication of these methods is illustrated through a number of physical examples.

**Mahmut Reyhanoglu**

Daytona Beach, Florida

USA

Embry-Riddle Aeronautical University Dynamical Systems and Control Laboratory

valued systems in modular metric spaces are presented.

VIII Preface

Provisional chapter

## **On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems** On Nonoscillatory Solutions of Two-Dimensional

Elvan Akın and Özkan Öztürk

Additional information is available at the end of the chapter Elvan Akın and Özkan Öztürk

Nonlinear Dynamical Systems

http://dx.doi.org/10.5772/67118 Additional information is available at the end of the chapter

#### Abstract

During the past years, there has been an increasing interest in studying oscillation and nonoscillation criteria for dynamical systems on time scales that harmonize the oscillation and nonoscillation theory for the continuous and discrete cases in order to combine them in one comprehensive theory and eliminate obscurity from both. We not only classify nonoscillatory solutions of two-dimensional systems of first-order dynamic equations on time scales but also guarantee the existence of such solutions using the Knaster, Schauder-Tychonoff and Schauder's fixed point theorems. The approach is based on the sign of components of nonoscillatory solutions. A short introduction to the time scale calculus is given as well. Examples are significant in order to see if nonoscillatory solutions exist or not. Therefore, we give several examples in order to highlight our main results for the set of real numbers R, the set of integers Z and qN<sup>0</sup> = {1, q, q<sup>2</sup> , q<sup>3</sup> , …}, q >1, which are the most well-known time scales.

Keywords: dynamical systems, dynamic equations, differential equations, difference equations, time scales, oscillation

## 1. Introduction

In this chapter, we investigate the existence and classification of nonoscillatory solutions of two-dimensional (2D) nonlinear time-scale systems of first-order dynamic equations. The method we follow is based on the sign of components of nonoscillatory solutions and the most well-known fixed point theorems. The motivation of studying dynamic equations on time scales is to unify continuous and discrete analysis and harmonize them in one comprehensive theory and eliminate obscurity from both. A time scale T is an arbitrary nonempty closed subset of the real numbers R. The most well-known examples for time scales are R (which leads to

© 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

differential equations, see [1]), Z (which leads to difference equations, see Refs. [2, 3]) and <sup>q</sup><sup>N</sup><sup>0</sup> :<sup>¼</sup> {1, <sup>q</sup>, <sup>q</sup><sup>2</sup>, <sup>⋯</sup>}, <sup>q</sup> <sup>&</sup>gt; 1 (which leads to <sup>q</sup>-difference equations, see Ref. [4]). In 1988, the theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis [5]. We assume that most readers are not familiar with the calculus of time scales and therefore we give a brief introduction to time scales calculus in Section 2. In fact, we refer readers books [6, 7] by Bohner and Peterson for more details.

The study of 2D dynamic systems in nature and society has been motivated by their applications. Especially, a system of delay dynamic equations, considered in Section 4, take a lot of attention in all areas such as population dynamics, predator-prey epidemics, genomic and neuron dynamics and epidemiology in biological sciences, see [8, 9]. For instance, when the birth rate of preys is affected by the previous values rather than current values, a system of delay dynamic equations is utilized, because the rate of change at any time depends on solutions at prior times. Another novel application of delay dynamical systems is time delays that often arise in feedback loops involving actuators. A major issue faced in engineering is an unavoidable time delay between measurement and the signal received by the controller. In fact, the delay should be taken into consideration at the design stage to avoid the risk of instability, see Refs. [10, 11].

Another special case of 2D systems of dynamic equations is the Emden-Fowler type, which is covered in Section 5 of this chapter. The equation has several interesting applications, such as in astrophysics, gas dynamics and fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting systems, see Refs. [12–15]. For example, the fundamental problem in studying the stellar structure for gaseous dynamics in astrophysics was to look into the equilibrium formation of the mass of spherical clouds of gas for the continuous case, proposed by Kelvin and Lane, see Refs. [16, 17]. Such an equation is called Lane-Emden equation in literature. Much information about the solutions of Lane-Emden equation was provided by Ritter, see Ref. [18], in a series of 18 papers, published during 1878–1889. The mathematical foundation for the study of such an equation was made by Fowler in a series of four papers during 1914–1931, see Refs. [19–22].

## 2. Preliminaries

The set of real numbers R, the set of integers Z, the natural numbers N, the nonnegative integers N<sup>0</sup> and the Cantor set, q<sup>N</sup><sup>0</sup> , q > 1 and ½0, 1�∪½2, 3� are some examples of time scales. However, the set of rational numbers Q, the set of irrational numbers R\Q, the complex numbers C, and the open interval ð0, 1Þ are not considered as time scales.

Definition 2.1. [6, Definition 1.1] Let T be a time scale. For t ∈T, the forward jump operator σ : T ! T is given by

$$\sigma(t) := \inf \{ \mathbf{s} \in \mathbb{T} : \quad s > t \} \quad \text{for all} \quad t \in \mathbb{T}^\*$$

whereas the backward jump operator ρ : T ! T is defined by

$$\rho(t) := \sup \{ \mathbf{s} \in \mathbb{T} \, : \quad s < t \} \quad \text{for all} \quad t \in \mathbb{T}.$$

Finally, the graininess function μ : T ! ½0, ∞Þ is given by μðtÞ :¼ σðtÞ−t for all t∈T:

We define inf∅ ¼ supT. If σðtÞ > t, then t is called right-scattered, whereas if ρðtÞ < t, t is called left-scattered. If t is right- and left-scattered at the same time, then we say that t is isolated. If t < supT and σðtÞ ¼ t, then t is called right-dense, while if t > inf T and ρðtÞ ¼ t, we say that t is left-dense. Also, if t is right- and left-dense at the same time, then we say that t is dense.

Table 1 shows some examples of the forward and backward jump operators and the graininess function for most known time scales.


Table 1. Examples of most known time scales.

differential equations, see [1]), Z (which leads to difference equations, see Refs. [2, 3]) and <sup>q</sup><sup>N</sup><sup>0</sup> :<sup>¼</sup> {1, <sup>q</sup>, <sup>q</sup><sup>2</sup>, <sup>⋯</sup>}, <sup>q</sup> <sup>&</sup>gt; 1 (which leads to <sup>q</sup>-difference equations, see Ref. [4]). In 1988, the theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis [5]. We assume that most readers are not familiar with the calculus of time scales and therefore we give a brief introduction to time scales calculus in Section 2. In fact, we refer readers books [6, 7] by Bohner and

The study of 2D dynamic systems in nature and society has been motivated by their applications. Especially, a system of delay dynamic equations, considered in Section 4, take a lot of attention in all areas such as population dynamics, predator-prey epidemics, genomic and neuron dynamics and epidemiology in biological sciences, see [8, 9]. For instance, when the birth rate of preys is affected by the previous values rather than current values, a system of delay dynamic equations is utilized, because the rate of change at any time depends on solutions at prior times. Another novel application of delay dynamical systems is time delays that often arise in feedback loops involving actuators. A major issue faced in engineering is an unavoidable time delay between measurement and the signal received by the controller. In fact, the delay should be taken into consideration at the design stage to avoid the risk of

Another special case of 2D systems of dynamic equations is the Emden-Fowler type, which is covered in Section 5 of this chapter. The equation has several interesting applications, such as in astrophysics, gas dynamics and fluid mechanics, relativistic mechanics, nuclear physics and chemically reacting systems, see Refs. [12–15]. For example, the fundamental problem in studying the stellar structure for gaseous dynamics in astrophysics was to look into the equilibrium formation of the mass of spherical clouds of gas for the continuous case, proposed by Kelvin and Lane, see Refs. [16, 17]. Such an equation is called Lane-Emden equation in literature. Much information about the solutions of Lane-Emden equation was provided by Ritter, see Ref. [18], in a series of 18 papers, published during 1878–1889. The mathematical foundation for the study of such an equation was made by Fowler in a series of four papers

The set of real numbers R, the set of integers Z, the natural numbers N, the nonnegative integers N<sup>0</sup> and the Cantor set, q<sup>N</sup><sup>0</sup> , q > 1 and ½0, 1�∪½2, 3� are some examples of time scales. However, the set of rational numbers Q, the set of irrational numbers R\Q, the complex

Definition 2.1. [6, Definition 1.1] Let T be a time scale. For t ∈T, the forward jump operator

σðtÞ :¼ inf{s ∈T : s > t} for all t∈ T

numbers C, and the open interval ð0, 1Þ are not considered as time scales.

whereas the backward jump operator ρ : T ! T is defined by

Peterson for more details.

4 Dynamical Systems - Analytical and Computational Techniques

instability, see Refs. [10, 11].

during 1914–1931, see Refs. [19–22].

2. Preliminaries

σ : T ! T is given by

If sup<sup>T</sup> <sup>&</sup>lt; <sup>∞</sup>, then <sup>T</sup><sup>κ</sup> <sup>¼</sup> <sup>T</sup>\ðρðsupTÞ, supT� and <sup>T</sup><sup>κ</sup> <sup>¼</sup> <sup>T</sup> if sup<sup>T</sup> <sup>¼</sup> <sup>∞</sup>. Suppose that <sup>f</sup> : <sup>T</sup> ! <sup>R</sup> is a function. Then f <sup>σ</sup> : <sup>T</sup> ! <sup>R</sup> is defined by <sup>f</sup> σ ðtÞ ¼ fðσðtÞÞ for all t∈ T:

Definition 2.2. [6, Definition 1.10] For any ε, if there exists a δ > 0 such that

$$|f(\sigma(t)) - f(s) - f^\Delta(t)(\sigma(t) - \mathbf{s})| \le \varepsilon |\sigma(t) - \mathbf{s}| \quad \text{ for all} \quad s \in (t - \delta, t + \delta) \cap \mathbb{T},$$

then f is called delta (or Hilger) differentiable on T<sup>κ</sup> and f <sup>Δ</sup> is called delta derivative of f .

Theorem 2.3 [6, Theorem 1.16] Let f : <sup>T</sup> ! <sup>R</sup> be a function with t <sup>∈</sup>T<sup>κ</sup>. Then


$$f^A(t) = \frac{f(\sigma(t)) \neg f(t)}{\mu(t)}.$$

c. If t is right dense, then f is differentiable at t if and only if

$$f^{\mathcal{A}}(t) = \lim\_{s \to t} \frac{f(t) \neg f(s)}{t \to s}$$

exists as a finite number.

d. If f is differentiable at t, then fðσðtÞÞ ¼ fðtÞ þ μðtÞf <sup>Δ</sup>ðtÞ:

If T ¼ R, then f <sup>Δ</sup> turns out to be the usual derivative f ′ while f <sup>Δ</sup> is reduced to forward difference operator Δf if T ¼ Z: Finally, if T ¼ q<sup>N</sup><sup>0</sup> , then the delta derivative turns out to be q-difference operator Δq: The following theorem presents the sum, product and quotient rules on time scales.

Theorem 2.4 [6, Theorem 1.20] Let f , <sup>g</sup> : <sup>T</sup> ! <sup>R</sup> be differentiable at t <sup>∈</sup>T<sup>κ</sup>. Then

a. The sum f þ g : T ! R is differentiable at t with

$$(f+g)^A(t) = f^A(t) + g^A(t).$$

b. If f g : T ! R is differentiable at t, then

$$f(fg)^{\Delta}(t) = f^{\Delta}(t)g(t) + f(\sigma(t))g^{\Delta}(t) = f(t)g^{\Delta}(t) + f^{\Delta}(t)g(\sigma(t)).$$

c. If gðtÞgðσðtÞÞ≠0, then <sup>f</sup> <sup>g</sup> is differentiable at t with

$$\left(\frac{f}{\mathcal{g}}\right)^A(t) = \frac{f^A(t)\mathcal{g}(t) \neg f(t)\mathcal{g}^A(t)}{\mathcal{g}(t)\mathcal{g}(\sigma(t))}.$$

The following concepts must be introduced in order to define delta-integrable functions.

Definition 2.5. [6, Definition 1.58] f : T ! R is called right dense continuous (rd-continuous), denoted by Crd, CrdðTÞ, or CrdðT, RÞ, if it is continuous at right dense points in T and its leftsided limits exist as a finite number at left dense points in T. We denote continuous functions by C throughout this chapter.

Theorem 2.6 [6, Theorem 1.60] Let f : T ! R:


Also, the Cauchy integral is defined by

$$\int\_{a}^{b} f(t)\Delta t = F(b) - F(a) \quad \text{ for all} \quad a, b \in \mathbb{T}.$$

The following theorem presents the existence of antiderivatives.

Theorem 2.7 [6, Theorem 1.74] Every rd-continuous function has an antiderivative. Moreover, F given by

$$F(t) = \int\_{t\_0}^{t} f(s) \Delta s \quad \text{for} \quad t \in \mathbb{T}$$

is an antiderivative of f .

Theorem 2.8 [6, Theorems 1.76–1.77] Let a, b, c∈T, α∈ R, and f , g ∈ Crd. Then we have:

1. If f <sup>Δ</sup>≥0, then f is nondecreasing.

$$\text{2. } \newline If (t) \newline \text{20 } for \text{ } all \; a \le t \le b, \; then \; \int\_a^b f(t) \newline \text{ $\forall t \ge 0$ . }$$

$$\begin{aligned} \text{3.} \quad &\int\_{a}^{b} [(\alpha f(t)) + (\alpha g(t))] \Delta t = \alpha \int\_{a}^{b} f(t) \Delta t + \alpha \int\_{a}^{b} g(t) \Delta t. \\ \text{4.} \quad &\int\_{a}^{b} f(t) \Delta t = -\int\_{b}^{a} f(t) \Delta t. \\ \text{5.} \quad &\int\_{a}^{b} f(t) \Delta t = \int\_{a}^{c} f(t) \Delta t + \int\_{c}^{b} f(t) \Delta t. \\ \text{6.} \quad &\int\_{a}^{b} f(t) g^{\Delta}(t) \Delta t = (fg)(b) - (fg)(a) - \int\_{a}^{b} f^{\Delta}(t) g(\sigma(t)) \Delta t \\ \text{7.} \quad &\int\_{a}^{b} (\sigma(t)) g^{\Delta}(t) \Delta t = (fg)(b) - (fg)(a) - \int\_{a}^{b} f^{\Delta}(t) g(t) \Delta t \\ \text{8.} \quad &\int\_{a}^{b} f(t) \Delta t = 0. \end{aligned}$$

q-difference operator Δq: The following theorem presents the sum, product and quotient rules

<sup>Δ</sup>ðtÞ ¼ <sup>f</sup>

<sup>Δ</sup>ðtÞgðtÞ þ <sup>f</sup>ðσðtÞÞg<sup>Δ</sup>ðtÞ ¼ <sup>f</sup>ðtÞg<sup>Δ</sup>ðtÞ þ <sup>f</sup>

Definition 2.5. [6, Definition 1.58] f : T ! R is called right dense continuous (rd-continuous), denoted by Crd, CrdðTÞ, or CrdðT, RÞ, if it is continuous at right dense points in T and its leftsided limits exist as a finite number at left dense points in T. We denote continuous functions

fðtÞΔt ¼ FðbÞ−FðaÞ for all a, b∈ T:

Theorem 2.7 [6, Theorem 1.74] Every rd-continuous function has an antiderivative. Moreover, F

fðsÞΔs f or t ∈T

<sup>Δ</sup>ðtÞ þ <sup>g</sup><sup>Δ</sup>ðtÞ:

<sup>Δ</sup>ðtÞgðtÞ−fðtÞg<sup>Δ</sup>ðt<sup>Þ</sup> <sup>g</sup>ðtÞgðσðtÞÞ : <sup>Δ</sup>ðtÞgðσðtÞÞ:

Theorem 2.4 [6, Theorem 1.20] Let f , <sup>g</sup> : <sup>T</sup> ! <sup>R</sup> be differentiable at t <sup>∈</sup>T<sup>κ</sup>. Then

ðf þ gÞ

<sup>g</sup> is differentiable at t with f g � �<sup>Δ</sup>

<sup>ð</sup>tÞ ¼ <sup>f</sup>

The following concepts must be introduced in order to define delta-integrable functions.

a. The sum f þ g : T ! R is differentiable at t with

6 Dynamical Systems - Analytical and Computational Techniques

<sup>Δ</sup>ðtÞ ¼ <sup>f</sup>

b. If f g : T ! R is differentiable at t, then ðf gÞ

c. If gðtÞgðσðtÞÞ≠0, then <sup>f</sup>

by C throughout this chapter.

given by

is an antiderivative of f .

1. If f <sup>Δ</sup>≥0, then f is nondecreasing.

2. If fðtÞ≥0 for all a ≤ t ≤ b, then

Theorem 2.6 [6, Theorem 1.60] Let f : T ! R: a. If f is continuous, then f is rd-continuous.

b. The jump operator σ is rd-continuous.

Also, the Cauchy integral is defined by

ðb a

The following theorem presents the existence of antiderivatives.

ðb a

FðtÞ ¼

fðtÞΔt≥0:

ðt t0

Theorem 2.8 [6, Theorems 1.76–1.77] Let a, b, c∈T, α∈ R, and f , g ∈ Crd. Then we have:

on time scales.

Table 2 shows the derivative and integral definitions for the most known time scales for a, b∈T.


Table 2. Derivatives and integrals for most common time scales.

Finally, we finish the section by the following fixed point theorems.

Theorem 2.9 (Schauder's Fixed Point Theorem) [23, Theorem 2.A] Let S be a nonempty, closed, bounded, convex subset of a Banach space X and suppose that T : S ! S is a compact operator. Then, T has a fixed point.

The Schauder fixed point theorem was proved by Juliusz Schauder in 1930. In 1934, Tychonoff proved the same theorem for the case when S is a compact convex subset of a locally convex space X. In the literature, this version is known as the Schauder-Tychonoff fixed point theorem, see Ref. [24].

Theorem 2.10 (Schauder-Tychonoff Fixed Point Theorem). Let S be a compact convex subset of a locally convex (linear topological) space X and T a continuous map of S into itself. Then, T has a fixed point.

Finally, we provide the Knaster fixed point theorem, see Ref. [25].

Theorem 2.11 (Knaster Fixed Point Theorem) If ðS, ≤ Þ is a complete lattice and T : S ! S is orderpreserving (also called monotone or isotone), then T has a fixed point. In fact, the set of fixed points of T is a complete lattice.

## 3. Dynamical Systems on Time Scales

In this section, we consider the following system

$$\begin{cases} \mathbf{x}^A(t) = a(t)f(y(t)) \\ \mathbf{y}^A(t) = -b(t)\mathbf{g}(\mathbf{x}(t)), \end{cases} \tag{1}$$

where f , g∈CðR, RÞ are nondecreasing such that ufðuÞ > 0, ugðuÞ > 0 for u≠0 and a, b∈ Crdð½t0,∞ÞT, R<sup>þ</sup> � . The main results in this section come from Ref. [26]. If T ¼ R and T ¼ Z, Eq. (1) turns out to be a system of first-order differential equations and difference equations, see Refs. [27] and [28], respectively. Recent advances in oscillation and nonoscillation criteria for two-dimensional time scale systems have been studied in Refs. [29–31].

Throughout this chapter, we assume that T is unbounded above. Whenever we write t≥t1, we mean t ∈½t1,∞Þ<sup>T</sup> :¼ ½t1, ∞Þ∩T. We call ðx, yÞ a proper solution if it is defined on ½t0,∞Þ<sup>T</sup> and sup{jxðsÞj, jyðsÞj : s∈½t,∞ÞT} > 0 for t≥t0: A solution ðx, yÞ of Eq. (1) is said to be nonoscillatory if the component functions x and y are both nonoscillatory, i.e., either eventually positive or eventually negative. Otherwise, it is said to be oscillatory. The definitions above are also valid for systems considered in the next sections.

Assume that ðx, yÞ is a nonoscillatory solution of system (1) such that x oscillates but y is eventually positive. Then the first equation of system (1) yields <sup>x</sup><sup>Δ</sup>ðtÞ ¼ <sup>a</sup>ðtÞfðyðtÞÞ <sup>&</sup>gt; 0 eventually one sign for all large t≥t0, a contradiction. The case where y is eventually negative is similar. Therefore, we have that the component functions x and y are themselves nonoscillatory. In other words, any nonoscillatory solution ðx, yÞ of system (1) belongs to one of the following classes:

$$M^+ := \{ (\mathbf{x}, y) \in M \, : \, \mathbf{x}y > 0 \quad \text{eventually} \},$$

$$M^- := \{ (\mathbf{x}, y) \in M \, : \, \mathbf{x}y < 0 \quad \text{eventually} \},$$

where M is the set of all nonoscillatory solutions of system (1).

In this section, we only focus on the existence of nonoscillatory solutions of system (1) in M<sup>−</sup> , whereas M<sup>þ</sup> is considered together with delay system (12) in the following section.

For convenience, let us set

$$Y(t) = \int\_{t}^{\infty} a(\mathbf{s}) \Delta \mathbf{s} \qquad \text{and} \qquad Z(t) = \int\_{t}^{\infty} b(\mathbf{s}) \Delta \mathbf{s}. \tag{2}$$

We begin with the following results playing an important role in this chapter.

Lemma 3.1 Let ðx, yÞ be a nonoscillatory solution of system (1) and t<sup>0</sup> ∈ T. Then we have the followings:

a. [29, Lemma 2.3] If Yðt0Þ < ∞ and Zðt0Þ < ∞, then system (1) is nonoscillatory.


3. Dynamical Systems on Time Scales

8 Dynamical Systems - Analytical and Computational Techniques

In this section, we consider the following system

�

for systems considered in the next sections.

a, b∈ Crdð½t0,∞ÞT, R<sup>þ</sup>

of the following classes:

For convenience, let us set

followings:

<sup>x</sup><sup>Δ</sup>ðtÞ ¼ <sup>a</sup>ðtÞfðyðtÞÞ <sup>y</sup><sup>Δ</sup>ðtÞ ¼ <sup>−</sup>bðtÞgðxðtÞÞ,

where f , g∈CðR, RÞ are nondecreasing such that ufðuÞ > 0, ugðuÞ > 0 for u≠0 and

T ¼ Z, Eq. (1) turns out to be a system of first-order differential equations and difference equations, see Refs. [27] and [28], respectively. Recent advances in oscillation and nonoscillation

Throughout this chapter, we assume that T is unbounded above. Whenever we write t≥t1, we mean t ∈½t1,∞Þ<sup>T</sup> :¼ ½t1, ∞Þ∩T. We call ðx, yÞ a proper solution if it is defined on ½t0,∞Þ<sup>T</sup> and sup{jxðsÞj, jyðsÞj : s∈½t,∞ÞT} > 0 for t≥t0: A solution ðx, yÞ of Eq. (1) is said to be nonoscillatory if the component functions x and y are both nonoscillatory, i.e., either eventually positive or eventually negative. Otherwise, it is said to be oscillatory. The definitions above are also valid

Assume that ðx, yÞ is a nonoscillatory solution of system (1) such that x oscillates but y is eventually positive. Then the first equation of system (1) yields <sup>x</sup><sup>Δ</sup>ðtÞ ¼ <sup>a</sup>ðtÞfðyðtÞÞ <sup>&</sup>gt; 0 eventually one sign for all large t≥t0, a contradiction. The case where y is eventually negative is similar. Therefore, we have that the component functions x and y are themselves nonoscillatory. In other words, any nonoscillatory solution ðx, yÞ of system (1) belongs to one

M<sup>þ</sup> :¼ {ðx, yÞ ∈ M : xy > 0 eventually }

<sup>M</sup><sup>−</sup> :<sup>¼</sup> {ðx, <sup>y</sup><sup>Þ</sup> <sup>∈</sup> <sup>M</sup> : xy <sup>&</sup>lt; 0 eventually },

In this section, we only focus on the existence of nonoscillatory solutions of system (1) in M<sup>−</sup>

aðsÞΔs and ZðtÞ ¼

Lemma 3.1 Let ðx, yÞ be a nonoscillatory solution of system (1) and t<sup>0</sup> ∈ T. Then we have the

ð∞ t

bðsÞΔs: (2)

whereas M<sup>þ</sup> is considered together with delay system (12) in the following section.

We begin with the following results playing an important role in this chapter.

a. [29, Lemma 2.3] If Yðt0Þ < ∞ and Zðt0Þ < ∞, then system (1) is nonoscillatory.

where M is the set of all nonoscillatory solutions of system (1).

ð∞ t

YðtÞ ¼

criteria for two-dimensional time scale systems have been studied in Refs. [29–31].

. The main results in this section come from Ref. [26]. If T ¼ R and

(1)

,

�


Proof. Here, we only prove (a), (c) and (e) and the reader is asked to finish the proof in Exercise 3.2. To prove (a), choose t<sup>1</sup> ∈½t0,∞Þ<sup>T</sup> such that

$$\int\_{t\_1}^{\infty} a(t)f(1+g(2)\int\_{t}^{\infty} b(s)\Delta s)\Delta t < 1.$$

Let X be the space of all continuous functions on T with the norm ‖x‖ ¼ sup t ∈½t1,∞Þ<sup>T</sup> jxðtÞj and with the usual point-wise ordering ≤ . Define a subset Ω of X as

$$
\Omega := \{ \mathfrak{x} \in X \, : \quad 1 \le \mathfrak{x}(t) \le 2, \quad t \ge t\_1 \}.
$$

For any subset S of Ω, we have infS∈ Ω and supS∈ Ω. Define an operator F : Ω ! X such that

$$(F\mathbf{x})(t) = \mathbf{1} + \int\_{t\_1}^{t} a(s)f\left(\mathbf{1} + \int\_{s}^{\infty} b(u)\mathbf{g}(\mathbf{x}(u))\Delta u\right)d\mathbf{s}, \quad t \ge t\_1.$$

By using the monotonicity and the fact that x ∈ Ω, we have

$$1 \le (F\mathfrak{x})(t) \le 1 + \int\_{t\_1}^t a(s)f\left(1 + g(2)\int\_s^\infty b(u)\Delta u\right)\Delta s \le 2, \quad t \ge t\_1.$$

It is also easy to show that F is an increasing mapping. So by Theorem 2.11, there exists x∈ Ω such that Fx ¼ x. Then we have

$$
\overline{\mathfrak{X}}^{\Lambda}(t) = a(t)f\left(1 + \int\_{t}^{\infty} b(u)g(\overline{\mathfrak{x}}(u))d\Lambda u\right).
$$

Setting

$$\overline{y}(t) = 1 + \int\_{t}^{\infty} b(u)g(\overline{x}(u)) \Delta u > 0, \quad t \ge t\_1$$

gives us

$$
\overline{y}^{\mathbb{A}}(t) = -b(t)g(\overline{x}(t)) \quad \text{and} \quad \overline{x}^{\mathbb{A}}(t) = a(t)f(\overline{y}(t)),
$$

that is, ðx, yÞ is a nonoscillatory solution of Eq. (1). In order to prove part (c), assume that there exists a nonoscillatory solution ðx, yÞ of system (1) in M<sup>þ</sup> such that xðtÞ > 0 for t≥t1. Then by monotonicity of x and g, there exists a number k > 0 such that gðxðtÞÞ≥k for t≥t1. Integrating the second equation of system from t<sup>1</sup> to t gives us

$$y(t) \le y(t\_1) \neg k \int\_{t\_1}^t b(s) \Delta s.$$

As t ! ∞, it follows yðtÞ ! −∞. But this contradicts that y is eventually positive. Finally for part (e), without loss of generality, we assume that there exists t1≥t<sup>0</sup> such that xðtÞ > 0 for t≥t1. If <sup>ð</sup>x, <sup>y</sup>Þ<sup>∈</sup> <sup>M</sup><sup>−</sup> , then by the first equation of system (1), <sup>x</sup><sup>Δ</sup>ðt<sup>Þ</sup> <sup>&</sup>lt; 0 for <sup>t</sup>≥t1. Hence, the limit of <sup>x</sup> exists. So let us show that the assertion follows if ðx, yÞ∈ Mþ. Suppose ðx, yÞ ∈ Mþ. Then from the first equation of system (1), we have <sup>x</sup><sup>Δ</sup>ðt<sup>Þ</sup> <sup>&</sup>gt; 0 for <sup>t</sup>≥t1. Now let us show that lim<sup>t</sup>!<sup>∞</sup>xðtÞ ¼ <sup>∞</sup> cannot happen. Integrating the first equation of system (1) from t<sup>1</sup> to t and using the monotonicity of y and f yield

$$
\varkappa(t) \le \varkappa(t\_1) + f(\lg(t\_1)) \int\_{t\_1}^t a(s) \Delta s.
$$

Taking the limit as t ! ∞, it follows that x has a finite limit. This completes the proof.

Exercise 3.2. Prove the remainder of Lemma 3.1.

Throughout this section, we assume Yðt0Þ < ∞ and Zðt0Þ ¼ ∞. Note that Lemma 3.1 (c) indicates <sup>M</sup><sup>þ</sup> <sup>¼</sup> <sup>∅</sup>. Therefore, every nonoscillatory solution of system (1) belongs to <sup>M</sup><sup>−</sup> . Let ðx, yÞ be a nonoscillatory solution of system (1) such that the component function x of solution ðx, yÞ is eventually positive. Then, the second equation of system (1) yields y < 0 and eventually decreasing. Then for k < 0, we have that y approaches k or −∞. In view of Lemma 3.1 (e), x has a finite limit. So in light of this information, any nonoscillatory solution of system (1) in M<sup>−</sup> belongs to one of the following subclasses for 0 < c < ∞ and 0 < d < ∞:

$$\begin{aligned} M\_{0,B}^- &= \{ (\mathbf{x}, y) \in M^- : \lim\_{t \to \infty} |\mathbf{x}(t)| = 0, \quad \lim\_{t \to \infty} |y(t)| = d \}, \\\\ M\_{B,B}^- &= \{ (\mathbf{x}, y) \in M^- : \lim\_{t \to \infty} |\mathbf{x}(t)| = c, \quad \lim\_{t \to \infty} |y(t)| = d \}, \\\\ M\_{0,\circ}^- &= \{ (\mathbf{x}, y) \in M^- : \lim\_{t \to \infty} |\mathbf{x}(t)| = 0, \; \lim\_{t \to \infty} |y(t)| = \infty \}, \\\\ M\_{B,\circ}^- &= \{ (\mathbf{x}, y) \in M^- : \lim\_{t \to \infty} |\mathbf{x}(t)| = c, \; \lim\_{t \to \infty} |y(t)| = \infty \}. \end{aligned}$$

Nonoscillatory solutions in M<sup>−</sup> <sup>0</sup>;<sup>∞</sup> is called slowly decaying solutions in literature, see [32]. The following theorems show the existence of nonoscillatory solutions in subclasses of M<sup>−</sup> given above. Our approach for the next two theorems is based on the Schauder fixed point theorem, see Theorem 2.9.

Theorem 3.3 M<sup>−</sup> <sup>0</sup>;<sup>B</sup>≠∅ if and only if

$$\int\_{t\_0}^{\infty} b(t) \mathbf{g}\left(\mathbf{c}\_1 \int\_t^{\infty} a(\mathbf{s}) \Delta \mathbf{s}\right) \Delta t < \infty, \quad \mathbf{c}\_1 \neq \mathbf{0}. \tag{3}$$

Proof. Suppose that there exists a solution <sup>ð</sup>x, <sup>y</sup>Þ<sup>∈</sup> <sup>M</sup><sup>−</sup> <sup>0</sup>;<sup>B</sup> such that xðtÞ > 0 for t≥t0, xðtÞ ! 0 and yðtÞ ! −d as t ! ∞, where d > 0. Integrating the first equation of system (1) from t to ∞ and the monotonicity of f yield that there exists c > 0 such that

$$\mathbf{x}(t) \succeq \mathbf{c} \int\_{t}^{\infty} \mathbf{a}(\mathbf{s}) \Delta \mathbf{s}, \quad t \succeq t\_{0}. \tag{4}$$

By integrating the second equation from t<sup>0</sup> to t, using inequality (4) with c ¼ c<sup>1</sup> and the monotonicity of g, we have

$$y(t) = y(t\_0) - \int\_{t\_0}^t b(s)g(\mathbf{x}(s))\Delta s \le -\int\_{t\_0}^t b(s)g\left(c\_1 \int\_s^\infty a(u)\Delta u\right)\Delta s.$$

So as t ! ∞, the assertion follows since y has a finite limit. (For the case x < 0 eventually, the proof can be shown similarly with c<sup>1</sup> < 0:Þ

Conversely, suppose that Eq. (3) holds for some c<sup>1</sup> > 0: ðFor the case c<sup>1</sup> < 0 can be shown similarly.Þ Then there exist t1≥t<sup>0</sup> and d > 0 such that

$$\int\_{t\_1}^{\infty} b(t)g\left(c\_1 \int\_t^{\infty} a(s)\Delta s\right) \Delta t < d, \quad t \ge t\_1,\tag{5}$$

where c<sup>1</sup> ¼ −fð−3dÞ. Let X be the space of all continuous and bounded functions on ½t1,∞Þ<sup>T</sup> with the norm ‖y‖ ¼ sup t ∈½t1,∞Þ<sup>T</sup> jyðtÞj. Then X is a Banach space, see Ref. [33]. Let Ω be the subset of X

such that

monotonicity of x and g, there exists a number k > 0 such that gðxðtÞÞ≥k for t≥t1. Integrating the

As t ! ∞, it follows yðtÞ ! −∞. But this contradicts that y is eventually positive. Finally for part (e), without loss of generality, we assume that there exists t1≥t<sup>0</sup> such that xðtÞ > 0 for t≥t1.

exists. So let us show that the assertion follows if ðx, yÞ∈ Mþ. Suppose ðx, yÞ ∈ Mþ. Then from the first equation of system (1), we have <sup>x</sup><sup>Δ</sup>ðt<sup>Þ</sup> <sup>&</sup>gt; 0 for <sup>t</sup>≥t1. Now let us show that lim<sup>t</sup>!<sup>∞</sup>xðtÞ ¼ <sup>∞</sup> cannot happen. Integrating the first equation of system (1) from t<sup>1</sup> to t and using the monoto-

ðt t1 bðsÞΔs:

, then by the first equation of system (1), <sup>x</sup><sup>Δ</sup>ðt<sup>Þ</sup> <sup>&</sup>lt; 0 for <sup>t</sup>≥t1. Hence, the limit of <sup>x</sup>

ðt t1 aðsÞΔs:

<sup>j</sup>xðtÞj ¼ <sup>0</sup>, lim<sup>t</sup>!<sup>∞</sup>

<sup>j</sup>xðtÞj ¼ <sup>c</sup>, lim<sup>t</sup>!<sup>∞</sup>

<sup>j</sup>xðtÞj ¼ <sup>0</sup>, lim<sup>t</sup>!<sup>∞</sup>

<sup>j</sup>xðtÞj ¼ <sup>c</sup>, lim<sup>t</sup>!<sup>∞</sup>

following theorems show the existence of nonoscillatory solutions in subclasses of M<sup>−</sup> given above. Our approach for the next two theorems is based on the Schauder fixed point theorem,

jyðtÞj ¼ dg,

jyðtÞj ¼ dg,

jyðtÞj ¼ ∞g,

jyðtÞj ¼ ∞g:

Δt < ∞, c1≠0: (3)

<sup>0</sup>;<sup>B</sup> such that xðtÞ > 0 for t≥t0, xðtÞ ! 0 and

<sup>0</sup>;<sup>∞</sup> is called slowly decaying solutions in literature, see [32]. The

. Let ðx, yÞ

yðtÞ ≤ yðt1Þ−k

xðtÞ ≤ xðt1Þ þ fðyðt1ÞÞ

Taking the limit as t ! ∞, it follows that x has a finite limit. This completes the proof.

cates <sup>M</sup><sup>þ</sup> <sup>¼</sup> <sup>∅</sup>. Therefore, every nonoscillatory solution of system (1) belongs to <sup>M</sup><sup>−</sup>

belongs to one of the following subclasses for 0 < c < ∞ and 0 < d < ∞:

<sup>0</sup>;<sup>B</sup> ¼ fðx, <sup>y</sup><sup>Þ</sup> <sup>∈</sup> <sup>M</sup><sup>−</sup> : lim<sup>t</sup>!<sup>∞</sup>

<sup>B</sup>,<sup>B</sup> ¼ fðx, <sup>y</sup><sup>Þ</sup> <sup>∈</sup> <sup>M</sup><sup>−</sup> : lim<sup>t</sup>!<sup>∞</sup>

<sup>0</sup>;<sup>∞</sup> ¼ fðx, <sup>y</sup>Þ<sup>∈</sup> <sup>M</sup><sup>−</sup> : lim<sup>t</sup>!<sup>∞</sup>

<sup>B</sup>,<sup>∞</sup> ¼ fðx, <sup>y</sup>Þ<sup>∈</sup> <sup>M</sup><sup>−</sup> : lim<sup>t</sup>!<sup>∞</sup>

Throughout this section, we assume Yðt0Þ < ∞ and Zðt0Þ ¼ ∞. Note that Lemma 3.1 (c) indi-

be a nonoscillatory solution of system (1) such that the component function x of solution ðx, yÞ is eventually positive. Then, the second equation of system (1) yields y < 0 and eventually decreasing. Then for k < 0, we have that y approaches k or −∞. In view of Lemma 3.1 (e), x has a finite limit. So in light of this information, any nonoscillatory solution of system (1) in M<sup>−</sup>

second equation of system from t<sup>1</sup> to t gives us

10 Dynamical Systems - Analytical and Computational Techniques

Exercise 3.2. Prove the remainder of Lemma 3.1.

M<sup>−</sup>

M<sup>−</sup>

M<sup>−</sup>

M<sup>−</sup>

<sup>0</sup>;<sup>B</sup>≠∅ if and only if

Proof. Suppose that there exists a solution <sup>ð</sup>x, <sup>y</sup>Þ<sup>∈</sup> <sup>M</sup><sup>−</sup>

ð∞ t0 bðtÞg � c1 ð∞ t aðsÞΔs �

Nonoscillatory solutions in M<sup>−</sup>

see Theorem 2.9. Theorem 3.3 M<sup>−</sup>

If <sup>ð</sup>x, <sup>y</sup>Þ<sup>∈</sup> <sup>M</sup><sup>−</sup>

nicity of y and f yield

$$\mathcal{Q} := \{ y \in X \, : \, -\mathfrak{B}d \le y(t) \le -2d, \quad t \ge t\_1 \}$$

and define an operator T : Ω ! X such that

$$(Ty)(t) = -3d + \int\_t^\infty b(s)g\left(-\int\_s^\infty a(\mu)f(y(\mu))\Delta\mu\right)d\nu.$$

It is easy to see that T maps into itself. Indeed, we have

$$- \mathfrak{Z}d \le (Ty)(t) \le -\mathfrak{Z}d + \int\_t^\infty b(s)g\left(-\int\_s^\infty a(u)f(-\mathfrak{Z}d)\,\Delta u\right) \Delta s \le -\mathfrak{Z}d$$

by Eq. (5). Let us show that T is continuous on Ω. To accomplish this, let yn be a sequence in Ω such that yn ! y∈ Ω ¼ Ω: Then

$$\begin{aligned} &|(Ty\_n)(t) - (Ty)(t)| \\ &\le \int\_{t\_1}^{\infty} b(s) |[g\left(-\int\_s^{\infty} a(u)f(y\_n(u))\Delta u\right) - g\left(-\int\_s^{\infty} a(u)f(y(u))\Delta u\right)| |\Delta s.| \end{aligned}$$

Then the Lebesque dominated convergence theorem and the continuity of g give ‖ðTynÞ −ðTyÞ‖ ! 0 as n ! ∞, i.e., T is continuous. Also, since

$$0 < -(Ty)^{\Delta}(t) = b(t)g\left(-\int\_{t}^{\infty} a(u)f(y(u))\Delta u\right) < \infty,$$

it follows that TðΩÞ is relatively compact. Then by Theorem 2.9, we have that there exists y∈ Ω such that y ¼ Ty: So as t ! ∞, we have yðtÞ ! −3d < 0. Setting

$$\overline{\mathfrak{X}}(t) = -\int\_{t}^{\infty} a(\mu) f(\overline{\mathfrak{Y}}(\mu)) \Delta \mu > 0, \quad t \mathbb{2}t\_1$$

gives that <sup>x</sup>ðtÞ ! 0 as <sup>t</sup> ! <sup>∞</sup> and implies <sup>x</sup><sup>Δ</sup> <sup>¼</sup> afðyÞ, i.e., <sup>ð</sup>x, <sup>y</sup><sup>Þ</sup> is a nonoscillatory solution in M<sup>−</sup> <sup>0</sup>;<sup>B</sup>.

In the following example, we apply Theorem 3.3 to show the nonemptiness of M<sup>−</sup> <sup>0</sup>;<sup>B</sup>.

Example 3.4 Let T ¼ q<sup>N</sup><sup>0</sup> , q > 1 and consider the system

$$\begin{cases} \Delta\_{\boldsymbol{q}}\boldsymbol{x}(t) = \frac{t^{\frac{1}{5}}}{(t+1)(tq+1)(2t-1)^{\frac{5}{5}}} y^{\frac{1}{5}}(t) \\ \Delta\_{\boldsymbol{q}}\boldsymbol{y}(t) = -\frac{(t+1)^{\frac{5}{5}}}{qt^2}\mathbf{x}^{\frac{5}{5}}(t). \end{cases} \tag{6}$$

Since

$$\int\_1^T a(s)\Delta s = (q-1)\sum\_{s\in[1,T)\_{\varrho^{\mathbb{N}}}} \frac{s^{\frac{4}{3}}}{(s+1)(sq+1)(2s-1)^{\frac{1}{3}}} \le (q-1)\sum\_{s\in[1,T)\_{\varrho^{\mathbb{N}}}} \frac{1}{s^{\frac{2}{3}}},$$

where t <sup>¼</sup> qn and s <sup>¼</sup> tqm, n, <sup>m</sup> <sup>∈</sup> <sup>N</sup>0, we obtain

$$Y(1) \le (q-1) \sum\_{n=0}^{\infty} \left(\frac{1}{q^{\frac{2}{3}}}\right)^n < \infty.$$

Also, ðT 1 <sup>b</sup>ðsÞΔ<sup>s</sup> <sup>¼</sup> <sup>X</sup> s∈ ½1,TÞ qN0 <sup>ð</sup>sþ1<sup>Þ</sup> <sup>5</sup> 3 qs<sup>2</sup> <sup>ð</sup>q−1Þs<sup>≥</sup> <sup>q</sup>−<sup>1</sup> q X s∈½1,TÞ qN0 s 2 <sup>3</sup> implies Zð1Þ<sup>≥</sup> <sup>q</sup>−<sup>1</sup> q X∞ m¼0 ðq 2 3Þ <sup>m</sup> <sup>¼</sup> <sup>∞</sup>: Now let us

show that Eq. (3) holds. First,

$$\int\_{t}^{T} a(\mathbf{s}) \Delta \mathbf{s} \leq (q-1) \sum\_{\mathbf{s} \in \left[t, T\right]\_{q}} \frac{1}{\mathbf{s}^{\frac{2}{3}}} \quad \text{implies} \quad \int\_{t}^{\prime} a(\mathbf{s}) \Delta \mathbf{s} \leq (q-1) \sum\_{\mathbf{s} \in \left[t, \infty\right)\_{q} \mathbb{K}\_{0}} \frac{1}{\mathbf{s}^{\frac{2}{3}}} = \frac{q^{\frac{2}{3}}(q-1)}{(q^{\frac{2}{3}}-1)t^{\frac{2}{3}}}.$$

Therefore,

$$\int\_{1}^{T} b(t) \lg\left(c\_{1} \int\_{t}^{\infty} a(s) \Delta s\right) \Delta t \le \alpha \sum\_{t \in \left[1, T\right)\_{q}^{\mathbb{N}\_{0}}} \frac{(t+1)^{\frac{5}{5}}}{t^{\frac{19}{10}}},$$

where <sup>α</sup> <sup>¼</sup> <sup>ð</sup>q−1<sup>Þ</sup> 2 q 1 9 ðq 2 <sup>3</sup>−1Þ 5 3 : So as T ! ∞, we have that Eq. (3) holds by the Ratio test. One can also show that 1 <sup>t</sup>þ<sup>1</sup> , <sup>−</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> t � � of system (6) such that xðtÞ ! <sup>0</sup> and yðtÞ ! <sup>−</sup><sup>2</sup> as t ! <sup>∞</sup>, i.e., M<sup>−</sup> <sup>0</sup>;<sup>B</sup>≠∅.

The proof of the following theorem is similar to the proof of Theorem 3.3.

Theorem 3.5 M<sup>−</sup> <sup>B</sup>,B≠∅ if and only if

0 < −ðTyÞ

12 Dynamical Systems - Analytical and Computational Techniques

Example 3.4 Let T ¼ q<sup>N</sup><sup>0</sup> , q > 1 and consider the system

8 >>>><

>>>>:

<sup>a</sup>ðsÞΔ<sup>s</sup> ¼ ðq−1<sup>Þ</sup> <sup>X</sup>

where t <sup>¼</sup> qn and s <sup>¼</sup> tqm, n, <sup>m</sup> <sup>∈</sup> <sup>N</sup>0, we obtain

<sup>ð</sup>sþ1<sup>Þ</sup> <sup>5</sup> 3 qs<sup>2</sup> <sup>ð</sup>q−1Þs<sup>≥</sup> <sup>q</sup>−<sup>1</sup>

<sup>a</sup>ðsÞΔ<sup>s</sup> <sup>≤</sup> <sup>ð</sup>q−1<sup>Þ</sup> <sup>X</sup>

s∈ ½t,TÞ q N0

> ðT 1 bðtÞg � c1 ð∞ t aðsÞΔs �

M<sup>−</sup> <sup>0</sup>;<sup>B</sup>.

Since

Also, ðT 1

Therefore,

where <sup>α</sup> <sup>¼</sup> <sup>ð</sup>q−1<sup>Þ</sup>

� �

1 <sup>t</sup>þ<sup>1</sup> , <sup>−</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> t

<sup>b</sup>ðsÞΔ<sup>s</sup> <sup>¼</sup> <sup>X</sup>

ðT t

s∈ ½1,TÞ q N0

show that Eq. (3) holds. First,

ðq 2 <sup>3</sup>−1Þ 5 3

ðT 1

such that y ¼ Ty: So as t ! ∞, we have yðtÞ ! −3d < 0. Setting

xðtÞ ¼ −

<sup>Δ</sup>ðtÞ ¼ <sup>b</sup>ðtÞ<sup>g</sup>

ð∞ t

In the following example, we apply Theorem 3.3 to show the nonemptiness of M<sup>−</sup>

<sup>Δ</sup>qxðtÞ ¼ <sup>t</sup>

ðt þ 1Þ 5 3 qt<sup>2</sup> <sup>x</sup> 5 <sup>3</sup>ðtÞ:

ΔqyðtÞ ¼ −

s ∈½1,TÞ q N0

q

Yð1Þ ≤ ðq−1Þ

X s∈½1,TÞ qN0 s 2

implies

of system (6) such that xðtÞ ! <sup>0</sup> and yðtÞ ! <sup>−</sup><sup>2</sup> as t ! <sup>∞</sup>, i.e., M<sup>−</sup>

� − ð∞ t

it follows that TðΩÞ is relatively compact. Then by Theorem 2.9, we have that there exists y∈ Ω

gives that <sup>x</sup>ðtÞ ! 0 as <sup>t</sup> ! <sup>∞</sup> and implies <sup>x</sup><sup>Δ</sup> <sup>¼</sup> afðyÞ, i.e., <sup>ð</sup>x, <sup>y</sup><sup>Þ</sup> is a nonoscillatory solution in

aðuÞfðyðuÞÞΔu > 0, t≥t<sup>1</sup>

1 3 ðt þ 1Þðtq þ 1Þð2t−1Þ

s 4 3 ðs þ 1Þðsq þ 1Þð2s−1Þ

!<sup>n</sup>

X∞ n¼0

> ð∞ t

Δt ≤ α X t∈ ½1,TÞ q N0

: So as T ! ∞, we have that Eq. (3) holds by the Ratio test. One can also show that

1 3 y 1 <sup>3</sup>ðtÞ

1 3

< ∞:

<sup>3</sup> implies Zð1Þ<sup>≥</sup> <sup>q</sup>−<sup>1</sup>

<sup>a</sup>ðsÞΔ<sup>s</sup> <sup>≤</sup> <sup>ð</sup>q−1<sup>Þ</sup> <sup>X</sup>

<sup>≤</sup> <sup>ð</sup>q−1<sup>Þ</sup> <sup>X</sup> s ∈½1,TÞ q N0

> q X∞ m¼0 ðq 2 3Þ

s∈ ½t,∞Þ qN0

ðt þ 1Þ 5 3

t 19 10 ,

1 s 2 3 ¼ q 2 <sup>3</sup>ðq−1Þ

ðq 2 <sup>3</sup>−1Þt 2 3 :

<sup>0</sup>;<sup>B</sup>≠∅.

1 s 2 3 ,

<sup>m</sup> <sup>¼</sup> <sup>∞</sup>: Now let us

aðuÞfðyðuÞÞΔu

� < ∞,

<sup>0</sup>;<sup>B</sup>.

(6)

$$\int\_{t\_0}^{\infty} b(t)g\left(d\_1 \cdots c\_1 \int\_t^{\infty} a(s)\Delta s\right) \Delta t < \infty$$

for some c<sup>1</sup> < 0 and d<sup>1</sup> > 0: ðOr c<sup>1</sup> > 0 and d<sup>1</sup> < 0:Þ

Exercise 3.6. Prove Theorem 3.5 by means of Theorem 2.9.

The following theorem follows from the Knaster fixed point theorem, see Theorem 2.11.

Theorem 3.7 M<sup>−</sup> <sup>B</sup>,∞≠∅ if and only if

$$\int\_{t\_0}^{\infty} a(s) f\left(c\_1 \int\_{t\_0}^s b(u) \Delta u\right) \Delta s < \infty \tag{7}$$

for some c1≠0, where f is an odd function.

Proof. Suppose that there exists a nonoscillatory solution <sup>ð</sup>x, <sup>y</sup>Þ<sup>∈</sup> <sup>M</sup><sup>−</sup> <sup>B</sup>,<sup>∞</sup> such that x > 0 eventually, xðtÞ ! c<sup>2</sup> and yðtÞ ! −∞ as t ! ∞, where 0 < c<sup>2</sup> < ∞. Because of the monotonicity of x and the fact that x has a finite limit, there exist t1≥t<sup>0</sup> and c<sup>3</sup> > 0 such that

$$
\mathfrak{c}\_2 \le \mathfrak{x}(t) \le \mathfrak{c}\_3 \quad \text{ for} \quad t \ge t\_1. \tag{8}
$$

Integrating the first equation from t<sup>1</sup> to t gives us

$$c\_2 \le \mathbf{x}(t) = \mathbf{x}(t\_1) + \int\_{t\_1}^t \mathbf{a}(s) f(\mathbf{y}(s)) \, \Delta s \le c\_3, \quad t \ge t\_1.$$

So by taking the limit as t ! ∞, we have

$$\int\_{t\_1}^{\infty} a(s) |f(y(s))| \, |\Delta s| < \infty. \tag{9}$$

The monotonicity of g, Eq. (8) and integrating the second equation from t<sup>1</sup> to t yield

$$y(t) \le y(t\_1) - \mathcal{g}(c\_2) \int\_{t\_1}^t b(s) \Delta s \le -\mathcal{g}(c\_2) \int\_{t\_1}^t b(s) \Delta s.$$

Since fð−uÞ ¼ −fðuÞ for u≠0 and by the monotonicity of f , we have

$$|f(\boldsymbol{y}(t))| \Big| \forall f \Big(\boldsymbol{g}(c\_2) \Big| \int\_{t\_1}^{t} b(s) \Delta s \Big), \quad t \succeq t\_1. \tag{10}$$

By Eqs. (9) and (10), we have

$$\int\_{t\_1}^{t} a(s)|f(\mathcal{y}(s))|\Delta s \ge \int\_{t\_1}^{t} a(s)f\left(\mathcal{g}(c\_2)\int\_{t\_1}^{s} b(u)\Delta u\right)\Delta s, \quad \text{where} \quad \mathcal{g}(c\_2) = c\_1.$$

As t ! ∞, the proof is finished. (The case x < 0 eventually can be proved similarly with c<sup>1</sup> < 0.)

Conversely, suppose <sup>ð</sup><sup>∞</sup> t0 aðsÞf � c1 ðs t0 bðuÞΔu � Δs < ∞ for some c1≠0. Without loss of generality, assume c<sup>1</sup> > 0. (The case c<sup>1</sup> < 0 can be done similarly.) Then, we can choose t1≥t<sup>0</sup> and d > 0 such that

$$\int\_{t\_1}^{\infty} a(s) f\left(c\_1 \int\_{t\_1}^{s} b(u) \Delta u\right) \Delta s < d, \quad t \ge t\_1,$$

where c<sup>1</sup> ¼ gð2dÞ > 0: Let X be the partially ordered Banach space of all real-valued continuous functions endowed with supremum norm ‖x‖ ¼ sup t∈½t1,∞Þ<sup>T</sup> jxðtÞj and with the usual pointwise

ordering ≤ . Define a subset Ω of X such that

$$\mathcal{Q} = \colon \{ \mathbf{x} \in \mathcal{X} \,:\, \quad d \le \mathbf{x}(t) \le 2d, \quad t \ge t\_1 \}.$$

For any subset B of Ω, infB∈ Ω and supB∈ Ω, i.e., ðΩ, ≤ Þ is complete. Define an operator F : Ω ! X as

$$(F\mathfrak{x})(t) = d + \int\_{t}^{\infty} a(\mathfrak{s}) f\left(\int\_{t\_1}^{s} b(\mathfrak{u}) \mathfrak{g}(\mathfrak{x}(\mathfrak{u})) \Delta \mathfrak{u}\right) \Delta \mathfrak{s}, \quad t \ge t\_1 \dots$$

The rest of the proof can be completed similar to the proof of Lemma 3.1(a). So, it is omitted.

Exercise 3.8 Let <sup>T</sup> <sup>¼</sup> <sup>Z</sup>: Use Theorem 3.7 to justify that <sup>ð</sup>xn, ynÞ¼ð<sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>−</sup><sup>n</sup>, <sup>−</sup>2<sup>n</sup><sup>Þ</sup> is a nonoscillatory solution in M<sup>−</sup> <sup>B</sup>,<sup>∞</sup> of

$$\begin{cases} \Delta \mathbf{x}\_n = 2^{\frac{-6n}{5} - 1} (y\_n)^{\frac{1}{5}} \\ \Delta y\_n = -\frac{4^n}{1 + 2^n} (\mathbf{x}\_n) . \end{cases}$$

For convenience, set

$$I = \int\_{t\_0}^{\infty} a(t) f\left(k \int\_t^{\infty} b(s) \Delta s\right) \Delta t, \quad k \neq 0. \tag{11}$$

In order to obtain the nonemptiness of M<sup>−</sup> <sup>0</sup>;<sup>∞</sup>, we apply Theorem 2.11 and use the similar discussion as in Lemma 3.1(a).

Theorem 3.9 M<sup>−</sup> <sup>0</sup>;<sup>∞</sup>≠∅ if for some k > 0 and any d<sup>1</sup> > 0 ðk < 0 and d<sup>1</sup> < 0Þ

$$I < \infty \qquad \text{and} \qquad \int\_{t\_0}^{\infty} b(t)g\left(d\_1 \int\_t^{\infty} a(s)\Delta s\right) \Delta t = \infty,$$

where I is defined as in Eq. (11) and f is an odd function.

Exercise 3.10. Prove Theorem 3.9.

We reconsider system (1) in the next section to emphasize the existence of nonoscillatory solutions in Mþ.

## 4. Delay Dynamical Systems on Time Scales

This section is concerned with the delay system

Conversely, suppose

such that

F : Ω ! X as

nonoscillatory solution in M<sup>−</sup>

discussion as in Lemma 3.1(a).

Exercise 3.10. Prove Theorem 3.9.

For convenience, set

Theorem 3.9 M<sup>−</sup>

solutions in Mþ.

ð∞ t0 aðsÞf � c1 ðs t0 bðuÞΔu �

14 Dynamical Systems - Analytical and Computational Techniques

ð∞ t1 aðsÞf � c1 ðs t1 bðuÞΔu �

functions endowed with supremum norm ‖x‖ ¼ sup

ðFxÞðtÞ ¼ d þ

<sup>B</sup>,<sup>∞</sup> of

I ¼ ð∞ t0 aðtÞf � k ð∞ t bðsÞΔs �

I < ∞ and

where I is defined as in Eq. (11) and f is an odd function.

In order to obtain the nonemptiness of M<sup>−</sup>

ð∞ t aðsÞf �ð<sup>s</sup> t1

8 ><

>:

ordering ≤ . Define a subset Ω of X such that

Δs < ∞ for some c1≠0. Without loss of generality,

jxðtÞj and with the usual pointwise

assume c<sup>1</sup> > 0. (The case c<sup>1</sup> < 0 can be done similarly.) Then, we can choose t1≥t<sup>0</sup> and d > 0

where c<sup>1</sup> ¼ gð2dÞ > 0: Let X be the partially ordered Banach space of all real-valued continuous

Ω ¼: {x ∈ X : d ≤ xðtÞ ≤ 2d, t≥t1}:

For any subset B of Ω, infB∈ Ω and supB∈ Ω, i.e., ðΩ, ≤ Þ is complete. Define an operator

The rest of the proof can be completed similar to the proof of Lemma 3.1(a). So, it is omitted.

Exercise 3.8 Let <sup>T</sup> <sup>¼</sup> <sup>Z</sup>: Use Theorem 3.7 to justify that <sup>ð</sup>xn, ynÞ¼ð<sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>−</sup><sup>n</sup>, <sup>−</sup>2<sup>n</sup><sup>Þ</sup> is a

−6n <sup>5</sup> <sup>−</sup> <sup>1</sup>ðyn<sup>Þ</sup>

Δxn¼ 2

<sup>0</sup>;<sup>∞</sup>≠∅ if for some k > 0 and any d<sup>1</sup> > 0 ðk < 0 and d<sup>1</sup> < 0Þ

We reconsider system (1) in the next section to emphasize the existence of nonoscillatory

ð∞ t0 bðtÞg � d1 ð∞ t aðsÞΔs � Δt ¼ ∞,

<sup>Δ</sup>yn <sup>¼</sup> <sup>−</sup> <sup>4</sup><sup>n</sup>

Δs < d, t≥t1,

t∈½t1,∞Þ<sup>T</sup>

bðuÞgðxðuÞÞΔu

1 5

<sup>1</sup> <sup>þ</sup> <sup>2</sup><sup>n</sup> <sup>ð</sup>xnÞ:

�

Δs, t≥t1:

Δt, k≠0: (11)

<sup>0</sup>;<sup>∞</sup>, we apply Theorem 2.11 and use the similar

$$\begin{cases} \mathbf{x}^{\Delta}(t) = a(t)f(y(t)) \\ \mathbf{y}^{\Delta}(t) = -b(t)g(\mathbf{x}(\tau(t))) \end{cases} \tag{12}$$

with a, b∈Crdð½t0,∞ÞT, RþÞ, τ∈Crdð½t0,∞ÞT, ½t0, ∞ÞTÞ, τðtÞ ≤ t and τðtÞ ! ∞ as t ! ∞, f , g ∈CðR, RÞ are nondecreasing functions such that ufðuÞ > 0 and ugðuÞ > 0 for u≠0. Motivated by Ref. [34] in which τðtÞ ¼ t−η, η > 0, our purpose in this section is to obtain the criteria for the existence of nonoscillatory solutions of Eq. (12) based on Yðt0Þ and Zðt0Þ. However, note that the results in Ref. [34] do not hold for any time scale, e.g., T ¼ q<sup>N</sup><sup>0</sup> , q > 1, because t−η is not necessarily in T. In fact, theoretical claims in this section follow from Ref. [35].

Since system (12) is oscillatory for the case Yðt0Þ ¼ ∞ and Zðt0Þ ¼ ∞, the existence results on any time scale are obtained in the next subsections based on the other three cases of Yðt0Þ and Zðt0Þ. Let ðx, yÞ be a nonoscillatory solution of system (12) in M<sup>þ</sup> such that the component function x is eventually positive. Then by the second equation of system (12), y is eventually decreasing. In addition, using the first equation of system (12), we have that xðtÞ ! c or ∞ and yðtÞ ! d or 0 as t ! ∞ for 0 < c < ∞ and 0 < d < ∞. Therefore, we have the following subclasses of Mþ:

$$M\_{B,B}^{+} = \{(x,y) \in M^{+} : \lim\_{t \to \infty} |x(t)| = c, \quad \lim\_{t \to \infty} |y(t)| = d\},$$

$$M\_{B,0}^{+} = \{(x,y) \in M^{+} : \lim\_{t \to \infty} |x(t)| = c, \quad \lim\_{t \to \infty} |y(t)| = 0\},$$

$$M\_{\circ \circ, \mathbb{B}}^{+} = \{(x,y) \in M^{+} : \lim\_{t \to \infty} |x(t)| = \circ, \; \lim\_{t \to \infty} |y(t)| = d\},$$

$$M\_{\circ, 0}^{+} = \{(x,y) \in M^{+} : \lim\_{t \to \infty} |x(t)| = \circ, \; \lim\_{t \to \infty} |y(t)| = 0\}.$$

In the literature, solutions in M<sup>þ</sup> <sup>B</sup>;<sup>0</sup>, M<sup>þ</sup> <sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup> are called subdominant, dominant and intermediate solutions, respectively, see Ref. [36]. Any nonoscillatory solution of system (12) belongs to M<sup>þ</sup> or M<sup>−</sup> given in Section 3. Also, it is important to emphasize that Lemma 3.1 holds for system (12) as well.

## 4.1. The case Yðt0Þ ¼ ∞ and Zðt0Þ < ∞

We restrict our attention to <sup>M</sup><sup>þ</sup> in this subsection because <sup>M</sup><sup>−</sup> <sup>¼</sup> <sup>∅</sup> when <sup>Y</sup>ðt0Þ ¼ <sup>∞</sup> and Zðt0Þ < ∞. The following lemma specifies the limit behavior of the component functions of nonoscillatory solutions ðx, yÞ under the case Yðt0Þ ¼ ∞ and Zðt0Þ < ∞.

Lemma 4.1 If jxðtÞj ! c, then yðtÞ ! 0 as t ! ∞ for 0 < c < ∞.

Proof. Assume to the contrary. So yðtÞ ! d for 0 < d < ∞ as t ! ∞. Then since yðtÞ > 0 and decreasing eventually, there exists t1≥t<sup>0</sup> such that fðyðτðtÞÞÞ≥fðdÞ ¼ k for t≥t1. By the same discussion as in the proof of Theorem 3.3, we obtain

$$\propto(t) \cong k \int\_{t\_1}^t a(s) \Delta s, \quad t \ge t\_1.$$

However, this gives us a contradiction to the fact that xðtÞ ! c as t ! ∞. So the assertion follows.

Remark 4.2. The discussion above and Lemma 4.1 yield us M<sup>þ</sup> <sup>B</sup>,<sup>B</sup> ¼ ∅:

Theorem 4.3. M<sup>þ</sup> <sup>B</sup>;<sup>0</sup>≠∅ if and only if I < ∞:

Proof. Suppose that there exists a solution ðx, yÞ∈ M<sup>þ</sup> <sup>B</sup>;<sup>0</sup> such that xðtÞ > 0, xðτðtÞÞ > 0 for t≥t0, xðtÞ ! c<sup>1</sup> and yðtÞ ! 0 as t ! ∞. Because x is eventually increasing, there exist t1≥t<sup>0</sup> and c<sup>2</sup> > 0 such that c<sup>2</sup> ≤ gðxðτðtÞÞÞ for t≥t1. Integrating the second equation from t to ∞ gives

$$y(t) = \int\_{t}^{\infty} b(s)g(\mathbf{x}(\tau(s))) \, \Delta s, \quad t \ge t\_1. \tag{13}$$

Also, integrating the first equation from t<sup>1</sup> to t, Eq. (13) and the monotonicity of g result in

$$\mathbf{x}(t) \ge \int\_{t\_1}^t a(\mathbf{s}) f\left(\int\_s^\infty b(\mathbf{u}) \mathbf{g}(\mathbf{x}(\tau(\mathbf{u}))) \Delta \mathbf{u}\right) \Delta \mathbf{s} \ge \int\_{t\_1}^t a(\mathbf{s}) f\left(c\_2 \int\_s^\infty b(\mathbf{u}) \Delta \mathbf{u}\right) \Delta \mathbf{s}.$$

Setting c<sup>2</sup> ¼ k and taking the limit as t ! ∞ prove the assertion. ðFor the case x < 0 eventually, the proof can be shown similarly with k < 0:Þ

Conversely, suppose I < ∞ for some k > 0: (For the case k < 0 can be shown similarly.) Then, choose t1≥t<sup>0</sup> so large that

$$\int\_{t\_1}^{\infty} a(t) f\left(k \int\_{t}^{\infty} b(s) \Delta s\right) \Delta t < \frac{c\_1}{2}, \quad t \ge t\_1,$$

where k ¼ gðc1Þ. Let X be the space of all continuous and bounded functions on ½t1,∞Þ<sup>T</sup> with the norm ‖y‖ ¼ sup t∈½t1,∞Þ<sup>T</sup> jyðtÞj. Then, X is a Banach space. Let Ω be the subset of X such that

$$
\Omega := \{ \mathfrak{x} \in X : \quad \frac{c\_1}{2} \le \mathfrak{x}(\tau(t)) \le c\_1, \quad \tau(t) \ge t\_1 \},
$$

and define an operator F : Ω ! X such that

$$(F\mathbf{x})(t) = c\_1 - \int\_t^\infty a(\mathbf{s}) f\left(\int\_s^\infty b(\boldsymbol{\mu}) g(\mathbf{x}(\boldsymbol{\pi}(\boldsymbol{\mu}))) \Delta \boldsymbol{\mu}\right) \Delta \mathbf{s}, \quad \boldsymbol{\pi}(t) \succeq t\_1.$$

It is easy to see that Ω is bounded, convex and a closed subset of X. It can also be shown that F maps into itself, relatively compact and continuous on Ω by the Lebesques dominated convergence theorem. Then, Theorem 2.9 gives that there exists x ∈ Ω such that x ¼ Fx: As t ! ∞, we get xðtÞ ! c<sup>1</sup> > 0. Setting

$$\overline{y}(t) = \int\_{t}^{\infty} b(u)g(\overline{x}(\tau(u))) \Delta u > 0, \quad \tau(t) \ge t\_1$$

shows yðtÞ ! 0 as t ! ∞: Taking the derivatives of x and y yield that ðx, yÞ is a solution of system (12). Hence, M<sup>þ</sup> <sup>B</sup>;<sup>0</sup>≠∅.

We demonstrate the following example to highlight Theorem 4.3.

Example 4.4 Let <sup>T</sup><sup>¼</sup> <sup>2</sup><sup>N</sup><sup>0</sup> and consider the system

$$\begin{cases} \Delta\_2 \mathbf{x}(t) = \frac{1}{2t^4} \left( y(t) \right)^{\frac{3}{5}} \\ \Delta\_2 y(t) = -\frac{3}{4t^2(8t-4)} \mathbf{x}(\frac{t}{4}). \end{cases} \tag{14}$$

First, it must be shown Yðt0Þ ¼ ∞ and Zðt0Þ < ∞. Indeed,

$$\int\_{t\_0}^{t} a(s) \Delta s = \frac{1}{2} \sum\_{s \in \left[4, t\right)\_{2^{N\_0}}} s^{\frac{1}{5}} \quad implies \quad Y(t\_0) = \frac{1}{2} \lim\_{n \to \infty} \sum\_{m=2}^{n-1} (2^m)^{\frac{1}{5}} = \infty$$

and

xðtÞ≥k ðt t1

Remark 4.2. The discussion above and Lemma 4.1 yield us M<sup>þ</sup>

yðtÞ ¼

�ð<sup>∞</sup> s

> ð∞ t1 aðtÞf � k ð∞ t bðsÞΔs � Δt < c1 <sup>2</sup> , <sup>t</sup>≥t1,

<sup>Ω</sup> :<sup>¼</sup> {x<sup>∈</sup> <sup>X</sup> : <sup>c</sup><sup>1</sup>

ð∞ t aðsÞf �ð<sup>∞</sup> s

<sup>B</sup>;<sup>0</sup>≠∅ if and only if I < ∞:

Proof. Suppose that there exists a solution ðx, yÞ∈ M<sup>þ</sup>

16 Dynamical Systems - Analytical and Computational Techniques

xðtÞ≥ ðt t1 aðsÞf

choose t1≥t<sup>0</sup> so large that

the norm ‖y‖ ¼ sup

get xðtÞ ! c<sup>1</sup> > 0. Setting

the proof can be shown similarly with k < 0:Þ

t∈½t1,∞Þ<sup>T</sup>

and define an operator F : Ω ! X such that

ðFxÞðtÞ ¼ c1−

follows.

Theorem 4.3. M<sup>þ</sup>

aðsÞΔs, t≥t1:

<sup>B</sup>,<sup>B</sup> ¼ ∅:

bðsÞgðxðτðsÞÞÞΔs, t≥t1: (13)

bðuÞΔu � Δs:

<sup>B</sup>;<sup>0</sup> such that xðtÞ > 0, xðτðtÞÞ > 0 for t≥t0,

However, this gives us a contradiction to the fact that xðtÞ ! c as t ! ∞. So the assertion

xðtÞ ! c<sup>1</sup> and yðtÞ ! 0 as t ! ∞. Because x is eventually increasing, there exist t1≥t<sup>0</sup> and c<sup>2</sup> > 0

Also, integrating the first equation from t<sup>1</sup> to t, Eq. (13) and the monotonicity of g result in

� Δs≥ ðt t1 aðsÞf � c2 ð∞ s

Setting c<sup>2</sup> ¼ k and taking the limit as t ! ∞ prove the assertion. ðFor the case x < 0 eventually,

Conversely, suppose I < ∞ for some k > 0: (For the case k < 0 can be shown similarly.) Then,

where k ¼ gðc1Þ. Let X be the space of all continuous and bounded functions on ½t1,∞Þ<sup>T</sup> with

jyðtÞj. Then, X is a Banach space. Let Ω be the subset of X such that

�

Δs, τðtÞ≥t1:

<sup>2</sup> <sup>≤</sup> <sup>x</sup>ðτðtÞÞ <sup>≤</sup> <sup>c</sup>1, <sup>τ</sup>ðtÞ≥t1},

bðuÞgðxðτðuÞÞÞΔu

It is easy to see that Ω is bounded, convex and a closed subset of X. It can also be shown that F maps into itself, relatively compact and continuous on Ω by the Lebesques dominated convergence theorem. Then, Theorem 2.9 gives that there exists x ∈ Ω such that x ¼ Fx: As t ! ∞, we

such that c<sup>2</sup> ≤ gðxðτðtÞÞÞ for t≥t1. Integrating the second equation from t to ∞ gives

bðuÞgðxðτðuÞÞÞΔu

ð∞ t

$$\int\_{t\_0}^t b(s)\,\Delta s \le \frac{3}{16} \sum\_{s \in \left[4, t\right)\_{2^{N\_0}}} \frac{1}{s} \quad implies \quad Z(t\_0) \le \frac{3}{16} \lim\_{n \to \infty} \sum\_{m=2}^{n-1} \frac{1}{2^m} < \infty$$

by the geometric series, where t<sup>¼</sup> <sup>2</sup><sup>n</sup>, <sup>s</sup><sup>¼</sup> <sup>2</sup><sup>m</sup>, <sup>m</sup>, <sup>n</sup>≥2. Note that

$$\int\_{t}^{T} b(s)\Delta s \leq \frac{3}{16} \sum\_{s \in \left[t, T\right)\_{2} \times\_{0}} \frac{1}{s} \quad \text{implies} \quad Z(t) \leq \frac{3}{16} \lim\_{n \to \infty} \sum\_{m=2}^{n-1} \frac{1}{2^{m}} = \frac{3}{8} \lim\_{n \to \infty} \left(\frac{1}{t} - \frac{1}{t2^{n}}\right) = \frac{3}{8t} \dots$$

Letting k ¼ 1 and using the last inequality gives

$$\int\_{t\_0}^T a(t) f\left(k \int\_{t}^\infty b(s) \Delta s\right) \Delta t \le \int\_{t\_0}^T \frac{1}{2t^{\frac{4}{5}}} \left(\frac{3}{8t}\right)^{\frac{3}{5}} \Delta t = \left(\frac{3}{8}\right)^{\frac{2}{5}} \frac{1}{2} \sum\_{t \in \left[1, T\right)\_{2^{\frac{10}{5}}}} \frac{1}{t\_5^{\frac{2}{5}}}.$$

Therefore, we have

$$\int\_{t\_0}^{\infty} a(t) f\left(k \int\_{t}^{\infty} b(s) \Delta s\right) \Delta t \le \left(\frac{3}{8}\right)^{\frac{3}{5}} \frac{1}{2} \sum\_{n=0}^{\infty} \frac{1}{2^{\frac{2n}{5}}} < \infty$$

by the geometric series. It can be seen that ðx, yÞ ¼ 8− 1 t , 1 t 2 � � is a nonoscillatory solution of Eq. (14) such that xðtÞ ! 8 and yðtÞ ! 0 as t ! ∞, i.e., M<sup>þ</sup> <sup>B</sup>;<sup>0</sup>≠∅.

The existence in subclasses M<sup>þ</sup> <sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup> is not obtained on general time scales. The main reason is that setting an operator including a delay function gives a struggle when the fixed points theorems are applied. In fact, when we restrict the delay function to τðtÞ ¼ t−η for η≥0, it was shown M<sup>þ</sup> <sup>∞</sup>,B≠∅, see Ref. [34]. Nevertheless, the existence in M<sup>þ</sup> <sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup> for system (1) is shown in Subsection 4.4.

## 4.2. The case Yðt0Þ < ∞ and Zðt0Þ < ∞

Because the component functions x and y have finite limits by Lemma 3.1(e) and (f), the subclasses M<sup>þ</sup> <sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup> are empty. Since the existence of nonoscillatory solutions in M<sup>þ</sup> <sup>B</sup>;<sup>0</sup> is shown in Theorem 4.3, we only focus on M<sup>þ</sup> <sup>B</sup>,<sup>B</sup> in this subsection.

The Knaster fixed point theorem is utilized in order to prove the following theorem.

Theorem 4.5 M<sup>þ</sup> <sup>B</sup>,B≠∅ if and only if

$$\int\_{t\_0}^{\infty} a(s) f\left(d\_1 + k \int\_s^{\infty} b(u) \Delta u\right) \Delta s < \infty, \quad k, d\_1 \neq 0. \tag{15}$$

Proof. The proof of the necessity part is very similar to those of previous theorems. So for sufficiency, suppose Eq. (15) holds. Choose t1≥t0, k > 0 and d<sup>1</sup> > 0 such that

$$\int\_{t\_1}^{\infty} a(s)f\left(d\_1 + k \int\_s^{\infty} b(u)\Delta u\right)ds < d\_1,$$

where k ¼ gð2d1Þ: (The case k, d<sup>1</sup> < 0 can be done similarly.) Let X be the Banach space of all continuous real-valued functions endowed with the norm ‖x‖ ¼ sup t∈ ½t1,∞Þ<sup>T</sup> jxðtÞj and with usual

point-wise ordering ≤ . Define a subset Ω of X as

$$\mathcal{Q} := \{ \mathbf{x} \in X \, : \, \quad d\_1 \le \mathfrak{x}(\mathfrak{r}(t)) \le 2d\_1, \quad \mathfrak{r}(t) \ge t\_1 \}.$$

For any subset B of Ω, it is clear that infB∈ Ω and supB∈ Ω. An operator F : Ω ! X is defined as

$$(F\mathbf{x})(t) = d\_1 + \int\_{t\_1}^t a(s)f\left(d\_1 + \int\_s^\infty b(u)g(\mathbf{x}(\tau(u)))\Delta u\right)d\mathbf{s}, \quad \tau(t) \succeq t\_1.$$

It is obvious that F is an increasing mapping into itself. Therefore,

$$d\_1 \le (F\mathbf{x})(t) \le d\_1 + \int\_{t\_1}^t a(s)f\left(d\_1 + g(2d\_1)\int\_s^\infty b(u)\Delta u\right)ds \le 2d\_1, \quad \pi(t) \succeq t\_1.$$

Then, by Theorem 2.11, there exists x∈ Ω such that x ¼ Fx. By setting

$$
\overline{y}(t) = d\_1 + \int\_t^\infty b(u)g(\overline{\mathfrak{x}}(\tau(u))), \quad \tau(t) \ge t\_1,
$$

we get that

On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems http://dx.doi.org/10.5772/67118 19

$$
\overline{y}^A(t) = -b(t)\overline{g}(\overline{\mathfrak{x}}(\pi(t))).\tag{16}
$$

Also taking the derivative of x and Eq. (16) give that ðx, yÞ is a solution of system (12). Hence, we conclude that xðtÞ ! α and yðtÞ ! d<sup>1</sup> as t ! ∞, where 0 < α < ∞, i.e., M<sup>þ</sup> <sup>B</sup>,B≠∅. Note that a similar proof can be done for the case k < 0 and d<sup>1</sup> < 0 with x < 0.

Example 4.6 Let <sup>T</sup><sup>¼</sup> <sup>2</sup><sup>N</sup><sup>0</sup> and consider the system

The existence in subclasses M<sup>þ</sup>

4.2. The case Yðt0Þ < ∞ and Zðt0Þ < ∞

<sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup>

shown in Theorem 4.3, we only focus on M<sup>þ</sup>

18 Dynamical Systems - Analytical and Computational Techniques

<sup>B</sup>,B≠∅ if and only if ð∞ t0 aðsÞf � d<sup>1</sup> þ k ð∞ s

point-wise ordering ≤ . Define a subset Ω of X as

ðFxÞðtÞ ¼ d<sup>1</sup> þ

d<sup>1</sup> ≤ ðFxÞðtÞ ≤ d<sup>1</sup> þ

we get that

was shown M<sup>þ</sup>

subclasses M<sup>þ</sup>

Theorem 4.5 M<sup>þ</sup>

shown in Subsection 4.4.

<sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup>

<sup>∞</sup>,B≠∅, see Ref. [34]. Nevertheless, the existence in M<sup>þ</sup>

The Knaster fixed point theorem is utilized in order to prove the following theorem.

sufficiency, suppose Eq. (15) holds. Choose t1≥t0, k > 0 and d<sup>1</sup> > 0 such that

continuous real-valued functions endowed with the norm ‖x‖ ¼ sup

ð∞ t1 aðsÞf � d<sup>1</sup> þ k ð∞ s

ðt t1 aðsÞf � d<sup>1</sup> þ ð∞ s

It is obvious that F is an increasing mapping into itself. Therefore,

ðt t1 aðsÞf �

Then, by Theorem 2.11, there exists x∈ Ω such that x ¼ Fx. By setting

ð∞ t

yðtÞ ¼ d<sup>1</sup> þ

reason is that setting an operator including a delay function gives a struggle when the fixed points theorems are applied. In fact, when we restrict the delay function to τðtÞ ¼ t−η for η≥0, it

Because the component functions x and y have finite limits by Lemma 3.1(e) and (f), the

bðuÞΔu �

Proof. The proof of the necessity part is very similar to those of previous theorems. So for

where k ¼ gð2d1Þ: (The case k, d<sup>1</sup> < 0 can be done similarly.) Let X be the Banach space of all

Ω :¼ {x ∈ X : d<sup>1</sup> ≤ xðτðtÞÞ ≤ 2d1, τðtÞ≥t1}:

For any subset B of Ω, it is clear that infB∈ Ω and supB∈ Ω. An operator F : Ω ! X is defined as

d<sup>1</sup> þ gð2d1Þ

bðuÞgðxðτðuÞÞÞΔu

bðuÞΔu �

ð∞ s

bðuÞgðxðτðuÞÞÞ, τðtÞ≥t1,

bðuÞΔu �

Δs < d1,

<sup>∞</sup>;<sup>0</sup> are empty. Since the existence of nonoscillatory solutions in M<sup>þ</sup>

<sup>B</sup>,<sup>B</sup> in this subsection.

<sup>∞</sup>;<sup>0</sup> is not obtained on general time scales. The main

<sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup>

Δs < ∞, k, d1≠0: (15)

t∈ ½t1,∞Þ<sup>T</sup>

Δs, τðtÞ≥t1:

Δs ≤ 2d1, τðtÞ≥t1:

�

<sup>∞</sup>;<sup>0</sup> for system (1) is

jxðtÞj and with usual

<sup>B</sup>;<sup>0</sup> is

$$\begin{cases} \Delta\_2 \mathbf{x}(t) = \frac{1}{2t^{\frac{5}{2}}(3t+1)^{\frac{1}{3}}} y^{\frac{1}{3}}(t) \\\ \Delta\_2 \mathbf{y}(t) = -\frac{1}{2t(6t-4)} \mathbf{x}\left(\frac{t}{4}\right). \end{cases} \tag{17}$$

We first demonstrate Yðt0Þ < ∞ and Zðt0Þ < ∞.

$$\int\_{t\_0}^t a(s)ds = \frac{1}{2} \sum\_{s \in \left[4, t\right]\_{2^{\mathbb{N}\_0}}} \frac{1}{s^{\frac{2}{3}}(3s+1)^{\frac{1}{3}}} \quad \text{implies} \quad Y(t\_0) = \frac{1}{2} \lim\_{n \to \infty} \sum\_{m=2}^{n-1} \frac{1}{\left(2^m\right)^{\frac{2}{3}}(3 \cdot 2^m + 1)^{\frac{1}{3}}} < \infty$$

by the Ratio test for t<sup>¼</sup> <sup>2</sup><sup>n</sup>, <sup>s</sup><sup>¼</sup> <sup>2</sup><sup>m</sup>, <sup>n</sup>≥2. Similarly,

$$\int\_{t\_0}^t b(s)ds = \frac{1}{2} \sum\_{s \in \{4, t\}\_{2} \times\_0} \frac{1}{6s - 4} \quad \text{implies} \quad Z(t\_0) = \frac{1}{2} \lim\_{n \to \infty} \sum\_{m=2}^{n-1} \frac{1}{6.2^m - 4} < \infty.$$

Because Yðt0Þ < ∞ and Zðt0Þ < ∞, it is easy to show that Eq. (15) holds. One can also verify that 6− <sup>1</sup> <sup>t</sup> ; <sup>3</sup> <sup>þ</sup> <sup>1</sup> t � � is a nonoscillatory solution of system (17) such that xðtÞ ! <sup>6</sup> and yðtÞ ! <sup>3</sup> as t ! <sup>∞</sup>, i.e., M<sup>þ</sup> <sup>B</sup>,B≠∅ by Theorem 4.5.

4.3. The case Yðt0Þ < ∞ and Zðt0Þ ¼ ∞

Lemma 3.1(c) yields M<sup>þ</sup> ¼ ∅ for the case Yðt0Þ < ∞ and Zðt0Þ ¼ ∞. Thus, we pay our attention to M<sup>−</sup> in this subsection. The proof of the following remark is similar to that of Theorem 3.7.

Remark 4.7 M<sup>−</sup> <sup>B</sup>,∞≠∅ if and only if integral condition (7) holds.

Exercise 4.8 Prove Remark 4.7 and also show that <sup>ð</sup><sup>3</sup> <sup>þ</sup> <sup>1</sup> <sup>t</sup> , −t− <sup>1</sup> t Þ is a nonoscillatory solution of

$$\begin{cases} \Delta\_2 \mathfrak{x}(t) = \frac{1}{2t^{\frac{7}{5}}(t^2+1)^{\frac{3}{5}}} (y(t))^{\frac{3}{5}} \\\\ \Delta\_2 y(t) = -\frac{2t^2 - 1}{2t^{\frac{9}{5}}(3t+4)^{\frac{1}{5}}} \left(\mathfrak{x}(\frac{t}{4})\right)^{\frac{1}{5}} \end{cases}$$

in M<sup>−</sup> <sup>B</sup>,∞≠<sup>∅</sup> when <sup>T</sup><sup>¼</sup> <sup>2</sup><sup>N</sup><sup>0</sup> .

#### 4.4. Dominant and intermediate solutions of Eq. (1)

Note that the existence of nonoscillatory solutions of system (1) in M<sup>−</sup> <sup>0</sup>;<sup>∞</sup>, <sup>M</sup><sup>−</sup> <sup>B</sup>,<sup>B</sup> and <sup>M</sup><sup>−</sup> <sup>0</sup>;<sup>B</sup> is not shown on a general time scale. In fact, the existence in these subclasses is obtained for system (1) in Section 3. Since system (12) is reduced to system (1) when τðtÞ ¼ t, notice that the results obtained for system (12) in Section 4 also hold for system (1). Therefore, we only need to show the existence of nonoscillatory solutions for Eq. (1) in M<sup>þ</sup> <sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup>, which are not acquired for Eq. (12) on a general time scale. To achieve the goal, we assume Yðt0Þ ¼ ∞ and Zðt0Þ < ∞.

Theorem 4.9 M<sup>þ</sup> <sup>∞</sup>,B≠∅ if and only if

$$\int\_{t\_0}^s b(s)g\left(c\_1\int\_{t\_0}^s a(u)\Delta u\right)d\mathbf{s} < \infty, \quad c\_1\forall 0. \tag{18}$$

Proof. The necessity part is left to readers as an exercise. Therefore, for sufficiency, suppose that Eq. (18) holds. Choose t1≥t0, c<sup>1</sup> > 0 and d<sup>1</sup> > 0 such that

$$\int\_{t\_1}^{s} b(s)g\left(c\_1 \int\_{t\_1}^{s} b(u)\Delta u\right) \Delta s < d\_1, \quad t \ge t\_1,\tag{19}$$

where c<sup>1</sup> ¼ fð2d1Þ > 0: (The case c<sup>1</sup> < 0 can be done similarly.) Let X be the partially ordered Banach space of all real-valued continuous functions endowed with supremum norm ‖x‖ ¼ sup t∈ ½t1,∞Þ<sup>T</sup> jxðtÞj ðt t1 aðsÞΔs and with the usual point-wise ordering ≤ . Define a subset Ω of X such that

$$\mathcal{Q} = \colon \{ \mathbf{x} \in \mathcal{X} : \quad f(d\_1) \int\_{t\_1}^t a(\mathbf{s}) \Delta \mathbf{s} \le \mathbf{x}(t) \preceq f(2d\_1) \int\_{t\_1}^t a(\mathbf{s}) \Delta \mathbf{s}, \quad t \ge t\_1 \}.$$

For any subset B of Ω, infB∈ Ω and supB∈ Ω, i.e., ðΩ, ≤ Þ is complete. Define an operator F : Ω ! X as

$$(F\mathbf{x})(t) = \int\_{t\_1}^{t} a(\mathbf{s}) f\left(d\_1 + \int\_{t}^{\infty} b(\mathbf{u}) \mathbf{g}(\mathbf{x}(\boldsymbol{\mu})) \Delta \boldsymbol{u} \right) \Delta \mathbf{s}, \quad t \ge t\_1.$$

It is obvious that it is an increasing mapping, so let us show F :¼ Ω ! Ω:

$$\begin{aligned} \left(f(d\_1)\right)^t\_{t\_1} a(s) \Delta s &\le (Fx)(t) \\ &\le \int\_{t\_1}^t a(s) f\left(d\_1 + \int\_s^\infty b(u) g\left(f(2d\_1) \int\_{t\_1}^u a(\lambda) \Delta \lambda\right) \Delta u\right) \Delta s \\ &\le f(2d\_1) \int\_{t\_1}^t a(s) \Delta s \end{aligned}$$

by Eq. (19). Then, by Theorem 2.11, there exists x∈ Ω such that x ¼ Fx and so

$$
\overline{\mathfrak{X}}^{\Lambda}(t) = a(t)f\left(d\_1 + \int\_t^{\infty} b(u)g(\overline{\mathfrak{X}}(u)) \, \Lambda u\right), \quad t \ge t\_1.
$$

Setting yðtÞ ¼ d<sup>1</sup> þ ð∞ t <sup>b</sup>ðuÞgðxðuÞÞΔ<sup>u</sup> leads us <sup>y</sup><sup>Δ</sup> <sup>¼</sup> <sup>−</sup>bgðx<sup>Þ</sup> and so, <sup>ð</sup>x, <sup>y</sup><sup>Þ</sup> is a solution of system (1) such that xðtÞ > 0 and yðtÞ > 0 for t≥t<sup>1</sup> and xðtÞ ! ∞ and yðtÞ ! d<sup>1</sup> > 0 as t ! ∞, i.e., M<sup>þ</sup> <sup>∞</sup>,B≠∅.

Theorem 4.10 M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup>≠∅ if

(1) in Section 3. Since system (12) is reduced to system (1) when τðtÞ ¼ t, notice that the results obtained for system (12) in Section 4 also hold for system (1). Therefore, we only need to show

for Eq. (12) on a general time scale. To achieve the goal, we assume Yðt0Þ ¼ ∞ and Zðt0Þ < ∞.

Proof. The necessity part is left to readers as an exercise. Therefore, for sufficiency, suppose that

where c<sup>1</sup> ¼ fð2d1Þ > 0: (The case c<sup>1</sup> < 0 can be done similarly.) Let X be the partially ordered Banach space of all real-valued continuous functions endowed with supremum norm

aðsÞΔs ≤ xðtÞ ≤ fð2d1Þ

bðuÞgðxðuÞÞΔu

bðuÞgðxðuÞÞΔu

(1) such that xðtÞ > 0 and yðtÞ > 0 for t≥t<sup>1</sup> and xðtÞ ! ∞ and yðtÞ ! d<sup>1</sup> > 0 as t ! ∞, i.e.,

� , t≥t1:

<sup>b</sup>ðuÞgðxðuÞÞΔ<sup>u</sup> leads us <sup>y</sup><sup>Δ</sup> <sup>¼</sup> <sup>−</sup>bgðx<sup>Þ</sup> and so, <sup>ð</sup>x, <sup>y</sup><sup>Þ</sup> is a solution of system

For any subset B of Ω, infB∈ Ω and supB∈ Ω, i.e., ðΩ, ≤ Þ is complete. Define an operator

ðt t1

and with the usual point-wise ordering ≤ . Define a subset Ω of X such that

ðt t1

�

<sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup>

<sup>∞</sup>;<sup>0</sup>, which are not acquired

Δs < ∞, c1≠0: (18)

Δs < d1, t≥t1, (19)

aðsÞΔs, t≥t1}:

Δs, t≥t1:

the existence of nonoscillatory solutions for Eq. (1) in M<sup>þ</sup>

ð∞ t0 bðsÞg � c1 ðs t0 aðuÞΔu �

Eq. (18) holds. Choose t1≥t0, c<sup>1</sup> > 0 and d<sup>1</sup> > 0 such that ð∞ t1 bðsÞg � c1 ðs t1 bðuÞΔu �

Ω ¼: {x∈ X : fðd1Þ

ðFxÞðtÞ ¼

ðt t1 aðsÞf � d<sup>1</sup> þ ð∞ t

aðsÞΔs ≤ ðFxÞðtÞ

≤ ðt t1 aðsÞf � d<sup>1</sup> þ ð∞ s bðuÞg � fð2d1Þ ðu t1 aðλÞΔλ � Δu � Δs

≤ fð2d1Þ

<sup>x</sup><sup>Δ</sup>ðtÞ ¼ <sup>a</sup>ðtÞ<sup>f</sup>

It is obvious that it is an increasing mapping, so let us show F :¼ Ω ! Ω:

ðt t1 aðsÞΔs

by Eq. (19). Then, by Theorem 2.11, there exists x∈ Ω such that x ¼ Fx and so

� d<sup>1</sup> þ ð∞ t

<sup>∞</sup>,B≠∅ if and only if

20 Dynamical Systems - Analytical and Computational Techniques

Theorem 4.9 M<sup>þ</sup>

‖x‖ ¼ sup

F : Ω ! X as

Setting yðtÞ ¼ d<sup>1</sup> þ

M<sup>þ</sup> <sup>∞</sup>,B≠∅.

t∈ ½t1,∞Þ<sup>T</sup>

jxðtÞj ðt t1 aðsÞΔs

> fðd1Þ ðt t1

> > ð∞ t

$$I = \Leftrightarrow \quad \text{and} \quad \int\_{t\_0}^{\infty} b(t)g\left(l \int\_{t\_0}^{\infty} a(s)\Delta s\right) \Delta t < \infty,$$

where I is defined as in Eq. (11), for any k > 0 and some l > 0 ðk < 0 and l < 0Þ.

Exercise 4.11 Prove Theorem 4.10 using Theorem 2.11.

## 5. Emden-Fowler Dynamical Systems on Time Scales

Motivated by the papers [28, 36, 37], we deal with the classification and existence of nonoscillatory solutions of the Emden-Fowler dynamical system

$$\begin{cases} \mathbf{x}^A(t) = a(t)|y(t)|^\frac{1}{\alpha}\text{sgn }y(t) \\ y^A(t) = -b(t)|\mathbf{x}^\sigma(t)|^\frac{\beta}{\beta}\text{sgn }\mathbf{x}^\sigma(t), \end{cases} \tag{20}$$

where <sup>α</sup>, <sup>β</sup> <sup>&</sup>gt; <sup>0</sup> <sup>a</sup>, <sup>b</sup>∈Crdð½t0,∞ÞT, <sup>R</sup>þÞ and <sup>x</sup><sup>σ</sup> (t) = <sup>x</sup> (σ(t)). The main results of this section follow from Ref. [38]. If T ¼ Z, system (20) is reduced to a Emden-Fowler system of difference equations while it is reduced to a Emden-Fowler system of differential equations when T ¼ R, see Refs. [32, 39, 40], respectively. We also refer readers to Refs. [41–46] for quasilinear and Emden-Fowler dynamic equations on time scales.

Note that any nonoscillatory solution of system (20) belongs to M<sup>þ</sup> or M<sup>−</sup> given in Section 3. Also, it could be shown that Lemma 3.1 holds for system (20) as well.

5.1. The case Yðt0Þ ¼ ∞ and Zðt0Þ < ∞

In this case, we have <sup>M</sup><sup>−</sup> <sup>¼</sup> <sup>∅</sup>, see Lemma 3.1(d). By a similar discussion as in Subsection 4.1, solutions in M<sup>þ</sup> belongs to one of the subclasses M<sup>þ</sup> <sup>B</sup>;<sup>0</sup>, M<sup>þ</sup> <sup>∞</sup>,<sup>B</sup> and M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup>:

Let us set

$$\begin{aligned} J\_{\alpha} &= \int\_{t\_0}^{\infty} a(t) \left( \int\_{t}^{\infty} b(s) \Delta s \right)^{\frac{1}{\alpha}} \Delta t \\ K\_{\beta} &= \int\_{t\_0}^{\infty} b(t) \left( \int\_{t\_0}^{v(t)} a(s) \Delta s \right)^{\beta} \Delta t . \end{aligned}$$

Note that integral I, defined as in Eq. (11), is reduced to J<sup>α</sup> by replacing fðzÞ ¼ z 1 <sup>α</sup> and <sup>g</sup>ðzÞ ¼ <sup>z</sup><sup>β</sup>. The following theorem can be proven similar to Theorem 4.3.

Theorem 5.1 M<sup>þ</sup> <sup>B</sup>;<sup>0</sup>≠∅ if and only if J<sup>α</sup> < ∞.

Exercise 5.2 Prove Theorem 5.1.

Next, we provide the existence of dominant and intermediate solutions of system (20) along with examples.

#### Theorem 5.3 M<sup>þ</sup> <sup>∞</sup>,B≠∅ if and only if K<sup>β</sup> < ∞:

Proof. Suppose that there exists ðx, yÞ∈ M<sup>þ</sup> such that x > 0 eventually, xðtÞ ! ∞ and yðtÞ ! d as t ! ∞ for 0 < d < ∞. Integrating the first equation from t<sup>1</sup> to σðtÞ, using the monotonicity of y and integrating the second equation from t<sup>1</sup> to t of system (20) give us

$$\mathbf{x}^{\sigma}(t) = \mathbf{x}^{\sigma}(t\_1) + \int\_{t\_1}^{\sigma(t)} a(\mathbf{s}) y^{\frac{1}{\alpha}}(\mathbf{s}) \Delta \mathbf{s} > d \frac{1}{a} \Big|\_{t\_1}^{\sigma(t)} a(\mathbf{s}) \Delta \mathbf{s}. \tag{21}$$

and

$$\mathcal{Y}(t\_1) \neg \mathcal{Y}(t) = \int\_{t\_1}^{t} b(s) \left(\mathbf{x}^{\boldsymbol{\sigma}}(s)\right)^{\boldsymbol{\beta}} \, \Delta s,\tag{22}$$

respectively. Then, by Eqs. (21) and (22), we have

$$\int\_{t\_1}^t b(s) \left( \int\_{t\_1}^{v(s)} a(u) \Delta u \right)^{\beta} \Delta s < d^{\frac{-\beta}{a}} \int\_{t\_1}^t b(s) \left( \mathbf{x}^{\sigma}(s) \right)^{\beta} \Delta s = d^{\frac{-\beta}{a}} \left( y(t\_1) \neg y(t) \right)^{\beta}$$

So as t ! ∞, it follows K<sup>β</sup> < ∞.

Conversely, suppose K<sup>β</sup> < ∞. Choose t1≥t<sup>0</sup> so large that

$$\int\_{t\_1}^{\infty} b(s) \left( \int\_{t\_1}^{\sigma(s)} a(u) \Delta u \right)^{\beta} \Delta s < \frac{d^{1-\beta}}{2^{\beta}}$$

for arbitrarily given d > 0. Let X be the partially ordered Banach Space of all real-valued continuous functions with the norm ∥x∥ ¼ sup t>t<sup>1</sup> jxðtÞj ðt t1 aðsÞΔs and the usual point-wise ordering ≤ .

Define a subset Ω of X as follows:

$$\mathcal{Q} : \{ \mathbf{x} \in \mathcal{X} : \quad d^{\frac{1}{\alpha}} \int\_{t\_1}^{t} a(s) \Delta s \le \mathbf{x}(t) \le (2d)^{\frac{1}{\alpha}} \int\_{t\_1}^{t} a(s) \Delta s \quad \text{for} \quad t > t\_1\}.$$

First, since every subset of Ω has a supremum and infimum in Ω, ðΩ, ≤ Þ is a complete lattice. Define an operator F : Ω ! X as

$$(F\mathfrak{x})(t) = \int\_{t\_1}^{t} a(\mathfrak{s}) \left( d + \int\_{\mathfrak{s}}^{\infty} b(\mathfrak{u}) \left( \mathfrak{x}^{\mathcal{I}}(\mathfrak{u}) \right)^{\mathcal{J}} \Delta \mathfrak{x} \right)^{\frac{1}{\alpha}} \Delta \mathfrak{s} \dots$$

The rest of the proof can be finished via the Knaster fixed point theorem, see Theorem 4.9 and thus is left to readers.

Example 5.4 Let T ¼ q<sup>N</sup><sup>0</sup> , q > 1 and consider the system

Theorem 5.3 M<sup>þ</sup>

and

<sup>∞</sup>,B≠∅ if and only if K<sup>β</sup> < ∞:

22 Dynamical Systems - Analytical and Computational Techniques

and integrating the second equation from t<sup>1</sup> to t of system (20) give us

ð<sup>σ</sup>ðt<sup>Þ</sup> t1

yðt1Þ−yðtÞ ¼

<sup>x</sup><sup>σ</sup>ðtÞ ¼ <sup>x</sup><sup>σ</sup>ðt1Þ þ

respectively. Then, by Eqs. (21) and (22), we have

�ð<sup>σ</sup>ðs<sup>Þ</sup> t1

aðuÞΔu

Conversely, suppose K<sup>β</sup> < ∞. Choose t1≥t<sup>0</sup> so large that

continuous functions with the norm ∥x∥ ¼ sup

Ω : {x∈ X : d

�β Δs < d −β α ðt t1 bðsÞ � <sup>x</sup><sup>σ</sup>ðs<sup>Þ</sup> �β

ð∞ t1 bðsÞ

1 α ðt t1

ðFxÞðtÞ ¼

ðt t1 aðsÞ � d þ ð∞ s bðuÞ � <sup>x</sup><sup>σ</sup>ðu<sup>Þ</sup> �β Δτ � 1 α Δs:

�ð<sup>σ</sup>ðs<sup>Þ</sup> t1

aðuÞΔu

for arbitrarily given d > 0. Let X be the partially ordered Banach Space of all real-valued

t>t<sup>1</sup>

aðsÞΔs ≤ xðtÞ ≤ ð2dÞ

First, since every subset of Ω has a supremum and infimum in Ω, ðΩ, ≤ Þ is a complete lattice.

The rest of the proof can be finished via the Knaster fixed point theorem, see Theorem 4.9 and

�β Δs <

jxðtÞj ðt t1 aðsÞΔs

> 1 α ðt t1

ðt t1 bðsÞ

So as t ! ∞, it follows K<sup>β</sup> < ∞.

Define a subset Ω of X as follows:

Define an operator F : Ω ! X as

thus is left to readers.

Proof. Suppose that there exists ðx, yÞ∈ M<sup>þ</sup> such that x > 0 eventually, xðtÞ ! ∞ and yðtÞ ! d as t ! ∞ for 0 < d < ∞. Integrating the first equation from t<sup>1</sup> to σðtÞ, using the monotonicity of y

> aðsÞy 1

> > ðt t1 bðsÞ � <sup>x</sup><sup>σ</sup>ðs<sup>Þ</sup> �β

<sup>α</sup>ðsÞΔs > d

1 α ð<sup>σ</sup>ðt<sup>Þ</sup> t1

Δs ¼ d −β α �

d<sup>1</sup>−<sup>β</sup> 2β

aðsÞΔs: (21)

Δs, (22)

�

yðt1Þ−yðtÞ

and the usual point-wise ordering ≤ .

aðsÞΔs for t > t1}:

$$\begin{cases} \mathbf{x}^{\Delta} = \frac{t}{1+2t} |y| \text{sgn } y \\ \mathbf{y}^{\Delta} = -\frac{1}{q^{1+\beta}t^{\beta+2}} |\mathbf{x}^{\sigma}|^{\beta} \text{sgn } \mathbf{x}. \end{cases} \tag{23}$$

It is left to readers to show Yðt0Þ ¼ ∞ and Zðt0Þ < ∞. In order to show K<sup>β</sup> < ∞, we first calculate

$$\begin{aligned} &\int\_{t\_0}^T b(t) \left(\int\_{t\_0}^{\sigma(t)} a(s) \Delta s\right)^{\beta} \Delta t = \sum\_{t \in [1, T]\_{q^{\mathbb{N}\_0}}} \frac{1}{q^{1+\beta} t^{\beta+2}} \left(\sum\_{s \in [1, \sigma(t)]\_{q^{\mathbb{N}\_0}}} \frac{s^2 (q-1)}{1+2s}\right)^{\beta} (q-1) t^{\beta} \\ &< \frac{(q-1)^{\beta+1}}{q^{1+\beta}} \sum\_{t \in [1, T]\_{q^{\mathbb{N}\_0}}} \frac{1}{t^{1+\beta}} \left(\sum\_{s \in [1, \sigma(t)]\_{q^{\mathbb{N}\_0}}} s\right)^{\beta} < \frac{q-1}{q} \sum\_{t \in [1, T]\_{q^{\mathbb{N}\_0}}} \frac{1}{t}, \end{aligned}$$

where s <sup>¼</sup> qm and t <sup>¼</sup> <sup>q</sup><sup>n</sup> for m, <sup>n</sup><sup>∈</sup> <sup>N</sup>0. Since

$$\lim\_{T \to \infty} \sum\_{t \in \left[1, T\right)\_q^{\mathbb{N}\_0}} \frac{1}{t} = \sum\_{n=0}^{\infty} \frac{1}{q^n} < \infty$$

by the geometric series, we have K<sup>β</sup> <sup>&</sup>lt; <sup>∞</sup>. It can be verified that <sup>ð</sup>t, <sup>1</sup> <sup>t</sup> þ 2Þ is a nonoscillatory solution of system (23) in M<sup>þ</sup> <sup>∞</sup>,B:

Theorem 5.5 M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup>≠∅ if J<sup>α</sup> ¼ ∞ and K<sup>β</sup> < ∞.

Proof. Suppose that J<sup>α</sup> ¼ ∞ and K<sup>β</sup> < ∞ hold. Since Yðt0Þ ¼ ∞, we can choose t<sup>1</sup> and t<sup>2</sup> so large that

$$\int\_{t\_2}^{\infty} b(t) \left( \int\_{t\_0}^{v(t)} a(s) \Delta s \right)^{\beta} \Delta t \le 1 \qquad \text{and} \qquad \int\_{t\_1}^{t\_2} a(s) \Delta s \ge 1, \quad t \ge t\_2 \ge t\_1.$$

Let X be the Fréchet Space of all continuous functions on ½t1,∞Þ<sup>T</sup> endowed with the topology of uniform convergence on compact subintervals of ½t1,∞ÞT: Set

$$\mathcal{Q} := \{ \mathbf{x} \in X : \quad 1 \le \mathbf{x}(t) \le \int\_{t\_1}^t a(s) \, \Delta s \quad \text{for} \quad t \ge t\_1 \}.$$

and define an operator T : Ω ! X by

$$(T\mathbf{x})(t) = \mathbf{1} + \int\_{t\_2}^{t} a(\mathbf{s}) \left( \int\_{s}^{\infty} b(\mathbf{u}) \left( \mathbf{x}^{\sigma}(\mathbf{u}) \right)^{\beta} \Delta \mathbf{u} \right)^{\frac{1}{\alpha}}.\tag{24}$$

We can show that T : Ω ! Ω is continuous on Ω⊂X by the Lebesque dominated convergence theorem. Since

$$\begin{aligned} 0 \le \left[ (T\boldsymbol{x})(t) \right]^\mathcal{A} &= a(t) \left( \int\_t^\infty b(\boldsymbol{u}) \left( \boldsymbol{x}^\boldsymbol{\boldsymbol{\sigma}}(\boldsymbol{u}) \right)^\beta \boldsymbol{\Delta} \boldsymbol{u} \right)^{\frac{1}{\alpha}} \\ \le a(t) \left( \int\_t^\infty b(\boldsymbol{u}) \left( \int\_{t\_1}^{\boldsymbol{\sigma}(\boldsymbol{u})} a(\boldsymbol{\Lambda}) \boldsymbol{\Delta} \boldsymbol{\lambda} \right)^\beta \boldsymbol{\Delta} \boldsymbol{u} \right)^{\frac{1}{\alpha}} < \infty, \end{aligned}$$

it follows that T is equibounded and equicontinuous. Then by Theorem 2.10, there exists x∈ Ω such that x ¼ Tx: Thus, it follows that x is eventually positive, i.e nonoscillatory. Then differentiating x and the first equation of system (20) give us

$$\overline{y}(t) = \left(\frac{1}{a(t)}\right)^a \left(\overline{\mathfrak{x}}^4(t)\right)^a = \int\_t^\infty b(u) \left(\overline{\mathfrak{x}}^v(u)\right)^\beta \Delta u > 0, \quad t \ge t\_1. \tag{25}$$

This results in that y is eventually positive and hence ðx, yÞ is a nonoscillatory solution of system (20) in Mþ. Also by monotonicity of x, we have

$$\overline{\mathfrak{X}}(t) = 1 + \int\_{t\_2}^t a(s) \left( \int\_s^\infty b(u) \left( \overline{\mathfrak{X}}'(u) \right)^\beta \Delta u \right)^{\frac{1}{\alpha}} \succeq \left( \overline{\mathfrak{X}}(t\_2) \right)^\beta \int\_{t\_2}^t a(s) \left( \int\_s^\infty b(u) \Delta u \right)^{\frac{1}{\alpha}}.$$

Hence as t ! ∞, it follows xðtÞ ! ∞. And by Eq. (25), we have yðtÞ ! 0 as t ! ∞. Therefore M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup>≠∅:

Example 5.6 Let T ¼ q<sup>N</sup><sup>0</sup> , q > 1 and β < 1: Consider the system

$$\begin{cases} \mathbf{x}^{4} = (1+t)|y|^{\frac{1}{\alpha}} \text{sgn } y\\ y^{4} = -\frac{1}{(1+t)(1+tq)^{\beta+1}} |\mathbf{x}^{o}|^{\beta} \text{sgn } \mathbf{x}. \end{cases} \tag{26}$$

It is easy to verify Yðt0Þ ¼ <sup>∞</sup> and Zðt0<sup>Þ</sup> <sup>&</sup>lt; <sup>∞</sup>. Letting s <sup>¼</sup> <sup>q</sup><sup>m</sup> and t <sup>¼</sup> <sup>q</sup>n, where m, <sup>n</sup><sup>∈</sup> <sup>N</sup><sup>0</sup> gives

$$\begin{aligned} \int\_{t\_0}^T a(t) \left( \int\_t^T b(s) \Delta s \right)^{\frac{1}{\sigma}} \Delta t &= \sum\_{t \in [1, T]\_{q^N}} (1 + t) \left( \sum\_{s \in [t, T]\_{q^N}} \frac{(q - 1)s}{(1 + s)(1 + s q)^{\beta + 1}} \right) (q - 1)t^{\beta} \\ \geq (q - 1)^2 \sum\_{t \in [1, T]\_{q^N}} (1 + t) \left( \frac{t}{(1 + t)(1 + t q)^{\beta + 1}} \right) t &= (q - 1)^2 \sum\_{t \in [1, T]\_{q^N}} \frac{t^2}{(1 + t q)^{\beta + 1}} . \end{aligned}$$

So we have

$$\lim\_{T \to \infty} \sum\_{t \in [1, T)\_{q^N 0}} \frac{t^2}{(1 + tq)^{\delta + 1}} = \sum\_{n=0}^{\infty} \frac{q^{2n}}{(1 + q^{n+1})^{\delta + 1}} = \infty$$

by the Test for Divergence and β < 1. Now let us show that K<sup>β</sup> < ∞. Since

$$\int\_{t\_0}^{\sigma(t)} a(s) \Delta s = \sum\_{s \in [1, t)\_{q^{\mathbb{N}\_0}}} (1 + s)(q - 1)s \le tq(1 + tq),$$

we have

On Nonoscillatory Solutions of Two-Dimensional Nonlinear Dynamical Systems http://dx.doi.org/10.5772/67118 25

$$\begin{aligned} \int\_{t\_0}^T b(t) \left( \int\_{t\_0}^{v(t)} a(s) \Delta s \right)^{\beta} \Delta t &\leq \sum\_{t \in [1, T]\_{q^{\mathbb{N}\_0}}} \frac{1}{(1+t)(1+tq)^{\beta+1}} \left( tq(1+tq) \right)^{\beta} t(q-1) \\ &\leq q^{\beta} (q-1) \sum\_{t \in [1, T)\_{q^{\mathbb{N}\_0}}} \frac{t^{\beta}}{1+t} . \end{aligned}$$

Therefore by the Ratio test,

<sup>0</sup> <sup>≤</sup> ½ðTxÞðtÞ�<sup>Δ</sup> <sup>¼</sup> <sup>a</sup>ðt<sup>Þ</sup>

<sup>x</sup><sup>Δ</sup>ðt<sup>Þ</sup> �α ¼ ð∞ t bðuÞ � <sup>x</sup><sup>σ</sup>ðu<sup>Þ</sup> �β

<sup>x</sup><sup>Δ</sup> ¼ ð<sup>1</sup> <sup>þ</sup> <sup>t</sup>Þjy<sup>j</sup>

<sup>y</sup><sup>Δ</sup> <sup>¼</sup> <sup>−</sup> <sup>1</sup>

≤ aðtÞ �ð<sup>∞</sup> t bðuÞ

entiating x and the first equation of system (20) give us

aðtÞ � �<sup>α</sup>�

system (20) in Mþ. Also by monotonicity of x, we have

Example 5.6 Let T ¼ q<sup>N</sup><sup>0</sup> , q > 1 and β < 1: Consider the system

8 ><

>:

<sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>X</sup> t∈ ½1,TÞ qN0

<sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>t</sup><sup>Þ</sup> <sup>t</sup>

X t∈ ½1,TÞ q N0

by the Test for Divergence and β < 1. Now let us show that K<sup>β</sup> < ∞. Since

<sup>a</sup>ðsÞΔ<sup>s</sup> <sup>¼</sup> <sup>X</sup>

<sup>y</sup>ðtÞ ¼ <sup>1</sup>

24 Dynamical Systems - Analytical and Computational Techniques

ðt t2 aðsÞ �ð<sup>∞</sup> s bðuÞ � <sup>x</sup><sup>σ</sup>ðu<sup>Þ</sup> �β Δu � 1 α ≥ � xðt2Þ �β ðt t2 aðsÞ �ð<sup>∞</sup> s

xðtÞ ¼ 1 þ

ðT t0 aðtÞ �ð<sup>T</sup> t bðsÞΔs �1 α

≥ðq−1Þ

So we have

we have

<sup>2</sup> X t∈ ½1,TÞ q N0

> lim T!∞

> > ð<sup>σ</sup>ðt<sup>Þ</sup> t0

M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup>≠∅: �ð<sup>∞</sup> t bðuÞ � <sup>x</sup><sup>σ</sup>ðu<sup>Þ</sup> �β Δu � 1 α

aðλÞΔλ

it follows that T is equibounded and equicontinuous. Then by Theorem 2.10, there exists x∈ Ω such that x ¼ Tx: Thus, it follows that x is eventually positive, i.e nonoscillatory. Then differ-

This results in that y is eventually positive and hence ðx, yÞ is a nonoscillatory solution of

Hence as t ! ∞, it follows xðtÞ ! ∞. And by Eq. (25), we have yðtÞ ! 0 as t ! ∞. Therefore

1 <sup>α</sup>sgn y

> <sup>β</sup>þ<sup>1</sup> <sup>j</sup>x<sup>σ</sup><sup>j</sup> β sgn x:

ð1 þ tÞð1 þ tqÞ

It is easy to verify Yðt0Þ ¼ <sup>∞</sup> and Zðt0<sup>Þ</sup> <sup>&</sup>lt; <sup>∞</sup>. Letting s <sup>¼</sup> <sup>q</sup><sup>m</sup> and t <sup>¼</sup> <sup>q</sup>n, where m, <sup>n</sup><sup>∈</sup> <sup>N</sup><sup>0</sup> gives

ð1 þ tÞð1 þ tqÞ

t 2

ð1 þ tqÞ

s∈½1,tÞ qN0

!

<sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>t</sup><sup>Þ</sup> <sup>X</sup>

βþ1

<sup>β</sup>þ<sup>1</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> n¼0

0 B@

s ∈½t,TÞ qN0

t ¼ ðq−1Þ

q<sup>2</sup><sup>n</sup> ð1 þ qnþ<sup>1</sup>Þ

ð1 þ sÞðq−1Þs ≤ tqð1 þ tqÞ,

ðq−1Þs ð1 þ sÞð1 þ sqÞ

> <sup>2</sup> X t∈½1,TÞ q N0

> > <sup>β</sup>þ<sup>1</sup> <sup>¼</sup> <sup>∞</sup>

βþ1

t 2

ð1 þ tqÞ

1

CAðq−1Þ<sup>t</sup>

<sup>β</sup>þ<sup>1</sup> :

�β Δu �1 α < ∞,

Δu > 0, t≥t1: (25)

bðuÞΔu � 1 α :

(26)

�ð<sup>σ</sup>ðu<sup>Þ</sup> t1

$$\lim\_{T \to \infty} q^{\beta}(q-1) \sum\_{t \in [1,T)\_{q^{\mathbb{N}}}} \frac{t^{\beta}}{1+t} = q^{\beta}(q-1) \sum\_{n=0}^{\infty} \frac{(q^n)^{\beta}}{(1+q^n)} < \infty$$

gives K<sup>β</sup> <sup>&</sup>lt; <sup>∞</sup>. It can also be verified that <sup>1</sup> <sup>þ</sup> <sup>t</sup>, <sup>1</sup> t þ 1 � � is a nonoscillatory solution of Eq. (26) in M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup>:

Exercise 5.7 Show that the following system

$$\begin{cases} \mathbf{x}' = e^{2t} |y|^{\frac{1}{\alpha}} \text{sgn } y \\ \mathbf{y}' = -\alpha e^{-t(\alpha+\beta)} |\mathbf{x}|^{\beta} \text{sgn } \mathbf{x} \end{cases}$$

has a nonoscillatory solution <sup>ð</sup>e<sup>t</sup> ,e<sup>−</sup>α<sup>t</sup> Þ in M<sup>þ</sup> <sup>∞</sup>;<sup>0</sup>.

Next, we intend to derive a conclusion for the existence of nonoscillatory solutions of system (20) based on α and β. The proof of the following lemma is similar to the proofs of Lemmas 1.1, 3.2, 3.3, 3.6 and 3.7 in [47].

#### Lemma 5.8


Exercise 5.9 Prove Lemma 5.8.

The following corollary summarizes the existence of subdominant and dominant solutions of system (20) in this subsection by means of Lemma 5.8.

Corollary 5.10 Suppose that Yðt0Þ ¼ ∞ and Zðt0Þ < ∞. Then

	- (i) J<sup>α</sup> < ∞, (ii) α < β, β≥1 and J<sup>β</sup> < ∞,

(iii) α < β and K<sup>β</sup> < ∞, (iv) α ≤ 1 and K<sup>α</sup> < ∞.

	- (i) K<sup>β</sup> < ∞, (ii) α≥1 and J<sup>β</sup> < ∞,
	- (iii) α > β and J<sup>α</sup> < ∞.

With the similar discussion as in Subsection 4.2, we concentrate on M<sup>þ</sup> <sup>B</sup>,<sup>B</sup> and M<sup>þ</sup> <sup>B</sup>;<sup>0</sup>. Actually, the existence in M<sup>þ</sup> <sup>B</sup>;<sup>0</sup> is shown in Subsection 5.1. Also, we use the same argument of the proof of Lemma 3.1(a) so that the criteria for the existence of nonoscillatory solutions of system (20) in M<sup>þ</sup> <sup>B</sup>,<sup>B</sup> is Yðt0Þ < ∞ and Zðt0Þ < ∞.

The most important question that arose in this section is about the existence of nonoscillatory solutions of the Emden-Fowler system in M<sup>−</sup> . The existence of such solutions in M<sup>−</sup> <sup>B</sup>,∞, <sup>M</sup><sup>−</sup> <sup>0</sup>;<sup>∞</sup> can similarly be shown as in Theorems 3.7 and 3.9. When concerns about and M<sup>−</sup> <sup>0</sup>;<sup>B</sup> come to our attention, we need to assume that σ must be differentiable, which is not necessarily true on arbitrary time scales, see Example 1.56 in [6]. The following exercise is a great observation about the discussion mentioned above.

Exercise 5.11 Consider the system

$$\begin{cases} \mathbf{x}^{\rm A}(t) = \frac{t^{\frac{1}{2}}}{2(t+1)(t+2)(3t-1)^{\frac{1}{2}}} |y(t)|^{\frac{1}{2}} \text{sgn } y(t) \\\ \mathbf{y}^{\rm A}(t) = -\frac{(t+1)^{\frac{1}{3}}}{2^{\frac{1}{3}}(4t+5)^{\frac{1}{3}}} |\mathbf{x}^{\boldsymbol{\sigma}}(t)|^{\frac{1}{3}} \text{sgn } \mathbf{x}^{\boldsymbol{\sigma}}(t) \end{cases} \tag{27}$$

in <sup>T</sup><sup>¼</sup> <sup>2</sup><sup>N</sup><sup>0</sup> and show that <sup>ð</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>t</sup> <sup>þ</sup> <sup>2</sup> , <sup>−</sup><sup>3</sup> <sup>þ</sup> <sup>1</sup> t <sup>Þ</sup> is a nonoscillatory solution of system (27) in <sup>M</sup><sup>−</sup> <sup>B</sup>,B. Note that <sup>σ</sup>ðtÞ ¼ <sup>2</sup><sup>t</sup> is differentiable on <sup>T</sup><sup>¼</sup> <sup>2</sup><sup>N</sup><sup>0</sup> :

## Author details

Elvan Akın\* and Özkan Öztürk

\*Address all correspondence to: akine@mst.edu

Missouri University of Science and Technology, Missouri, USA

#### References

[1] Kelley W. G., Peterson A. C. The Theory of Differential Equations: Classical and Qualitative, Springer, 2010. 413 p. DOI: 10.1007/978-1-4419-5783-2

[2] Kelley W. G., Peterson A. C. Difference Equations, Second Edition: An Introduction with Applications, Academic Press, 2001. 403 p.

(iii) α < β and K<sup>β</sup> < ∞, (iv) α ≤ 1 and K<sup>α</sup> < ∞.

With the similar discussion as in Subsection 4.2, we concentrate on M<sup>þ</sup>

<sup>B</sup>;<sup>0</sup> is shown in Subsection 5.1. Also, we use the same argument of the proof of

1 2 jyðtÞj 1 2sgn yðtÞ

. The existence of such solutions in M<sup>−</sup>

<sup>Þ</sup> is a nonoscillatory solution of system (27) in <sup>M</sup><sup>−</sup>

Lemma 3.1(a) so that the criteria for the existence of nonoscillatory solutions of system (20) in

The most important question that arose in this section is about the existence of nonoscillatory

attention, we need to assume that σ must be differentiable, which is not necessarily true on arbitrary time scales, see Example 1.56 in [6]. The following exercise is a great observation

> 1 2 2ðt þ 1Þðt þ 2Þð3t−1Þ

1 3

[1] Kelley W. G., Peterson A. C. The Theory of Differential Equations: Classical and Qualita-

<sup>2</sup>ð4<sup>t</sup> <sup>þ</sup> <sup>5</sup><sup>Þ</sup> 1 3 <sup>j</sup>x<sup>σ</sup>ðtÞj 1 3sgn <sup>x</sup><sup>σ</sup>ðt<sup>Þ</sup>

t

similarly be shown as in Theorems 3.7 and 3.9. When concerns about and M<sup>−</sup>

<sup>x</sup><sup>Δ</sup>ðtÞ ¼ <sup>t</sup>

<sup>y</sup><sup>Δ</sup>ðtÞ ¼ <sup>−</sup> <sup>ð</sup><sup>t</sup> <sup>þ</sup> <sup>1</sup><sup>Þ</sup>

<sup>t</sup> <sup>þ</sup> <sup>2</sup> , <sup>−</sup><sup>3</sup> <sup>þ</sup> <sup>1</sup>

2 2 3t <sup>B</sup>,<sup>B</sup> and M<sup>þ</sup>

<sup>B</sup>;<sup>0</sup>. Actually, the

<sup>B</sup>,∞, <sup>M</sup><sup>−</sup>

<sup>0</sup>;<sup>B</sup> come to our

<sup>0</sup>;<sup>∞</sup> can

(27)

<sup>B</sup>,B.

<sup>∞</sup>,B≠∅ if any of the followings hold:

26 Dynamical Systems - Analytical and Computational Techniques

(i) K<sup>β</sup> < ∞, (ii) α≥1 and J<sup>β</sup> < ∞,

5.2. The Case Yðt0Þ < ∞ and Zðt0Þ < ∞

<sup>B</sup>,<sup>B</sup> is Yðt0Þ < ∞ and Zðt0Þ < ∞.

solutions of the Emden-Fowler system in M<sup>−</sup>

about the discussion mentioned above.

8 >>>><

>>>>:

Note that <sup>σ</sup>ðtÞ ¼ <sup>2</sup><sup>t</sup> is differentiable on <sup>T</sup><sup>¼</sup> <sup>2</sup><sup>N</sup><sup>0</sup> :

\*Address all correspondence to: akine@mst.edu

Missouri University of Science and Technology, Missouri, USA

tive, Springer, 2010. 413 p. DOI: 10.1007/978-1-4419-5783-2

Exercise 5.11 Consider the system

in <sup>T</sup><sup>¼</sup> <sup>2</sup><sup>N</sup><sup>0</sup> and show that <sup>ð</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup>

Elvan Akın\* and Özkan Öztürk

Author details

References

(iii) α > β and J<sup>α</sup> < ∞.

existence in M<sup>þ</sup>

M<sup>þ</sup>

b. M<sup>þ</sup>


[34] Zhang X. Nonoscillation Criteria for Nonlinear Delay Dynamic Systems on Time Scales. Int. J. Math. Comput. Natural Phys. Eng., 2014; 8(1):222–226.

[18] Ritter A. Untersuchungen über die Höhe der Atmosphäre und die Konstitution gasförmiger Weltkörper, 18 articles, Wiedemann Annalender Physik, 5–20 (1878–1883).

[19] Fowler R. H. The Form Near Infinity of Real, Continuous Solutions of a Certain Differen-

[20] Fowler R. H. The Solution of Emden's and Similar Differential Equations. Monthly

[21] Fowler R. H. Some Results on the Form Near Infinity of Real Continuous Solutions of a Certain Type of Second Order Differential Equations. Proc. London Math. Soc.,

[22] Fowler R. H. Further Studies of Emden's and Similar Differential Equations. Quart. J.

[23] Zeidler E. Nonlinear Functional Analysis and its Applications - I: Fixed Point Theorems,

[24] Sidney M. A., Noussair E. S. The Schauder-Tychonoff Fixed Point Theorem and Applica-

[25] Knaster B. Un théorème sur les fonctions d'ensembles. Ann. Soc. Polon. Math., 1928;

[26] Öztürk Ö., Akın E. Nonoscillation Criteria for Two Dimensional Timeâ€"-Scale Systems.

[27] Li W. T., Cheng S. Limiting Behaviors of Non-oscillatory Solutions of a Pair of Coupled

[28] Li W. T. Classification Schemes for Nonoscillatory Solutions of Two-Dimensional Nonlinear Difference Systems. Comput. Math. Appl., 2001; 42:341–355. DOI:10.1016/

[29] Anderson D. R. Oscillation and Nonoscillation Criteria for Two-dimensional Time-Scale Systems of First Order Nonlinear Dynamic Equations. Electron. J. Differential Equations,

[30] Hassan T. S. Oscillation Criterion for Two-Dimensional Dynamic Systems on Time Scales.

[31] Zhu S., Sheng C. Oscillation and Nonoscillation Criteria for Nonlinear Dynamic Systems on Time Scales. Discrete Dyn. Nature Soc., 2012; 2012:1–14. Article ID 137471.

[32] Agarwal R. P., Manojlovic′ J. V. On the Existence and the Asymptotic Behavior of Nonoscillatory Solutions of Second Order Quasilinear Difference Equations. Funkcialaj

[33] Ciarlet P. G. Linear and Nonlinear Functional Analysis with Applications, Siam, 2013.

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#### **Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument** Oscillation Criteria for Second-Order Neutral Damped Differential Equations with Delay Argument

Said R. Grace and Irena Jadlovská Said R. Grace and Irena Jadlovská

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/65909

#### Abstract

The chapter is devoted to study the oscillation of all solutions to second-order nonlinear neutral damped differential equations with delay argument. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and integral averaging techniques.

2010 Mathematics Subject Classification: 34C10, 34K11.

Keywords: neutral differential equation, damping, delay, second-order, generalized Riccati technique, oscillation

## 1. Introduction

In the chapter, we are mainly concerned with the oscillatory behavior of solutions to secondorder nonlinear neutral damped differential equations with delay argument of the form

$$\left(r(t)\left(\dot{z'}(t)\right)^{\alpha}\right)' + p(t)\left(\dot{z'}(t)\right)^{\alpha} + q(t)f\left(\mathbf{x}(\sigma(t))\right) = 0, \quad t \ge t\_0,\tag{1}$$

where α≥1 is a quotient of positive odd integers and

$$z(t) = \mathbf{x}(t) + a(t)\mathbf{x}(\tau(t)).\tag{2}$$

Throughout, we suppose that the following hypotheses hold:

$$\text{i.i.}\quad r,\ p,\ q \in \mathbb{C}(\mathcal{F}, \mathbb{R}^+), \\
\text{where } \mathcal{F} = [t\_0, \ast) \text{ and } \mathbb{R}^+ = (0, \ast);$$

ii. a∈Cðℐ, ℝÞ, 0 ≤ a ðtÞ ≤ 1;

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.


By a solution of Eq. (1), we mean a nontrivial real-valued function xðtÞ, which has the property <sup>z</sup>ðtÞ∈C<sup>1</sup> ð½Tx, ∞ÞÞ, rðtÞ z′ ðtÞ α ∈C<sup>1</sup> ð½Tx, ∞ÞÞ, Tx≥t0, and satisfies Eq. (1) on ½Tx, ∞Þ. In the sequel, we will restrict our attention to those solutions xðtÞ of Eq. (1) that satisfy the condition

$$\sup\left\{|\mathbf{x}(t)|: T \le t < \ast\right\} > 0 \quad \text{for} \quad T \ge T\_x. \tag{3}$$

We make the standing hypothesis that Eq. (1) admits such a solution. As is customary, a solution of Eq. (1) is said to be oscillatory if it is neither eventually positive nor eventually negative on ½Tx, ∞Þ and otherwise, it is termed nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

Remark 1. All the functional inequalities considered in the sequel are assumed to hold eventually, that is, they are satisfied for all t large enough.

Oscillation theory was created in 1836 with a paper of Jacques Charles François Sturm published in Journal des Mathematiqués Pures et Appliqueés. His long and detailed memoir [1] was one of the first contributions in Liouville's newly founded journal and initiated a whole new research into the qualitative analysis of differential equations. Heretofore, the theory of differential equations was primarily about finding solutions of a given equation and so was very limited. Contrarily, the main idea of Sturm was to obtain geometric properties of solutions (such as sign changes, zeros, boundaries, and oscillation) directly from the differential equation, without benefit of solutions themselves.

Henceforth, the oscillation theory for ordinary differential equations has undergone a significant development. Nowadays, it is considered as coherent, self-contained domain in the qualitative theory of differential equations that is turning mainly toward the study of solution properties of functional differential equations (FDEs).

The problem of obtaining sufficient conditions for asymptotic and oscillatory properties of different classes of FDEs has experienced long-term interest of many researchers. This is caused by the fact that differential equations, especially those with deviating argument, are deemed to be adequate in modeling of the countless processes in all areas of science. For a summary of the most significant efforts and recent findings in the oscillation theory of FDEs and vast bibliography therein, we refer the reader to the excellent monographs [2–6].

In a neutral delay differential equation the highest-order derivative of the unknown function appears both with and without delay. The study of qualitative properties of solutions of such equations has, besides its theoretical interest, significant practical importance. This is due to the fact that neutral differential equations arise in various phenomena including problems concerning electric networks containing lossless transmission lines (as in high-speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar or in the solution of variational problems with time delays. We refer the reader to the monograph [7] for further applications in science and technology.

So far, most of the results obtained in the literature has centered around the special undamped form of Eq. (1), i.e., when pðtÞ ¼ 0 (for example, see Refs. [8–18]). For instance, in one of the pioneering works on the subject, Grammatikopoulos et al. [8] studied the second-order neutral differential equation with constant delay of the form

$$\mathbf{x}(\mathbf{x}(t) + a(t)\mathbf{x}(t-\tau)\stackrel{\*}{\ast} + q(t)\mathbf{x}(t-\tau) = \mathbf{0} \tag{4}$$

and proved that Eq. (4) is oscillatory if

$$\int\_{t\_0}^{\infty} q(s) \left( 1 \text{-} a(s - \tau) \right) \text{ds} = \text{\textquotedbl{}s\textquotedbl{}} \tag{5}$$

Later on, Grace and Lalli [9] extended the results from [8] to the more general equation

$$\left(r(t)(\mathbf{x}(t) + a(t)\mathbf{x}(t-\tau))\right)' + q(t)f\left(\mathbf{x}(t-\tau)\right) = 0,\tag{6}$$

with

iii. τ∈Cðℐ, ℝÞ, τðtÞ≤t, τðtÞ ! ∞ as t ! ∞;

32 Dynamical Systems - Analytical and Computational Techniques

the ratio of odd positive integers.

 z′ ðtÞ α ∈C<sup>1</sup>

oscillatory if all its solutions are oscillatory.

tion, without benefit of solutions themselves.

properties of functional differential equations (FDEs).

tually, that is, they are satisfied for all t large enough.

ðtÞ≥0, σðtÞ ! ∞ as t ! ∞;

v. <sup>f</sup>∈Cðℝ, <sup>ℝ</sup>Þ, such that xfðx<sup>Þ</sup> <sup>&</sup>gt; 0 and <sup>f</sup>ðxÞ=x<sup>β</sup>≥<sup>k</sup> <sup>&</sup>gt; 0 for <sup>x</sup>=<sup>¼</sup> 0, where <sup>k</sup> is a constant and <sup>β</sup> is

By a solution of Eq. (1), we mean a nontrivial real-valued function xðtÞ, which has the property

We make the standing hypothesis that Eq. (1) admits such a solution. As is customary, a solution of Eq. (1) is said to be oscillatory if it is neither eventually positive nor eventually negative on ½Tx, ∞Þ and otherwise, it is termed nonoscillatory. The equation itself is called

Remark 1. All the functional inequalities considered in the sequel are assumed to hold even-

Oscillation theory was created in 1836 with a paper of Jacques Charles François Sturm published in Journal des Mathematiqués Pures et Appliqueés. His long and detailed memoir [1] was one of the first contributions in Liouville's newly founded journal and initiated a whole new research into the qualitative analysis of differential equations. Heretofore, the theory of differential equations was primarily about finding solutions of a given equation and so was very limited. Contrarily, the main idea of Sturm was to obtain geometric properties of solutions (such as sign changes, zeros, boundaries, and oscillation) directly from the differential equa-

Henceforth, the oscillation theory for ordinary differential equations has undergone a significant development. Nowadays, it is considered as coherent, self-contained domain in the qualitative theory of differential equations that is turning mainly toward the study of solution

The problem of obtaining sufficient conditions for asymptotic and oscillatory properties of different classes of FDEs has experienced long-term interest of many researchers. This is caused by the fact that differential equations, especially those with deviating argument, are deemed to be adequate in modeling of the countless processes in all areas of science. For a summary of the most significant efforts and recent findings in the oscillation theory of FDEs

In a neutral delay differential equation the highest-order derivative of the unknown function appears both with and without delay. The study of qualitative properties of solutions of such equations has, besides its theoretical interest, significant practical importance. This is due to the fact that neutral differential equations arise in various phenomena including problems concerning electric networks containing lossless transmission lines (as in high-speed computers

and vast bibliography therein, we refer the reader to the excellent monographs [2–6].

we will restrict our attention to those solutions xðtÞ of Eq. (1) that satisfy the condition

ð½Tx, ∞ÞÞ, Tx≥t0, and satisfies Eq. (1) on ½Tx, ∞Þ. In the sequel,

sup {jxðtÞj : T≤t < ∞} > 0 for T≥Tx: (3)

<sup>ð</sup>ℐ, <sup>ℝ</sup>Þ, <sup>σ</sup>ðtÞ≤t, <sup>σ</sup>′

ð½Tx, ∞ÞÞ, rðtÞ

iv. σ∈C<sup>1</sup>

<sup>z</sup>ðtÞ∈C<sup>1</sup>

$$\frac{f(\mathbf{x})}{\mathbf{x}} \succeq k, \quad k > 0 \quad \text{and} \quad \int\_{t\_0}^{\boldsymbol{\alpha}} \frac{\mathbf{ds}}{r(\mathbf{s})} = \boldsymbol{\alpha} \tag{7}$$

and showed that Eq. (6) is oscillatory if there exists a continuously differentiable function ρðtÞ such that

$$\int\_{t\_0}^{\infty} \left( \rho(s) q(s) (1 - a(s - \tau)) - \frac{\left(\rho'(s)\right)^2 r(s - \tau)}{4k\rho(s)} \right) d\mathbf{s} = \text{\textquotedbl{}} \tag{8}$$

In Ref. [10], Dong has involved to study the oscillation problem for a half-linear case of Eq. (1) and by defining a sequence of continuous functions has obtained various kinds of better results. Afterward, his approach has been further developed by several authors, see, e.g., [11– 14]. However, it appears that very little is known regarding the oscillation of Eq. (1) with pðtÞ≠0 and α≠β. Motivated by the results of Ref. [10], this chapter presents some new oscillation criteria, which are applicable on Eq. (1).

On the other hand, Eq. (1) can be considered as a natural generalization of the second-order delay differential equation of the form

$$
\left(r(t)\left(\mathbf{x'}(t)\right)^{\alpha}\right) + p(t)\left(\mathbf{x'}(t)\right)^{\alpha} + q(t)f\left(\mathbf{x}(\sigma(t))\right) = 0. \tag{9}
$$

Very recently, the authors of [19] studied the oscillation problem of Eq. (9) with pðtÞ ¼ 0 and α ¼ β. Their ideas, which are based on careful investigation of classical techniques covering Riccati transformations and integral averages, will be extended to the more general equation (1).

#### 2. Main results

For the simplicity and without further mention, we use the following notations:

$$A(t) = \exp\left(-\int\_{t\_0}^t \frac{p(s)}{r(s)} \, \mathrm{d}s\right), \quad Q(t) = kq(t) \Big(1 - a(\sigma(t))\Big)^\beta,\tag{10}$$

$$R(t) = \int\_{t}^{\infty} \left(\frac{A(s)}{r(s)}\right)^{\frac{1}{a}} ds, \quad \tilde{Q}(t) = q(t) \Big(1 - a(\sigma(t)) \frac{R(\pi(\sigma(t)))}{R(\sigma(t))}\Big)^{\beta},\tag{11}$$

$$P(t) = \frac{\phi^{\prime}(t)}{\phi(t)} \text{-} \frac{p(t)}{r(t)}, \quad \tilde{q}(t) = Q(t) + \frac{p(t)A(t)}{r(t)} \int\_{t}^{\infty} \frac{Q(s)}{A(s)} ds,\tag{12}$$

where <sup>φ</sup>ðtÞ∈C<sup>1</sup> ðℐ, ℝÞ is a given function and will be specified later.

The organization of this chapter is as follows. Before stating our main results, we present two lemmas that ensure that any solution xðtÞ of Eq. (1) satisfies the condition

$$z(t) > 0, \quad \dot{z'}(t) > 0, \quad \left(r(t)\left(\dot{z'}(t)\right)^a\right)' < 0,\tag{13}$$

for t sufficiently large. Next, we get our main oscillation results for Eq. (1) by employing the generalized Riccati transformations and integral averaging techniques. We base our arguments on the assumption that the function PðtÞ is positive or negative.

Lemma 1. Assume that

$$\int\_{t\_0}^{\infty} \left(\frac{A(s)}{r(s)}\right)^{\frac{1}{\pi}} \mathrm{ds} = \mathrm{\bf \,} \tag{14}$$

holds and Eq. (1) has a positive solution xðtÞ on ℐ. Then there exists a T∈ℐ, sufficiently large, such that

$$z(t) > 0, \quad \dot{z'}(t) > 0, \quad \left(r(t)\left(z'(t)\right)^a\right)' < 0,\tag{15}$$

on ½T, ∞Þ.

Proof. Since, xðtÞ is a positive solution of Eq. (1) on ℐ, then, by the assumptions ðiiiÞ and ðivÞ, there exists a t1∈ℐ such that xðτðtÞÞ > 0 and xðσðtÞÞ > 0 on ½t1, ∞Þ. Define the function zðtÞ as in Eq. (2). Then it is easy to see that zðtÞ≥xðtÞ > 0, for t≥t1, and at the same time, from Eq. (1), we get

Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument http://dx.doi.org/10.5772/65909 35

$$\left(r(t)\left(z'(t)\right)^a\right)' + p(t)\left(z'(t)\right)^a = -q(t)f\left(\mathbf{x}(\sigma(t))\right) < 0. \tag{16}$$

We assert that <sup>r</sup>ðt<sup>Þ</sup> AðtÞ � z′ ðtÞ �α is decreasing. Clearly, by writing the left-hand side of Eq. (16) in the form

$$\left(r(t)\left(\dot{z'}(t)\right)^{\alpha}\right)' + \frac{p(t)}{r(t)}r(t)\left(\dot{z'}(t)\right)^{\alpha} < 0,\tag{17}$$

we get

Riccati transformations and integral averages, will be extended to the more general equa-

, QðtÞ ¼ kqðtÞ

<sup>d</sup>s, <sup>Q</sup><sup>~</sup> <sup>ð</sup>tÞ ¼ <sup>q</sup>ðt<sup>Þ</sup> <sup>1</sup>−aðσðtÞÞRðτðσðtÞÞÞ

rðtÞ

, <sup>~</sup>qðtÞ ¼ <sup>Q</sup>ðtÞ þ <sup>p</sup>ðtÞAðt<sup>Þ</sup>

The organization of this chapter is as follows. Before stating our main results, we present two

for t sufficiently large. Next, we get our main oscillation results for Eq. (1) by employing the generalized Riccati transformations and integral averaging techniques. We base our arguments

� rðtÞ � z′ ðtÞ �<sup>α</sup>�′

�

1−aðσðtÞÞ

� �<sup>β</sup>

ð∞ t

RðσðtÞÞ

QðsÞ AðsÞ �β

, (10)

, (11)

ds, (12)

< 0; (13)

< 0; (15)

ds ¼ ∞ (14)

For the simplicity and without further mention, we use the following notations:

ðℐ, ℝÞ is a given function and will be specified later.

ðtÞ > 0;

ð∞ t0

AðsÞ rðsÞ � �<sup>1</sup> α

ðtÞ > 0;

holds and Eq. (1) has a positive solution xðtÞ on ℐ. Then there exists a T∈ℐ, sufficiently large,

Proof. Since, xðtÞ is a positive solution of Eq. (1) on ℐ, then, by the assumptions ðiiiÞ and ðivÞ, there exists a t1∈ℐ such that xðτðtÞÞ > 0 and xðσðtÞÞ > 0 on ½t1, ∞Þ. Define the function zðtÞ as in Eq. (2). Then it is easy to see that zðtÞ≥xðtÞ > 0, for t≥t1, and at the same time, from Eq. (1), we

� rðtÞ � z′ ðtÞ �<sup>α</sup>�′

lemmas that ensure that any solution xðtÞ of Eq. (1) satisfies the condition

<sup>z</sup>ðt<sup>Þ</sup> <sup>&</sup>gt; <sup>0</sup>; <sup>z</sup>′

on the assumption that the function PðtÞ is positive or negative.

<sup>z</sup>ðt<sup>Þ</sup> <sup>&</sup>gt; <sup>0</sup>; <sup>z</sup>′

ðt t0 pðsÞ rðsÞ ds � �

AðtÞ ¼ exp −

AðsÞ rðsÞ � �<sup>1</sup> α

ðtÞ φðtÞ − pðtÞ rðtÞ

ð∞ t

34 Dynamical Systems - Analytical and Computational Techniques

<sup>P</sup>ðtÞ ¼ <sup>φ</sup>′

RðtÞ ¼

tion (1).

2. Main results

where <sup>φ</sup>ðtÞ∈C<sup>1</sup>

Lemma 1. Assume that

such that

on ½T, ∞Þ.

get

$$\left(\frac{r(t)}{A(t)} \left(\dot{z}(t)\right)^a\right)' = -\frac{q(t)}{A(t)} f(\mathbf{x}(\sigma(t))) < 0\tag{18}$$

and so the assertion is proved.

Now, we claim that z′ <sup>ð</sup>t<sup>Þ</sup> <sup>&</sup>gt; 0 on <sup>½</sup>t1, <sup>∞</sup>Þ. If not, then there exists <sup>t</sup>2∈½t1, <sup>∞</sup><sup>Þ</sup> such that <sup>z</sup>′ ðt2Þ < 0. Using the fact that <sup>r</sup>ðt<sup>Þ</sup> AðtÞ � z′ ðtÞ �α is decreasing, we obtain, for t≥t2,

$$\frac{r(t)}{A(t)} \left( \dot{z'}(t) \right)^{\alpha} < c := \frac{r(t\_2)}{A(t\_2)} \left( \dot{z'}(t\_2) \right)^{\alpha} < 0. \tag{19}$$

Integrating the above inequality from t<sup>2</sup> to t, we find that

$$z(t) < z(t\_2) + c^{\frac{1}{\kappa}} \Big|\_{t\_2}^{t} \left(\frac{A(s)}{r(s)}\right)^{\frac{1}{\kappa}} \text{ds} \tag{20}$$

for t≥t2: By condition (14), zðtÞ approaches to −∞ as t ! ∞, which contradicts the fact that zðtÞ is eventually positive. Therefore, z′ <sup>ð</sup>t<sup>Þ</sup> <sup>&</sup>gt; 0 and from Eq. (1), we have that � rðtÞ � z′ ðtÞ �<sup>α</sup>�′ < 0. The proof is complete.

Lemma 2. Assume that

$$\int\_{t\_0}^{\infty} \left( \frac{A(u)}{r(u)} \int\_{t\_0}^{u} \frac{\tilde{Q}(s)R^\delta(\sigma(s))}{A(s)} ds \right)^{\frac{1}{u}} \mathrm{d}u = \text{\textquotedblleft}, \tag{21}$$

holds and Eq. (1) has a positive solution xðtÞ on ℐ. Then there exists T∈ℐ, sufficiently large, such that

$$z(t) > 0, \quad \dot{z'}(t) > 0, \quad \left(r(t)\left(z'(t)\right)^a\right)' < 0,\tag{22}$$

on ½T, ∞Þ.

Proof. Similarly to the proof of Lemma 1, we assume that there exists t2∈ℐ such that z′ ðtÞ < 0 on ½t2, ∞Þ. Taking Eq. (18) into account, we have

$$z'(s) \le \left(\frac{r(t)}{A(t)} \frac{A(s)}{r(s)}\right)^{\frac{1}{a}} z'(t),\tag{23}$$

for s≥t≥t2. Integrating the above inequality from t to t ′ , t ′ ≥t≥t2, we get

$$z(t') \lesssim z(t) + \left(\frac{r(t)}{A(t)}\right)^{\frac{1}{\alpha}} z'(t) \int\_{t}^{t} \left(\frac{r(s)}{A(s)}\right)^{-\frac{1}{\alpha}} \mathrm{d}s.\tag{24}$$

Letting t ′ ! <sup>∞</sup>, we have

$$z(t) \succeq -\mathcal{R}(t) \left(\frac{r(t)}{A(t)}\right)^{\frac{1}{a}} z'(t),\tag{25}$$

which yields

$$\left(\frac{z(t)}{R(t)}\right) \ge 0\tag{26}$$

and hence we see that <sup>z</sup>ðt<sup>Þ</sup> <sup>R</sup>ðt<sup>Þ</sup> is nondecreasing. By Eq. (2) and <sup>ð</sup>iiiÞ, we have

$$\begin{aligned} x(t) &= z(t) \neg a(t)x(\tau(t)) \\ &\ge z(t) \neg a(t)z(\tau(t)) \\ &\ge \left(1 \neg a(t) \frac{R(\tau(t))}{R(t)}\right) z(t), \end{aligned} \tag{27}$$

which together with Eq. (1) and the assumption ðvÞ yields

$$\begin{aligned} \left(r(t)\left(z'(t)\right)^a\right)' + p(t)\left(z'(t)\right)^a &\leq -k\eta(t)\left(1 - a(\sigma(t))\frac{\mathcal{R}(\tau(\sigma(t)))}{\mathcal{R}(\sigma(t))}\right)^\beta \mathcal{z}^\delta(\sigma(t)) \\ &= -k\breve{\mathcal{Q}}(t)z^\delta(\sigma(t)). \end{aligned} \tag{28}$$

On the other hand, from Eq. (23), we have

$$\frac{r(t)\left(z'(t)\right)^{\alpha}}{A(t)} \le \frac{r(t\_2)\left(z'(t\_2)\right)^{\alpha}}{A(t\_2)},\tag{29}$$

that is,

$$\frac{r(t)}{A(t)} \left( -z'(t) \right)^a \ge \frac{r(t\_2)}{A(t\_2)} \left( -z'(t\_2) \right)^a \; : \; \; = \gamma^a \tag{30}$$

for some positive constant γ. Setting Eq. (30) into Eq. (25), we obtain

Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument http://dx.doi.org/10.5772/65909 37

$$z(t) \succeq\_{\mathcal{V}} \mathcal{R}(t) \tag{31}$$

and so, Eq. (28) becomes

$$\left(r(t)\left(z'(t)\right)^a\right)' + p(t)\left(z'(t)\right)^a \hookrightarrow \tilde{\gamma}\tilde{Q}(t)R^\S(\sigma(t)),\tag{32}$$

where <sup>γ</sup><sup>~</sup> : <sup>¼</sup> <sup>k</sup>γ<sup>β</sup>. Now, if we define the function

$$\mathcal{U}I(t) = r(t)\left(-z'(t)\right)^{\alpha} > 0,\tag{33}$$

then

ðtÞ < 0

(27)

(28)

ðtÞ, (23)

ðtÞ, (25)

≥0 (26)

ds: (24)

Proof. Similarly to the proof of Lemma 1, we assume that there exists t2∈ℐ such that z′

rðtÞ AðtÞ

AðtÞ � �<sup>1</sup> α z′ ðtÞ ðt ′

<sup>z</sup>ðtÞ≥−Rðt<sup>Þ</sup> <sup>r</sup>ðt<sup>Þ</sup>

AðtÞ � �<sup>1</sup> α z′

zðtÞ RðtÞ � �

<sup>R</sup>ðt<sup>Þ</sup> is nondecreasing. By Eq. (2) and <sup>ð</sup>iiiÞ, we have

RðτðtÞÞ RðtÞ

zðtÞ,

≤−kqðt<sup>Þ</sup> <sup>1</sup>−aðσðtÞÞ <sup>R</sup>ðτðσðtÞÞÞ

<sup>¼</sup> <sup>−</sup>kQ<sup>~</sup> <sup>ð</sup>tÞz<sup>β</sup>ðσðtÞÞ:

RðσðtÞÞ

<sup>z</sup><sup>β</sup>ðσðtÞÞ

<sup>A</sup>ðt2<sup>Þ</sup> , (29)

: <sup>¼</sup> γα (30)

� �<sup>β</sup>

� �

rðt2Þ � z′ ðt2Þ �α

> � −z′ ðt2Þ �α

xðtÞ ¼ zðtÞ−aðtÞxðτðtÞÞ ≥zðtÞ−aðtÞzðτðtÞÞ

≥ 1−aðtÞ

AðsÞ rðsÞ � �<sup>1</sup>

α z′

′ , t ′

t

rðsÞ AðsÞ � �<sup>−</sup><sup>1</sup> α

≥t≥t2, we get

z′ ðsÞ≤

<sup>Þ</sup>≤zðtÞ þ <sup>r</sup>ðt<sup>Þ</sup>

on ½t2, ∞Þ. Taking Eq. (18) into account, we have

36 Dynamical Systems - Analytical and Computational Techniques

for s≥t≥t2. Integrating the above inequality from t to t

Letting t

which yields

that is,

and hence we see that <sup>z</sup>ðt<sup>Þ</sup>

� rðtÞ � z′ ðtÞ �<sup>α</sup>�′

On the other hand, from Eq. (23), we have

′ ! <sup>∞</sup>, we have

zðt ′

which together with Eq. (1) and the assumption ðvÞ yields

þ pðtÞ � z′ ðtÞ �α

> rðtÞ � z′ ðtÞ �α

for some positive constant γ. Setting Eq. (30) into Eq. (25), we obtain

rðtÞ AðtÞ � −z′ ðtÞ �α ≥ rðt2Þ Aðt2Þ

<sup>A</sup>ðt<sup>Þ</sup> <sup>≤</sup>

$$
\mathcal{U}\mathcal{U}(t) + \frac{p(t)}{r(t)}\mathcal{U}(t) \ni \tilde{\mathcal{V}}\tilde{\mathcal{Q}}(t)\mathcal{R}^{\emptyset}(\sigma(t)),\tag{34}
$$

or, equally

$$\left(\frac{\mathcal{U}(t)}{A(t)}\right)' \succeq \tilde{\mathcal{V}} \frac{\tilde{\mathcal{Q}}(t) R^{\beta}(\sigma(t))}{A(t)}.\tag{35}$$

Integrating the above inequality from t<sup>2</sup> to t, we get

$$\left[\boldsymbol{U}(t)\succeq\widecheck{\gamma}\boldsymbol{A}(t)\right]\_{t\_2}^{t} \frac{\tilde{\boldsymbol{Q}}(\boldsymbol{s})\boldsymbol{R}^{\boldsymbol{\beta}}(\boldsymbol{\sigma}(\boldsymbol{s}))}{\boldsymbol{A}(\boldsymbol{s})}\operatorname{d\boldsymbol{s}}\tag{36}$$

or

$$\left[r(t)\left(-\stackrel{\cdot}{z'}(t)\right)^{\alpha}\succeq\tilde{\gamma}\!/A(t)\right]\_{t\_2}^{t}\frac{\tilde{Q}(s)\mathcal{R}^{\beta}(\sigma(s))}{A(s)}\,\mathrm{d}s.\tag{37}$$

It follows from this last inequality that

$$0 < z(t) \Longleftarrow z(t\_2) - \tilde{\boldsymbol{\gamma}} \int\_{t\_2}^{t} \left( \frac{A(\boldsymbol{u})}{r(\boldsymbol{u})} \int\_{t\_2}^{u} \frac{\tilde{Q}(\boldsymbol{s}) R^{\boldsymbol{\delta}}(\boldsymbol{\sigma}(\boldsymbol{s}))}{A(\boldsymbol{s})} \, \mathrm{d}s \right)^{\frac{1}{\boldsymbol{\sigma}}} \, \mathrm{d}u \tag{38}$$

for t≥t2: As t ! ∞, then by condition Eq. (21), zðtÞ approaches to −∞, which contradicts the fact that <sup>z</sup>ðt<sup>Þ</sup> is eventually positive. Therefore, <sup>z</sup>′ ðtÞ > 0 and from Eq. (1), we have � rðtÞ � z′ ðtÞ �<sup>α</sup>�′ < 0. The proof is complete.

Lemma 3. Assume that

$$\int\_{t\_0}^{\infty} \left( \frac{A(u)}{r(u)} \int\_{t\_0}^{u} \frac{\tilde{Q}(s)}{A(s)} \mathrm{d}s \right)^{\frac{1}{\pi}} \mathrm{d}u = \text{\textquotedblleft}, \tag{39}$$

holds and Eq. (1) has a positive solution xðtÞ on ℐ. Then there exists T∈ℐ, sufficiently large, such that either

$$z(t) > 0, \quad \dot{z'}(t) > 0, \quad \left(r(t)\left(\dot{z'}(t)\right)^\alpha\right)' < 0,\tag{40}$$

on <sup>½</sup>T, <sup>∞</sup><sup>Þ</sup> or lim<sup>t</sup>!<sup>∞</sup> <sup>x</sup>ðtÞ ¼ 0.

Proof. As in the proof of Lemma 1, we assume that there exists t2∈ℐ such that z′ ðtÞ < 0 on ½t2, ∞Þ. So, zðtÞ is decreasing and

$$\lim\_{t \to \infty} z(t) =: b \ge 0 \tag{41}$$

exists. Therefore, there exists t3∈½t2, ∞Þ such that

$$z(\sigma(t)) > z(t) \exists b > 0. \tag{42}$$

As in the proof of Lemma 2, we obtain Eq. (27), i.e.,

$$\begin{array}{ll} \text{x}(\sigma(t)) & \ni \left( 1 \neg a(\sigma(t)) \frac{R(\tau(\sigma(t)))}{R(\sigma(t))} \right) \text{z}(\sigma(t)) \\ & \ni b \left( 1 \neg a(\sigma(t)) \frac{R(\tau(\sigma(t)))}{R(\sigma(t))} \right), \quad \text{for} \quad t \sharp t\_3. \end{array} \tag{43}$$

Thus,

$$\left(r(t)\left(\dot{z}'(t)\right)^a\right)' + p(t)\left(\dot{z'}(t)\right)^a \le -\tilde{b}\cdot q(t)\left(1 - a(\sigma(t))\frac{R(\tau(\sigma(t)))}{R(\sigma(t))}\right)^{\tilde{\beta}}$$

$$= -\tilde{b}\cdot \tilde{Q}(t),\tag{44}$$

where <sup>~</sup><sup>b</sup> :<sup>¼</sup> kb<sup>β</sup> .

Define the function UðtÞ as in Eq. (103). Then Eq. (44) becomes

$$
\left(\frac{\mathcal{U}(t)}{A(t)}\right)' \cong \tilde{b} \frac{\tilde{Q}(t)}{A(t)}.\tag{45}
$$

Integrating the above inequality twice from t<sup>3</sup> to t, one gets

$$0 < z(t) \lhd z(t\_3) \lhd \int\_{t\_3}^{t} \left(\frac{A(u)}{r(u)} \int\_{t\_3}^{u} \frac{\tilde{Q}(s)}{A(s)} \mathrm{d}s\right)^{\frac{1}{\pi}} \mathrm{d}u,\tag{46}$$

for t≥t3: As t ! ∞, then by condition (39), zðtÞ approaches to −∞, which contradicts the fact that <sup>z</sup>ðt<sup>Þ</sup> is eventually positive. Thus, <sup>b</sup> <sup>¼</sup> 0 and from 0≤xðtÞ≤zðtÞ, we see that lim<sup>t</sup>!<sup>∞</sup> <sup>x</sup>ðtÞ ¼ 0. The proof is complete.

Using results of Lemmas 1 and 2, we can obtain the following oscillation criteria for Eq. (1).

Theorem 1. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. Furthermore, assume that there exists a positive continuously differentiable function φðtÞ such that, for all sufficiently large, T, T1≥T,

Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument http://dx.doi.org/10.5772/65909 39

$$P(t) \succeq 0 \tag{47}$$

$$\begin{aligned} \text{on } [T, \infty) \text{ and} \\ \lim\_{t \to \infty} & \left\{ \phi(t) A(t) \int\_{t}^{\infty} \frac{Q(s)}{A(s)} ds \; + \int\_{T\_1}^{t} \left[ \phi(s) Q(s) - \frac{a^{a}}{(a+1)^{a+1}} \frac{\phi(s) r(\sigma(s)) \left( P(s) \right)^{a+1}}{\left( \beta \sigma'(s) \psi(s) \right)^{a}} \right] ds \right\} = \text{o}, \end{aligned} \tag{48}$$

where

<sup>z</sup>ðt<sup>Þ</sup> <sup>&</sup>gt; <sup>0</sup>; <sup>z</sup>′

on <sup>½</sup>T, <sup>∞</sup><sup>Þ</sup> or lim<sup>t</sup>!<sup>∞</sup> <sup>x</sup>ðtÞ ¼ 0.

Thus,

where <sup>~</sup><sup>b</sup> :<sup>¼</sup> kb<sup>β</sup>

proof is complete.

sufficiently large, T, T1≥T,

� rðtÞ � z′ ðtÞ �<sup>α</sup>�′

½t2, ∞Þ. So, zðtÞ is decreasing and

þ pðtÞ � z′ ðtÞ �α

.

exists. Therefore, there exists t3∈½t2, ∞Þ such that

38 Dynamical Systems - Analytical and Computational Techniques

As in the proof of Lemma 2, we obtain Eq. (27), i.e.,

Define the function UðtÞ as in Eq. (103). Then Eq. (44) becomes

Integrating the above inequality twice from t<sup>3</sup> to t, one gets

<sup>0</sup> <sup>&</sup>lt; <sup>z</sup>ðtÞ≤zðt3Þ−~<sup>b</sup>

ðtÞ > 0;

Proof. As in the proof of Lemma 1, we assume that there exists t2∈ℐ such that z′

<sup>x</sup>ðσðtÞÞ <sup>≥</sup> <sup>1</sup>−aðσðtÞÞ <sup>R</sup>ðτðσðtÞÞÞ

� rðtÞ � z′ ðtÞ �<sup>α</sup>�′

RðσðtÞÞ

RðσðtÞÞ

� �<sup>β</sup>

<sup>≥</sup>~<sup>b</sup> <sup>Q</sup><sup>~</sup> <sup>ð</sup>t<sup>Þ</sup> AðtÞ

AðuÞ rðuÞ

for t≥t3: As t ! ∞, then by condition (39), zðtÞ approaches to −∞, which contradicts the fact that <sup>z</sup>ðt<sup>Þ</sup> is eventually positive. Thus, <sup>b</sup> <sup>¼</sup> 0 and from 0≤xðtÞ≤zðtÞ, we see that lim<sup>t</sup>!<sup>∞</sup> <sup>x</sup>ðtÞ ¼ 0. The

Using results of Lemmas 1 and 2, we can obtain the following oscillation criteria for Eq. (1).

Theorem 1. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. Furthermore, assume that there exists a positive continuously differentiable function φðtÞ such that, for all

ðu t3

!<sup>1</sup>

<sup>Q</sup><sup>~</sup> <sup>ð</sup>s<sup>Þ</sup> AðsÞ ds

RðσðtÞÞ

� �

� �

≤−~<sup>b</sup> <sup>q</sup>ðt<sup>Þ</sup> <sup>1</sup>−aðσðtÞÞRðτðσðtÞÞÞ

UðtÞ AðtÞ � �′

> ðt t3

<sup>≥</sup><sup>b</sup> <sup>1</sup>−aðσðtÞÞRðτðσðtÞÞÞ

< 0; (40)

lim<sup>t</sup>!<sup>∞</sup> <sup>z</sup>ðtÞ ¼: <sup>b</sup>≥<sup>0</sup> (41)

zðσðtÞÞ > zðtÞ≥b > 0: (42)

, for t≥t3:

<sup>¼</sup> <sup>−</sup>~bQ<sup>~</sup> <sup>ð</sup>tÞ, (44)

α

: (45)

du, (46)

zðσðtÞÞ

ðtÞ < 0 on

(43)

$$\psi(t) = \begin{cases} c\_1, & c\_1 \text{ is some positive constant if } \beta > a \\ 1, & \text{if } \beta = a \\ c\_2 \left( \int\_T^t r^{\frac{1}{a}}(s) ds \right)^{\frac{\beta - a}{a}}, & c\_2 \text{ is some positive constant if } \beta < a. \end{cases} \tag{49}$$

Then, Eq. (1) is oscillatory.

Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the conclusions of Lemma 1 or 2 on ½T, ∞Þ with

$$\mathbf{x}(t) > \mathbf{0}, \quad \mathbf{x}(\pi(t)) > \mathbf{0}, \quad \mathbf{x}(\sigma(t)) > \mathbf{0} \tag{50}$$

on ½T, ∞Þ. In particular, we have

$$z(t) > 0, \quad \dot{z'}(t) > 0, \quad \left(r(t)\left(\dot{z'}(t)\right)^{\alpha}\right)' < 0, \quad \text{for} \quad t \ge T. \tag{51}$$

By Eq. (2) and the assumption ðiiiÞ, we get

$$\begin{aligned} \mathbf{x}(t) &= \mathbf{z}(t) \mathbf{-} a(t) \mathbf{x}(\tau(t)) \\ &\ge \mathbf{z}(t) \mathbf{-} a(t) \mathbf{z}(\tau(t)) \\ &\ge (1 - a(t)) \mathbf{z}(t), \end{aligned} \tag{52}$$

which together with Eq. (1) implies

$$\begin{aligned} \left(r(t)\left(z'(t)\right)^{\alpha}\right)' + \frac{p(t)}{r(t)}\left(z'(t)\right)^{\alpha} &\leq -k\eta(t)\left(1 - a(\sigma(t))\right)^{\beta}z^{\beta}(\sigma(t)) \\ &= -Q(t)z^{\beta}(\sigma(t)). \end{aligned} \tag{53}$$

We consider the generalized Riccati substitution

$$w(t) = \phi(t) \frac{r(t) \left(z'(t)\right)^{\alpha}}{z^{\beta}(\sigma(t))} > 0, \quad \text{for} \quad t \ge T. \tag{54}$$

As in the proof of Lemma 1, we get Eq. (18), which in view of the assumption ðvÞ yields

$$\left(\frac{r(t)}{A(t)}\left(z'(t)\right)^{\alpha}\right) \le -\frac{Q(t)}{A(t)}\varepsilon^{\beta}(\sigma(t)).\tag{55}$$

Integrating Eq. (55) from t to ∞ and using the fact that zðtÞ is increasing, we have

$$\begin{split} \frac{r(t)}{A(t)} \left( z'(t) \right)^{\alpha} &\geq \int\_{t}^{\infty} \frac{Q(s)}{A(s)} z^{\delta}(\sigma(s)) \mathrm{d}s \\ &\geq \underline{\sigma}^{\delta}(\sigma(t)) \int\_{t}^{\infty} \frac{Q(s)}{A(s)} \mathrm{d}s. \end{split} \tag{56}$$

So it follows from Eq. (56) and the definition (54) of wðtÞ that

$$w(t) = \phi(t) \frac{r(t) \left(z'(t)\right)^a}{z^\beta(\sigma(t))} \succeq \phi(t) A(t) \Big|\_t^\infty \frac{Q(s)}{A(s)} \text{d}s.\tag{57}$$

By Eq. (53) we can easily prove that

$$\begin{split} w^{'}(t) &= \left( r(t) \left( z^{'}(t) \right)^{\alpha} \right)^{\alpha} \frac{\phi(t)}{z^{\beta}(\sigma(t))} + \left( \frac{\phi(t)}{z^{\beta}(\sigma(t))} \right)^{\alpha} r(t) \left( z^{'}(t) \right)^{\alpha} \\ &\leq - \frac{\phi(t)}{z^{\beta}(\sigma(t))} \left( p(t) \left( z^{'}(t) \right)^{\beta} + Q(t) z^{\beta}(\sigma(t)) \right) \\ &\quad + r(t) \left( z^{'}(t) \right)^{\alpha} \left( \frac{\phi^{'}(t)}{z^{\beta}(\sigma(t))} - \frac{\phi(t) \left( z^{\beta}(\sigma(t)) \right)}{z^{\beta+1}(\sigma(t))} \right) \\ &\leq - \phi(t) Q(t) + w(t) \left( \frac{\phi^{'}(t)}{\phi(t)} - \frac{p(t)}{r(t)} \right) \\ &\quad - \beta \phi(t) \frac{r(t) \left( z^{\prime}(t) \right)^{\beta} z^{\prime}(\sigma(t)) \sigma^{\prime}(t)}{z^{\beta+1}(\sigma(t))} .\end{split} \tag{58}$$

On the other hand, since rðtÞ � z′ ðtÞ �α is decreasing, we have

$$\frac{z'(\sigma(t))}{z'(t)} \ge \left(\frac{r(t)}{r(\sigma(t))}\right)^{\frac{1}{\sigma}}\tag{59}$$

and thus Eq. (58) becomes

$$\begin{split} \boldsymbol{w}'(t) &\leq \quad -\phi(t)\boldsymbol{Q}(t) + P(t)\boldsymbol{w}(t) \\ &-\frac{\beta\phi(t)\boldsymbol{\sigma}'(t)}{r^{\frac{1}{a}}(\boldsymbol{\sigma}(t))} \left(\frac{\boldsymbol{w}(t)}{\phi(t)}\right)^{\frac{a+1}{a}} \boldsymbol{z}^{\frac{\beta+a}{a}}(\boldsymbol{\sigma}(t)). \end{split} \tag{60}$$

Now, we consider the following three cases:

Case I: β > α.

In this case, since z′ ðtÞ > 0 for t≥T, then there exists T1≥T such that zðσðtÞÞ≥c for t≥T1. This implies that

$$z^{\frac{\beta - a}{a}}(\sigma(t)) \succeq\_{c}^{\frac{\beta - a}{a}} := c\_1 \tag{61}$$

Case II: β ¼ α.

rðtÞ AðtÞ � z′ ðtÞ �<sup>α</sup> � �′

40 Dynamical Systems - Analytical and Computational Techniques

rðtÞ AðtÞ � z′ ðtÞ �α ≥ ð∞ t

So it follows from Eq. (56) and the definition (54) of wðtÞ that

wðtÞ ¼ φðtÞ

� rðtÞ � z′ ðtÞ

≤− <sup>φ</sup>ðt<sup>Þ</sup> z<sup>β</sup>ðσðtÞÞ

> þ rðtÞ � z′ ðtÞ

−βφðtÞ

� z′ ðtÞ �α

w′

Now, we consider the following three cases:

By Eq. (53) we can easily prove that

On the other hand, since rðtÞ

and thus Eq. (58) becomes

Case I: β > α.

w′ ðtÞ ¼

Integrating Eq. (55) from t to ∞ and using the fact that zðtÞ is increasing, we have

rðtÞ � z′ ðtÞ �α

� pðtÞ � z′ ðtÞ �β

≤−φðtÞQðtÞ þ <sup>w</sup>ðt<sup>Þ</sup> <sup>φ</sup>′

<sup>r</sup>ðtÞðz′ ðtÞ β z′ <sup>ð</sup>σðtÞσ′ ðtÞ

> z′ ðσðtÞÞ z′ <sup>ð</sup>t<sup>Þ</sup> <sup>≥</sup>

− βφðtÞσ′ ðtÞ

r 1 <sup>α</sup>ðσðtÞÞ

ðtÞ≤ −φðtÞQðtÞ þ PðtÞwðtÞ

�<sup>α</sup>�′ φðtÞ

�<sup>α</sup> φ′

0 @

ðtÞ <sup>z</sup><sup>β</sup>ðσðtÞÞ <sup>−</sup>

> ðtÞ φðtÞ − pðtÞ rðtÞ

<sup>z</sup><sup>β</sup>þ<sup>1</sup>ðσðtÞÞ :

is decreasing, we have

rðtÞ rðσðtÞÞ � �<sup>1</sup>

> wðtÞ φðtÞ � �<sup>α</sup>þ<sup>1</sup> α z β−α

α

� �

≤− <sup>Q</sup>ðt<sup>Þ</sup> AðtÞ

> QðsÞ AðsÞ

ð∞ t

<sup>≥</sup>z<sup>β</sup>ðσðtÞÞ

<sup>z</sup><sup>β</sup>ðσðtÞÞ <sup>≥</sup>φðtÞAðt<sup>Þ</sup>

<sup>z</sup><sup>β</sup>ðσðtÞÞ <sup>þ</sup> <sup>φ</sup>ðt<sup>Þ</sup>

<sup>þ</sup> <sup>Q</sup>ðtÞz<sup>β</sup>

φðtÞ � <sup>z</sup><sup>β</sup>ðσðtÞÞ �

<sup>z</sup><sup>β</sup>ðσðsÞÞd<sup>s</sup>

QðsÞ AðsÞ ds:

> ð∞ t

z<sup>β</sup>ðσðtÞÞ � �′

> ðσðtÞÞ �

z<sup>β</sup>þ<sup>1</sup>ðσðtÞÞ

QðsÞ AðsÞ

> rðtÞ � z′ ðtÞ �α

> > 1 A

<sup>α</sup> <sup>ð</sup>σðtÞÞ: (60)

<sup>z</sup><sup>β</sup>ðσðtÞÞ: (55)

ds: (57)

(56)

(58)

(59)

In this case, we see that z β−α <sup>α</sup> ðσðtÞÞ ¼ 1:

Case III: β < α.

Since rðtÞ � z′ ðtÞ �α is decreasing, there exists a constant d such that

$$r(t)\left(z'(t)\right)^{\alpha} \not\simeq d\tag{62}$$

for t≥T. Integrating the above inequality from T to t, we have

$$z(t) \lhd z(T) + \int\_{T}^{t} \left(\frac{d}{r(s)}\right)^{\frac{1}{\pi}} \mathrm{ds}.\tag{63}$$

Hence, there exists T1≥T and a constant d<sup>1</sup> depending on d such that

$$z(t) \le d\_1 \int\_T^t r^{\frac{1}{d}}(s) \, \text{ds}, \quad \text{for} \quad t \ge T\_1 \tag{64}$$

and thus

$$\mathbb{E}^{\frac{\beta \cdot a}{a}}(\sigma(t)) \mathbb{E} d\_1^{\frac{\beta \cdot a}{a}} \left( \int\_T^t r^{-\frac{1}{a}}(s) \, \mathrm{ds} \right)^{\frac{\beta \cdot a}{a}} = d\_2 \left( \int\_T^t r^{-\frac{1}{a}}(s) \, \mathrm{ds} \right)^{\frac{\beta \cdot a}{a}} \tag{65}$$

for some positive constant d2.

Using these three cases and the definition of ψðtÞ, we get

$$w'(t) \Leftarrow \phi(t)Q(t) + P(t)w(t) - \frac{\beta \sigma'(t)\psi(t)}{\left(\phi(t)r(\sigma(t))\right)^{\frac{1}{a}}} w^{\frac{1+a}{a}}(t) \tag{66}$$

for t≥T1≥T. Setting

$$A := P(t),\tag{67}$$

$$B := \frac{\beta \sigma'(t)\psi(t)}{\left(\phi(t)r(\sigma(t))\right)^{\frac{1}{\sigma}}},\tag{68}$$

and using the inequality

$$A\mu - Bu^{\frac{1+\alpha}{\alpha}} \le \frac{\alpha^{\alpha}}{\left(\alpha+1\right)^{\alpha+1}} \frac{A^{\alpha+1}}{B^{\alpha}}\,,\tag{69}$$

we obtain

$$w'(t) \preceq \phi(t)Q(t) + \frac{\alpha^a}{\left(\alpha + 1\right)^{a+1}} \frac{\phi(t)r(\sigma(t))\left(P(t)\right)^{a+1}}{\left(\beta \sigma'(t)\psi(t)\right)^a} \,. \tag{70}$$

Integrating the above inequality from T<sup>1</sup> to t, we have

$$w(t) \lhd w(T\_1) - \int\_{T\_1}^t \left(\phi(s)Q(s) - \frac{\alpha^a}{\left(\alpha + 1\right)^{a+1}} \frac{\phi(s)r(\sigma(s))\left(P(s)\right)^{a+1}}{\left(\beta \sigma'(s)\psi(s)\right)^a}\right)ds.\tag{71}$$

Taking Eq. (57) into account, we get

$$\begin{split} &w(T\_1) \quad \succeq \phi(t)A(t) \Big|\_{t}^{\alpha} \frac{Q(s)}{A(s)} \mathrm{d}s \\ &\quad + \int\_{T\_1}^{t} \left( \phi(s)Q(s) - \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}} \frac{\phi(s)r(\sigma(s)) \left(P(s)\right)^{\alpha+1}}{\left(\beta\sigma'(s)\psi(s)\right)^{\alpha}} \right) \mathrm{d}s. \end{split} \tag{72}$$

Taking the lim sup on both sides of the above inequality as t ! ∞, we obtain a contradiction to the condition (48). This completes the proof.

Remark 2. Note that the presence of the term φðtÞAðtÞ ð∞ t QðsÞ <sup>A</sup>ðs<sup>Þ</sup> <sup>d</sup><sup>s</sup> in Eq. (57) improves a number of related results in, e.g., [9, 13–18, 20].

Setting φðtÞ ¼ t in Eq. (57), then the following corollary becomes immediate.

Corollary 1. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. Assume that, for all sufficiently large, T, T1≥T,

$$tp(t) \le r(t) \tag{73}$$

on ½T, ∞Þ and

$$\limsup\_{t \to \infty} \left\{ tA(t) \Big|\_{t}^{\infty} \frac{\mathcal{Q}(s)}{A(s)} \mathrm{d}s \right. \\ \left. + \int\_{T\_1}^{t} \left[ s\mathcal{Q}(s) - \frac{a^a}{(a+1)^{a+1}} \frac{sr(\sigma(s)) \left( \frac{1}{s} - \frac{p(s)}{r(s)} \right)^{a+1}}{\left(\beta \sigma'(s)\psi(s)\right)^a} \right] \mathrm{d}s \right\} = \infty, \tag{74}$$

where ψðtÞ is as in Theorem 1. Then Eq. (1) is oscillatory.

Corollary 2. Assume that the conditions (39) and (74) hold. Then Eq. (1) is oscillatory or lim<sup>t</sup>!<sup>∞</sup> <sup>x</sup>ðtÞ ¼ 0.

Next, we present some complementary oscillation results for Eq. (1) by using an integral averaging technique due to Philos. We need the class of functions F. Let

$$\mathcal{D}\_0 = \{(t, s) : t > s \sharp t\_0\} \quad \text{and} \quad \mathcal{D} = \{(t, s) : t > s \sharp t\_0\} \tag{75}$$

The function Hðt, sÞ∈CðD, ℝÞ is said to belong to a class F if

(a) Hðt, tÞ ¼ 0 for t≥T, Hðt, sÞ > 0 for ðt, sÞ∈D<sup>0</sup>

(b) Hðt, sÞ has a continuous and nonpositive partial derivative on D<sup>0</sup> with respect to the second variable such that

$$\frac{\partial}{\partial s}\left(H(t,s)\phi(s)\right) - H(t,s)\frac{\phi(s)p(s)}{r(s)} = -h(t,s)\left(H(t,s)\phi(s)\right)^{\frac{d}{s+1}}\tag{76}$$

for all ðt, sÞ∈D0.

Au−Bu<sup>1</sup>þ<sup>α</sup> <sup>α</sup> ≤

<sup>ð</sup>tÞ≤−φðtÞQðtÞ þ <sup>α</sup><sup>α</sup>

ð∞ t

QðsÞ AðsÞ ds

<sup>φ</sup>ðsÞQðsÞ<sup>−</sup> <sup>α</sup><sup>α</sup>

Setting φðtÞ ¼ t in Eq. (57), then the following corollary becomes immediate.

ðα þ 1Þ

Taking the lim sup on both sides of the above inequality as t ! ∞, we obtain a contradiction to

Corollary 1. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. Assume that, for

sQðsÞ<sup>−</sup> <sup>α</sup><sup>α</sup>

ðα þ 1Þ

αþ1

srðσðsÞÞ <sup>1</sup>

� βσ′

αþ1

ðα þ 1Þ

<sup>φ</sup>ðsÞQðsÞ<sup>−</sup> <sup>α</sup><sup>α</sup>

ðα þ 1Þ

αþ1

w′

42 Dynamical Systems - Analytical and Computational Techniques

wðtÞ≤wðT1Þ−

Taking Eq. (57) into account, we get

Integrating the above inequality from T<sup>1</sup> to t, we have

wðT1Þ ≥φðtÞAðtÞ

þ ðt T1

Remark 2. Note that the presence of the term φðtÞAðtÞ

QðsÞ AðsÞ

ds þ

where ψðtÞ is as in Theorem 1. Then Eq. (1) is oscillatory.

ðt T1 2 6 4

the condition (48). This completes the proof.

of related results in, e.g., [9, 13–18, 20].

all sufficiently large, T, T1≥T,

( tAðtÞ ð∞ t

on ½T, ∞Þ and

lim sup t!∞

0 B@

ðt T1 0 B@

we obtain

αα ðα þ 1Þ

αþ1

αþ1

A<sup>α</sup>þ<sup>1</sup>

φðtÞrðσðtÞÞ

� βσ′ � PðtÞ �<sup>α</sup>þ<sup>1</sup>

ðtÞψðtÞ

� PðsÞ �<sup>α</sup>þ<sup>1</sup>

ðsÞψðsÞ

� PðsÞ �<sup>α</sup>þ<sup>1</sup>

ðsÞψðsÞ

�α

tpðtÞ≤rðtÞ (73)

<sup>s</sup> <sup>−</sup> <sup>p</sup>ðs<sup>Þ</sup> rðsÞ � �<sup>α</sup>þ<sup>1</sup>

�α

)

¼ ∞, (74)

ðsÞψðsÞ

�α

φðsÞrðσðsÞÞ

� βσ′

φðsÞrðσðsÞÞ

� βσ′

ð∞ t QðsÞ

<sup>B</sup><sup>α</sup> , (69)

1 CA

1 CA ds:

<sup>A</sup>ðs<sup>Þ</sup> <sup>d</sup><sup>s</sup> in Eq. (57) improves a number

�<sup>α</sup> : (70)

ds: (71)

(72)

Theorem 2. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. Furthermore, assume that there exist functions Hðt, sÞ, hðt, sÞ∈F such that, for all sufficiently large, T, for T1≥T,

$$\begin{split} \limsup\_{t \to \infty} \frac{1}{H(t, T\_1)} \Big|\_{T\_1}^t & \quad \left( H(t, s) \Big( \phi(s) Q(s) + \rho(s) \phi(s) p(s) \Big) \right. \\ & \quad - \frac{\alpha^a}{(\alpha + 1)^{a+1}} \frac{h^{a+1}(t, s) r(\sigma(s))}{\theta^a(\sigma(s) \psi(s))^a} \text{ds} = \text{as} \end{split} \tag{77}$$

where φðtÞ and ρðtÞ are continuously differentiable functions and ψðtÞ is as in Theorem 1. Then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the conclusions of Lemma 1 or 2 on ½T, ∞Þ with

$$\mathbf{x}(t) > \mathbf{0}, \quad \mathbf{x}(\tau(t)) > \mathbf{0}, \quad \mathbf{x}(\sigma(t)) > \mathbf{0} \tag{78}$$

on ½T, ∞Þ. In particular, we have

$$z(t) > 0, \quad z'(t) > 0, \quad \left(r(t)\left(z'(t)\right)^a\right)' < 0, \quad \text{for} \quad t \ge T. \tag{79}$$

Define the function wðtÞ as

$$w(t) = \phi(t)r(t)\left(\frac{\left(z'(t)\right)^a}{\mathcal{D}^\delta(\sigma(t))} + \rho(t)\right) \rhd\phi(t)r(t)\rho(t),\tag{80}$$

where <sup>ρ</sup>ðtÞ∈C<sup>1</sup> ðℐ, ℝÞ. Similarly to the proof of Theorem 1, we obtain the inequality

$$\begin{split} \boldsymbol{w}^{\prime}(t) \leq & \quad \neg \phi(t) \boldsymbol{Q}(t) + \phi(t) \left( r(t) \rho(t) \right)^{\prime} + \left( \frac{\phi^{\prime}(t)}{\phi(t)} \text{-} \frac{p(t)}{r(t)} \right) \boldsymbol{w}(t) \\ & \quad \cdot \frac{\beta \boldsymbol{\sigma}^{\prime}(t) \boldsymbol{\psi}(t)}{\left( \phi(t) r(\boldsymbol{\sigma}(t)) \right)^{\frac{1}{\pi}}} \left( \boldsymbol{w}(t) \boldsymbol{-\phi}(t) r(t) \rho(t) \right)^{\frac{1+a}{a}} . \end{split} \tag{81}$$

Multiplying Eq. (81) by Hðt, sÞ, integrating with respect to s from T<sup>1</sup> to t for t≥T1≥T, and using ðaÞ and ðbÞ, we find that

ðt T1 Hðt, sÞφðsÞ � QðsÞ− � rðsÞρðsÞ ��′ ds ≤−ðt T <sup>H</sup>ðt, <sup>s</sup>Þw′ ðsÞds þ ðt T1 <sup>H</sup>ðt, <sup>s</sup><sup>Þ</sup> <sup>φ</sup>′ ðsÞ φðsÞ − pðsÞ rðsÞ � �wðsÞd<sup>s</sup> − ðt T1 <sup>β</sup>Hðt, <sup>s</sup>Þσ′ ðsÞψðsÞ � <sup>φ</sup>ðsÞrðσðsÞÞ�<sup>1</sup> α � wðsÞ−φðsÞrðsÞρðsÞ �<sup>1</sup>þ<sup>α</sup> α ds ¼ Hðt, sÞwðsÞ T1t <sup>j</sup> þ ðt T1 ∂ ∂s <sup>H</sup>ðt, <sup>s</sup>Þ þ <sup>H</sup>ðt, <sup>s</sup><sup>Þ</sup> <sup>φ</sup>′ ðsÞ φðsÞ − pðsÞ rðsÞ � � � � <sup>w</sup>ðsÞd<sup>s</sup> − ðt T1 <sup>β</sup>Hðt, <sup>s</sup>Þσ′ ðsÞψðsÞ � <sup>φ</sup>ðsÞrðσðsÞÞ�<sup>1</sup> α � wðsÞ−φðsÞrðsÞρðsÞ �<sup>1</sup>þ<sup>α</sup> α ds <sup>¼</sup> <sup>H</sup>ðt, <sup>T</sup>1ÞwðT1Þ þ <sup>ð</sup><sup>t</sup> T1 − hðt, sÞ φðsÞ � Hðt, sÞφðsÞ � <sup>α</sup> αþ1 wðsÞds − ðt T1 <sup>β</sup>Hðt, <sup>s</sup>Þσ′ ðsÞψðsÞ � <sup>φ</sup>ðsÞrðσðsÞÞ�<sup>1</sup> α � wðsÞ−φðsÞrðsÞρðsÞ �<sup>1</sup>þ<sup>α</sup> α ds (82)

Setting

$$A := -\frac{h(t,s)}{\phi(s)} \left[ H(t,s)\phi(s) \right]^{\frac{a}{a+1}}, \quad B := \frac{\beta H(t,s)\sigma'(s)\psi(s)}{\left(\phi(s)r(\sigma(s))\right)^{\frac{1}{a}}} \tag{83}$$

and

$$\mathcal{C} := \phi(s)r(s)\rho(s) \tag{84}$$

and using the inequality

$$A\mu \text{--} B(\mu \text{--} \mathbb{C})^{\frac{1 \pm \omega}{\alpha}} \triangle A \mathbb{C} + \frac{\alpha^{\alpha}}{(\alpha + 1)^{\alpha + 1}} \frac{A^{\alpha + 1}}{B^{\alpha}},\tag{85}$$

we obtain

$$\begin{split} & \int\_{T\_1}^t H(t,s)\phi(s) \left( Q(s) - \left( r(s)\rho(s) \right)^\prime \right) \mathrm{d}s \\ & \leq \!\!H(t,T\_1)w(T\_1) + \int\_{T\_1}^t \!\!-h(t,s)r(s)\rho(s)[H(t,s)\phi(s)]^\prime \right\prime \mathrm{d}s \\ & + \int\_{T\_1}^t \frac{\alpha^\alpha}{\left(\alpha+1\right)^{\alpha+1}} \frac{h^{\alpha+1}(t,s)r(\sigma(s))}{\beta^\alpha \left(\sigma'(s)\psi(s)\right)^\alpha} \mathrm{d}s \end{split} \tag{86}$$

Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument http://dx.doi.org/10.5772/65909 45

Thus,

w′

44 Dynamical Systems - Analytical and Computational Techniques

ðaÞ and ðbÞ, we find that

ðt T1

≤− ðt T

− ðt T1

− ðt T1

− ðt T1

and using the inequality

Setting

and

we obtain

Hðt, sÞφðsÞ

<sup>H</sup>ðt, <sup>s</sup>Þw′

<sup>β</sup>Hðt, <sup>s</sup>Þσ′

<sup>β</sup>Hðt, <sup>s</sup>Þσ′

¼ Hðt, T1ÞwðT1Þ þ

<sup>β</sup>Hðt, <sup>s</sup>Þσ′

A :¼ −

ðt T1

þ ðt T1

φðsÞrðσðsÞÞ

φðsÞrðσðsÞÞ

φðsÞrðσðsÞÞ

T1t <sup>j</sup> þ ðt T1

�

¼ Hðt, sÞwðsÞ

�

�

ðtÞ≤ −φðtÞQðtÞ þ φðtÞ

ðtÞψðtÞ

φðtÞrðσðtÞÞ

<sup>−</sup> βσ′

�

� QðsÞ− � rðsÞρðsÞ

ðsÞds þ

ðsÞψðsÞ

ðsÞψðsÞ

ðsÞψðsÞ

hðt, sÞ

�1 α �

Au−Bðu−CÞ

� QðsÞ− � rðsÞρðsÞ

αþ1

Hðt, sÞφðsÞ

≤Hðt, T1ÞwðT1Þ þ

αα ðα þ 1Þ

<sup>φ</sup>ðs<sup>Þ</sup> <sup>½</sup>Hðt, <sup>s</sup>ÞφðsÞ� <sup>α</sup>

1þα <sup>α</sup> ≤AC þ

ðt T1

h<sup>α</sup>þ<sup>1</sup>

βα � σ′ ðsÞψðsÞ

�1 α �

> ðt T1 − hðt, sÞ φðsÞ

�1 α �

ðt T1

> ∂ ∂s

� rðtÞρðtÞ �′ <sup>þ</sup> <sup>φ</sup>′ ðtÞ φðtÞ − pðtÞ rðtÞ

Multiplying Eq. (81) by Hðt, sÞ, integrating with respect to s from T<sup>1</sup> to t for t≥T1≥T, and using

��′ ds

wðsÞ−φðsÞrðsÞρðsÞ

wðsÞ−φðsÞrðsÞρðsÞ

�

wðsÞ−φðsÞrðsÞρðsÞ

ðsÞ φðsÞ − pðsÞ rðsÞ

<sup>H</sup>ðt, <sup>s</sup>Þ þ <sup>H</sup>ðt, <sup>s</sup><sup>Þ</sup> <sup>φ</sup>′

Hðt, sÞφðsÞ

� �

�<sup>1</sup>þ<sup>α</sup> α ds

�<sup>1</sup>þ<sup>α</sup> α ds

> � <sup>α</sup> αþ1 wðsÞds

�<sup>1</sup>þ<sup>α</sup> α ds

<sup>α</sup>þ<sup>1</sup>, <sup>B</sup> :<sup>¼</sup> <sup>β</sup>Hðt, <sup>s</sup>Þσ′

αα ðα þ 1Þ

> �′ � ds

�<sup>α</sup> <sup>d</sup><sup>s</sup>

ðt, sÞrðσðsÞÞ

αþ1

<sup>−</sup>hðt, <sup>s</sup>ÞrðsÞρðsÞ½Hðt, <sup>s</sup>ÞφðsÞ� <sup>α</sup>

A<sup>α</sup>þ<sup>1</sup>

�

� � � �

<sup>H</sup>ðt, <sup>s</sup><sup>Þ</sup> <sup>φ</sup>′

wðtÞ−φðtÞrðtÞρðtÞ

�1 α � � �

�<sup>1</sup>þ<sup>α</sup> α

wðsÞds

ðsÞ φðsÞ − pðsÞ rðsÞ wðtÞ

wðsÞds

ðsÞψðsÞ

�1 α

<sup>α</sup>þ1ds

<sup>B</sup><sup>α</sup> , (85)

φðsÞrðσðsÞÞ

C :¼ φðsÞrðsÞρðsÞ (84)

(82)

(83)

(86)

: (81)

$$\begin{split} H(t,T\_1)\nu(T\_1) &\quad \geq \int\_{T\_1}^t H(t,s)\phi(s) \left( Q(s) - \left( r(s)\rho(s) \right)' \right) \mathrm{d}s \\ &\quad + \int\_{T\_1}^t h(t,s)r(s)\rho(s)[H(t,s)\phi(s)]^{\frac{\alpha}{\alpha+1}} \mathrm{d}s \\ &\quad - \int\_{T\_1}^t \frac{\alpha^{\alpha}}{\left(\alpha+1\right)^{\alpha+1}} \frac{h^{\alpha+1}(t,s)r(\sigma(s))}{\beta^{\alpha}\left(\sigma'(s)\psi(s)\right)^{\alpha}} \mathrm{d}s. \end{split} \tag{87}$$

That is,

$$\begin{split} &H(t,T\_{1})w(T\_{1}) \\ &\geq \int\_{T\_{1}}^{t} H(t,s)\phi(s) \left(Q(s) - \left(r(s)\rho(s)\right)^{\prime}\right) \mathrm{d}s \\ &+ \int\_{T\_{1}}^{t} -r(s)\rho(s) \left(\frac{\partial}{\partial s}\left(H(t,s)\phi(s)\right) - H(t,s)\frac{\phi(s)p(s)}{r(s)}\right) \mathrm{d}s \\ &- \int\_{T\_{1}}^{t} \frac{a^{\alpha}}{\left(a+1\right)^{\alpha+1}} \frac{h^{\alpha+1}(t,s)r(\sigma(s))}{\beta^{\alpha}\left(\sigma(s)\psi(s)\right)^{\alpha}} \mathrm{d}s \\ &= \int\_{T\_{1}}^{t} H(t,s)\left(\phi(s)Q(s) + \rho(s)\phi(s)p(s)\right) \mathrm{d}s \\ &- H(t,s)\phi(s)r(s)\rho(s)^{T\_{1}t} - \int\_{T\_{1}}^{t} \frac{a^{\alpha}}{\left(\alpha+1\right)^{\alpha+1}} \frac{h^{\alpha+1}(t,s)r(\sigma(s))}{\beta^{\alpha}\left(\sigma(s)\psi(s)\right)^{\alpha}} \mathrm{d}s \end{split} \tag{88}$$

It follows that

$$\begin{split} & \quad \int\_{T\_1}^t H(t,s) \Big( \phi(s)Q(s) + \rho(s)\phi(s)p(s) \Big) \mathrm{d}s \\ & - \quad \int\_{T\_1}^t \frac{a^\alpha}{(\alpha+1)^{\alpha+1}} \frac{h^{\alpha+1}(t,s)r(\sigma(s))}{\beta^\alpha \left(\sigma'(s)\psi(s)\right)^\alpha} \mathrm{d}s \\ & \leq \mathcal{H}(t,T\_1) \Big( w(T\_1) \neg \phi(T\_1) r(T\_1) \rho(T\_1) \Big), \end{split} \tag{89}$$

which is a contradiction to Eq. (77). The proof is complete.

Remark 3. Authors in [15, 20] studied a partial case of Eq. (1) by employing the generalized Riccati substitution (80). Note that the function ρðtÞ used in the generalized Riccati substitution (80) finally becomes unimportant. Thus, we can put ρðtÞ ¼ 0 and obtain similar results to those from [15, 20].

In the next part, we provide several oscillation results for Eq. (1) under the assumption that the function PðtÞ is nonpositive. These results generalize those from [10] for Eq. (1) in such sense that α≠β and pðtÞ≠0.

Theorem 3. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. Furthermore, assume that there exists a continuously differentiable function φðtÞ such that, for all sufficiently large, T, T1≥T,

$$P(t) \le 0\tag{90}$$

on ½T, ∞Þ and

$$\limsup\_{t \to \infty} \left[ \phi(t) A(t) \right]\_t^\infty \frac{Q(s)}{A(s)} \mathrm{d}s + \int\_{T\_1}^t \phi(s) \left( Q(s) - A(s) P(s) \right)\_s^\infty \frac{Q(u)}{A(u)} \mathrm{d}u \right] \mathrm{d}s = \mathrm{\(\sigma\)}$$

Then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the conclusions of Lemma 1 or 2 on ½T, ∞Þ with

$$\mathbf{x}(t) > \mathbf{0}, \quad \mathbf{x}(\pi(t)) > \mathbf{0}, \quad \mathbf{x}(\sigma(t)) > \mathbf{0} \tag{92}$$

on ½T, ∞Þ. In particular, we have

$$z(t) > 0, \quad z'(t) > 0, \quad \left(r(t)\left(z'(t)\right)^a\right)' < 0, \quad \text{for} \quad t \ge T. \tag{93}$$

Proceeding as in the proof of Theorem 1, we obtain the inequality (66), i.e.,

$$w'(t) \le -\phi(t)Q(t) + P(t)w(t) - \frac{\beta \sigma'(t)\psi(t)}{\left(\phi(t)r(\sigma(t))\right)^{\frac{1}{\sigma}}} w^{\frac{1+\omega}{\sigma}}(t) \tag{94}$$

for t≥T1≥T. Using Eq. (90), and setting Eq. (57) in Eq. (94), we get

$$\begin{split} \boldsymbol{w}^{\prime}(t) &\quad \mathop{\rm s-\phi}\limits\_{t} (t) \boldsymbol{Q}(t) + \phi(t) \boldsymbol{A}(t) \boldsymbol{P}(t) \int\_{t}^{\infty} \frac{\boldsymbol{Q}(s)}{\boldsymbol{A}(s)} \, \mathrm{d}s \\ &\quad - \frac{\beta \boldsymbol{\sigma}^{\prime}(t) \boldsymbol{\psi}(t)}{\left(\phi(t) \boldsymbol{r}(\boldsymbol{\sigma}(t))\right)^{\frac{1}{\alpha}}} \boldsymbol{w}^{\frac{1+\alpha}{\alpha}}(t) \\ &\leq -\phi(t) \boldsymbol{Q}(t) + \phi(t) \boldsymbol{A}(t) \boldsymbol{P}(t) \int\_{t}^{\infty} \frac{\boldsymbol{Q}(s)}{\boldsymbol{A}(s)} \, \mathrm{d}s, \end{split} \tag{95}$$

that is,

$$\int w'(t) + \phi(t)Q(t) - \phi(t)A(t)P(t) \int\_{t}^{\infty} \frac{Q(s)}{A(s)} \, \text{ds} \lesssim 0. \tag{96}$$

Integrating the above inequality from T<sup>1</sup> to t, we have

$$\begin{split} \left\| w(T\_1) \quad \succeq w(t) + \int\_{T\_1}^{t} \left( \phi(s) Q(s) - \phi(s) A(s) P(s) \right)^{\alpha}\_s \frac{Q(u)}{A(u)} \mathrm{d}u \right) \mathrm{d}s \\ \left\| \phi(t) A(t) \right\|\_{t}^{\alpha} \frac{Q(s)}{A(s)} \mathrm{d}s + \int\_{T\_1}^{t} \left( \phi(s) Q(s) - \phi(s) A(s) P(s) \right)^{\alpha}\_s \frac{Q(u)}{A(u)} \mathrm{d}u \right) \mathrm{d}s \end{split} \tag{97}$$

Taking the lim sup on both sides of the above inequality as t ! ∞, we obtain a contradiction to condition Eq. (91). This completes the proof.

Setting φðtÞ ¼ 1, we have the following consequence.

Corollary 3. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. Assume that

$$\limsup\_{t \to \infty} \left[ A(t) \right]\_t^\infty \frac{Q(s)}{A(s)} \mathrm{d}s + \int\_{T\_1}^t \tilde{\eta}(s) \mathrm{d}s \right] = \infty,\tag{98}$$

for all sufficiently large T, for T1≥T. Then Eq. (1) is oscillatory.

Define a sequence of functions {ynðtÞ} ∞ <sup>n</sup>¼<sup>0</sup> as

$$y\_0(t) = \int\_t^\infty \check{q}(s) \, \mathrm{ds}, \qquad t \ge T \tag{99}$$

$$y\_n(t) = \int\_t^{\infty} \frac{\beta \sigma'(s)\psi(s)}{r^\sharp(\sigma(s))} \left( y\_{n-1}(s) \right)^{\frac{1+a}{a}} ds + y\_0(t), \quad t \ge T, \quad n = 1, 2, 3, \dots, \tag{100}$$

for T≥t<sup>0</sup> sufficiently large.

By induction, we can see that yn≤ynþ<sup>1</sup>, <sup>n</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup>;….

Lemma 4. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. Assume that xðtÞ is a positive solution of Eq. (1) on ℐ. Then there exists T∈ℐ, sufficiently large, such that

$$w(t) \succeq y\_n(t),\tag{101}$$

where wðtÞ and ynðtÞ are defined as Eqs. (54) and (100), respectively. Furthermore, there exists a positive function yðtÞ on ½T1, ∞Þ, T1≥T, such that

$$\lim\_{n \to \infty} y\_n(t) = y(t) \tag{102}$$

and

PðtÞ≤0 (90)

QðuÞ AðuÞ du

ds

< 0; for t≥T: (93)

<sup>α</sup> <sup>ð</sup>t<sup>Þ</sup> (94)

ds≤0: (96)

ds

(95)

(97)

¼ ∞: (91)

ð∞ s

� �

xðtÞ > 0; xðτðtÞÞ > 0; xðσðtÞÞ > 0 (92)

ðtÞψðtÞ

�1 α w<sup>1</sup>þ<sup>α</sup>

φðtÞrðσðtÞÞ

ð∞ t

ð∞ t

ð∞ t

ð∞ s

φðsÞQðsÞ−φðsÞAðsÞPðsÞ

QðsÞ AðsÞ

QðuÞ AðuÞ du

� �

ds

ð∞ s

QðuÞ AðuÞ du

QðsÞ AðsÞ ds

QðsÞ AðsÞ ds,

�

on ½T, ∞Þ and

that is,

lim sup t!∞

Then Eq. (1) is oscillatory.

on ½T, ∞Þ. In particular, we have

φðtÞAðtÞ

46 Dynamical Systems - Analytical and Computational Techniques

conclusions of Lemma 1 or 2 on ½T, ∞Þ with

w′

<sup>z</sup>ðt<sup>Þ</sup> <sup>&</sup>gt; <sup>0</sup>; <sup>z</sup>′

w′

w′

Integrating the above inequality from T<sup>1</sup> to t, we have

≥φðtÞAðtÞ

condition Eq. (91). This completes the proof.

ðt T1

> ð∞ t

QðsÞ AðsÞ ds þ ðt T1

wðT1Þ ≥wðtÞ þ

ð∞ t

QðsÞ AðsÞ ds þ ðt T1

ðtÞ > 0;

Proceeding as in the proof of Theorem 1, we obtain the inequality (66), i.e.,

for t≥T1≥T. Using Eq. (90), and setting Eq. (57) in Eq. (94), we get

<sup>ð</sup>tÞ≤ −φðtÞQðtÞ þ <sup>P</sup>ðtÞwðtÞ<sup>−</sup> βσ′

ðtÞ ≤−φðtÞQðtÞ þ φðtÞAðtÞPðtÞ

φðtÞrðσðtÞÞ

ðtÞ þ φðtÞQðtÞ−φðtÞAðtÞPðtÞ

φðsÞQðsÞ−φðsÞAðsÞPðsÞ

� �

Taking the lim sup on both sides of the above inequality as t ! ∞, we obtain a contradiction to

ðtÞψðtÞ

≤−φðtÞQðtÞ þ φðtÞAðtÞPðtÞ

�1 α w<sup>1</sup>þ<sup>α</sup> <sup>α</sup> ðtÞ

<sup>−</sup> βσ′

�

� rðtÞ � z′ ðtÞ �<sup>α</sup>�′

φðsÞ QðsÞ−AðsÞPðsÞ

� �

Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the

$$y(t) = \int\_{t}^{\infty} \frac{\beta \sigma'(s)\psi(s)}{r^{\frac{1}{a}}(\sigma(s))} \left( y(s) \right)^{\frac{1+a}{a}} ds + y\_0(t). \tag{103}$$

Proof. Similarly to the proof of Theorem 3, we obtain Eq. (95). Setting φðtÞ ¼ 1 in Eq. (95), we get

$$\left(w'(t) + Q(t) + \frac{p(t)A(t)}{r(t)}\int\_{t}^{\infty} \frac{Q(s)}{A(s)} ds + \frac{\beta \sigma'(t)\psi(t)}{r\_{\pi}^{\hbar}(\sigma(t))} w^{\frac{1+a}{a}}(t) \tag{104}$$

for t≥T1≥T. Integrating Eq. (104) from t to t ′ , we get

$$w(t') - w(t) + \int\_{t}^{t} \tilde{q}(s) \mathrm{d}s + \int\_{t}^{t} \frac{\beta \sigma'(s) \psi(s)}{r\_{\circ}^{\frac{1}{a}}(\sigma(s))} w^{\frac{1+a}{a}}(s) \mathrm{d}s \leq 0 \tag{105}$$

or

wðt ′ <sup>Þ</sup>−wðtÞ þ <sup>ð</sup><sup>t</sup> ′ t βσ′ ðsÞψðsÞ r 1 <sup>α</sup>ðσðsÞÞ w<sup>1</sup>þ<sup>α</sup> <sup>α</sup> ðsÞds≤0: (106)

We assert that

$$\int\_{t}^{\infty} \frac{\beta \sigma'(s)\psi(s)}{r\_{\pi}^{1}(\sigma(s))} w^{\frac{1+\omega}{\alpha}}(s) \mathrm{d}s < \infty. \tag{107}$$

If not, then

$$w(t') \lhd w(t) - \int\_{t}^{t} \frac{\beta \sigma'(s) \psi(s)}{r^{\frac{1}{a}}(\sigma(s))} w^{\frac{1+a}{a}}(s) \text{ds} \to -\infty \tag{108}$$

as t ′ ! <sup>∞</sup>, which contradicts to the positivity of <sup>w</sup>ðt<sup>Þ</sup> and thus the assertion is proved. By Eq. (104), we see that wðtÞ is decreasing that means

$$\lim\_{t \to \infty} w(t) = k, \quad k \ge 0. \tag{109}$$

By virtue of Eq. (107), we have k ¼ 0. Thus, letting t ′ ! <sup>∞</sup> in Eq. (105), we get

$$\begin{split} w(t) &\geq \int\_{t}^{\infty} \tilde{q}(s) \mathrm{ds} + \int\_{t}^{\infty} \frac{\beta \sigma'(s) \psi(s)}{r\_{\pi}^{1}(\sigma(s))} w^{\frac{1+s}{a}}(s) \mathrm{ds} \\ &= y\_{0}(t) + \int\_{t}^{\infty} \frac{\beta \sigma'(s) \psi(s)}{r\_{\pi}^{1}(\sigma(s))} w^{\frac{1+s}{a}}(s) \mathrm{ds}, \end{split} \tag{110}$$

that is,

$$w(t) \ge \int\_t^\infty \tilde{q}(s) \, \mathrm{d}s = y\_0(t). \tag{111}$$

Moreover, by induction, we have that

$$w(t) \succeq y\_n(t), \quad \text{for} \quad t \ge T\_1, \ n = 1, 2, 3, \dots \tag{112}$$

Thus, since the sequence {ynðtÞ} ∞ <sup>n</sup> <sup>¼</sup> <sup>0</sup> is monotone increasing and bounded above, it converges to yðtÞ. Letting n ! ∞ and using Lebesgue monotone convergence theorem in Eq. (100), we get Eq. (103). The proof is complete.

Theorem 4. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. If

$$\liminf\_{t \to \infty} \left( \frac{1}{y\_0(t)} \int\_t^{\infty} \frac{\beta \sigma'(s) \psi(s)}{r\_\*^{\frac{1}{a}}(\sigma(s))} \left( y\_0(s) \right)^{\frac{1+a}{a}} ds \right) > \frac{a}{(a+1)^{\frac{1+a}{a}}},\tag{113}$$

where ψðtÞ is as in Theorem 1, then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the conclusions of Lemma 1 or 2 on ½T, ∞Þ with

$$\mathbf{x}(t) > \mathbf{0}, \quad \mathbf{x}(\pi(t)) > \mathbf{0}, \quad \mathbf{x}(\sigma(t)) > \mathbf{0} \tag{114}$$

on ½T, ∞Þ. In particular, we have

Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument http://dx.doi.org/10.5772/65909 49

$$z(t) > 0, \quad \dot{z}'(t) > 0, \quad \left(r(t)\left(\dot{z}'(t)\right)^a\right)' < 0, \quad \text{for} \quad t \ge T. \tag{115}$$

By Eq. (113), there exists a constant γ > <sup>α</sup> ðαþ1Þ 1þα α such that

$$\lim\_{t \to \infty} \inf \frac{1}{y\_0(t)} \int\_t^{\infty} \frac{\beta \sigma'(s) \psi(s)}{r\_\pi^\dagger(\sigma(s))} \left( y\_0(s) \right)^{\frac{1+\mu}{\alpha}} \mathrm{d}s > \gamma. \tag{116}$$

Proceeding as in the proof of Lemma 4, we obtain Eq. (110) and from that, we have

$$\frac{w(t)}{y\_0(t)} \ge 1 + \frac{1}{y\_0(t)} \int\_t^u \frac{\beta \sigma'(s) \psi(s)}{r\_s^{\frac{1}{a}}(\sigma(s))} \left( y\_0(s) \right)^{\frac{1+a}{a}} \left( \frac{w(s)}{y\_0(s)} \right)^{\frac{1+a}{a}} ds \tag{117}$$

Let

ð∞ t

wðt ′ Þ≤wðtÞ−

Eq. (104), we see that wðtÞ is decreasing that means

48 Dynamical Systems - Analytical and Computational Techniques

By virtue of Eq. (107), we have k ¼ 0. Thus, letting t

Moreover, by induction, we have that

Thus, since the sequence {ynðtÞ}

Eq. (103). The proof is complete.

lim inf t!∞

conclusions of Lemma 1 or 2 on ½T, ∞Þ with

on ½T, ∞Þ. In particular, we have

wðtÞ ≥

ð∞ t

¼ y0ðtÞ þ

wðtÞ≥ ð∞ t

Theorem 4. Let conditions ðiÞ–ðvÞ and one of the conditions (14) or (21) hold. If

ðsÞψðsÞ

!

βσ′

r 1 <sup>α</sup>ðσðsÞÞ

∞

1 y0ðtÞ

where ψðtÞ is as in Theorem 1, then Eq. (1) is oscillatory.

ð∞ t

~qðsÞds þ

ð∞ t

If not, then

as t

that is,

βσ′

r 1 <sup>α</sup>ðσðsÞÞ

> ðt ′

t βσ′

r 1 <sup>α</sup>ðσðsÞÞ

ðsÞψðsÞ

w<sup>1</sup>þ<sup>α</sup>

ðsÞψðsÞ

′ ! <sup>∞</sup>, which contradicts to the positivity of <sup>w</sup>ðt<sup>Þ</sup> and thus the assertion is proved. By

ð∞ t

βσ′

r 1 <sup>α</sup>ðσðsÞÞ

to yðtÞ. Letting n ! ∞ and using Lebesgue monotone convergence theorem in Eq. (100), we get

� y0ðsÞ �<sup>1</sup>þ<sup>α</sup> α ds

Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the

βσ′

r 1 <sup>α</sup>ðσðsÞÞ

ðsÞψðsÞ

ðsÞψðsÞ

w<sup>1</sup>þ<sup>α</sup>

<sup>α</sup> ðsÞds < ∞: (107)

<sup>α</sup> ðsÞds ! −∞ (108)

(110)

lim<sup>t</sup>!<sup>∞</sup> <sup>w</sup>ðtÞ ¼ <sup>k</sup>, <sup>k</sup>≥0: (109)

′ ! <sup>∞</sup> in Eq. (105), we get

~qðsÞds ¼ y0ðtÞ: (111)

w<sup>1</sup>þ<sup>α</sup> <sup>α</sup> ðsÞds

wðtÞ≥ynðtÞ, for t≥T1, n ¼ 1; 2; 3;…: (112)

<sup>n</sup> <sup>¼</sup> <sup>0</sup> is monotone increasing and bounded above, it converges

>

xðtÞ > 0; xðτðtÞÞ > 0; xðσðtÞÞ > 0 (114)

α ðα þ 1Þ

1þα α

, (113)

w<sup>1</sup>þ<sup>α</sup> <sup>α</sup> ðsÞds,

$$\lambda = \inf\_{t \ge t\_1} \frac{w(t)}{y\_0(t)}.\tag{118}$$

Then it is easy to see that λ≥1 and

$$
\lambda \mathbb{1} + \lambda^{\frac{1+a}{a}} \mathbb{1},\tag{119}
$$

which contradicts the admissible value of λ and γ, and thus completes the proof.

Theorem 5. Let conditions ðiÞ–ðvÞ, one of the conditions (14) or (21) hold, and ynðtÞ be defined as in Eq. (100). If there exists some ynðtÞ such that, for T sufficiently large,

$$\limsup\_{t \to \infty} \left. y\_n(t) \left( \int\_{\mathcal{T}}^{\upsilon(t)} r^{\frac{1}{\alpha}}(s) ds \right)^{\alpha} > \frac{1}{\psi(t)}, \tag{120}$$

where ψðtÞ is as in Theorem 1, then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the conclusions of Lemma 1 or 2 on ½T, ∞Þ with

$$\mathbf{x}(t) > \mathbf{0}, \quad \mathbf{x}(\pi(t)) > \mathbf{0}, \quad \mathbf{x}(\sigma(t)) > \mathbf{0} \tag{121}$$

on ½T, ∞Þ. In particular, we have

$$z(t) > 0, \quad \dot{z'}(t) > 0, \quad \left(r(t)\left(\dot{z'}(t)\right)^a\right)' < 0, \quad \text{for} \quad t \ge T. \tag{122}$$

Proceeding as in the proof of Theorem 3 and using defining wðtÞ as in Eq. (54), for T1≥T, we get

$$\begin{aligned} \frac{1}{w(t)} &= \frac{z^{\delta}(\sigma(t))}{r(t)\left(z'(t)\right)^{\alpha}}\\ &\geq \frac{\psi(t)}{r(t)} \left(\frac{z(\sigma(t))}{z'(t)}\right)^{\alpha} \\ &= \frac{\psi(t)}{r(t)} \left(\frac{z(T\_1) + \int\_{T\_1}^{\sigma(t)} r^{\frac{1}{\alpha}}(s) r^{\frac{1}{\alpha}}(s) z'(s) ds}{z'(t)}\right)^{\alpha} \\ &\geq \psi(t) \left(\int\_{T\_1}^{\sigma(t)} r^{\frac{1}{\alpha}}(s) ds\right)^{\alpha} \end{aligned} \tag{123}$$

Thus,

$$w(t)\left(\int\_{T}^{\sigma(t)} r^{\frac{1}{a}}(s)\mathrm{d}s\right)^{a} \leq \frac{1}{\psi(t)}\left(\frac{\int\_{T}^{\sigma(t)} r^{\frac{1}{a}}(s)\mathrm{d}s}{\int\_{T\_{1}}^{\sigma(t)} r^{\frac{1}{a}}(s)\mathrm{d}s}\right)^{a} \tag{124}$$

And therefore,

$$\limsup\_{t \to \infty} w(t) \left( \int\_{T}^{\sigma(t)} r^{\perp}(s) ds \right)^{a} \leq \frac{1}{\psi(t)},\tag{125}$$

which contradicts Eq. (120). The proof is complete.

Theorem 6. Let conditions ðiÞ–ðvÞ, one of the conditions (14) or (21) hold, and ynðtÞ be defined as in Eq. (100). If there exists some ynðtÞ such that

$$\int\_{T\_1}^{\bullet} \tilde{q}(t) \exp\left(\int\_{T\_1}^{t} \frac{\beta \sigma'(s) \psi(s)}{r\_n^{\dagger}(\sigma(s))} y\_n^{\dagger}(s) ds\right) dt = \circ \tag{126}$$

or

$$\int\_{T\_1}^{\approx} \frac{\beta \sigma'(t)\psi(t)y\_n^{\frac{1}{2}}(t)y\_0(t)}{r^{\frac{1}{\alpha}}(\sigma(t))} \exp\left(\int\_{T\_1}^t \frac{\beta \sigma'(s)\psi(s)}{r^{\frac{1}{\alpha}}(\sigma(s))} y\_n^{\frac{1}{\alpha}}(s)ds\right) dt = \approx,\tag{127}$$

for T sufficiently large and T1≥T, where ψðtÞ is as in Theorem 1, then Eq. (1) is oscillatory.

Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the conclusions of Lemma 1 or 2 on ½T, ∞Þ with

$$\mathbf{x}(t) > \mathbf{0}, \quad \mathbf{x}(\pi(t)) > \mathbf{0}, \quad \mathbf{x}(\sigma(t)) > \mathbf{0} \tag{128}$$

Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument http://dx.doi.org/10.5772/65909 51

on ½T, ∞Þ. In particular, we have

$$z(t) > 0, \quad z'(t) > 0, \quad \left(r(t)\left(z'(t)\right)^a\right)' < 0, \quad \text{for} \quad t \ge T. \tag{129}$$

From Eq. (103), we have

1

50 Dynamical Systems - Analytical and Computational Techniques

Thus,

or

And therefore,

<sup>w</sup>ðt<sup>Þ</sup> <sup>¼</sup> <sup>z</sup><sup>β</sup>ðσðtÞÞ rðtÞ � z′ ðtÞ �α

> <sup>¼</sup> <sup>ψ</sup>ðt<sup>Þ</sup> rðtÞ

≥ψðtÞ

ð<sup>σ</sup>ðt<sup>Þ</sup> T r −1 <sup>α</sup>ðsÞds !<sup>α</sup>

lim sup t!∞

~qðtÞ exp

<sup>α</sup>ðσðtÞÞ exp

wðtÞ

ðt T1 βσ′

r 1 <sup>α</sup>ðσðsÞÞ <sup>y</sup>

wðtÞ

which contradicts Eq. (120). The proof is complete.

as in Eq. (100). If there exists some ynðtÞ such that

ð∞ T1 βσ′

conclusions of Lemma 1 or 2 on ½T, ∞Þ with

ð∞ T1

ðtÞψðtÞy 1 α <sup>n</sup>ðtÞy0ðtÞ

> r 1

zðσðtÞÞ z′ ðtÞ � �<sup>α</sup>

0

BBB@

ð<sup>σ</sup>ðt<sup>Þ</sup> T1 r−1 <sup>α</sup>ðsÞds !<sup>α</sup>

zðT1Þ þ

ð<sup>σ</sup>ðt<sup>Þ</sup> T1 r−1 <sup>α</sup>ðsÞr 1 <sup>α</sup>ðsÞz′

≤ 1 ψðtÞ

ð<sup>σ</sup>ðt<sup>Þ</sup> T r −1 <sup>α</sup>ðsÞds !<sup>α</sup>

Theorem 6. Let conditions ðiÞ–ðvÞ, one of the conditions (14) or (21) hold, and ynðtÞ be defined

ðsÞψðsÞ

ðt T1

Proof. Suppose to the contrary that xðtÞ is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists T∈ℐ large enough, so that xðtÞ satisfies the

for T sufficiently large and T1≥T, where ψðtÞ is as in Theorem 1, then Eq. (1) is oscillatory.

βσ′

r 1 <sup>α</sup>ðσðsÞÞ <sup>y</sup>

!

1 α <sup>n</sup>ðsÞds

ðsÞψðsÞ

!

1 α <sup>n</sup>ðsÞds

xðtÞ > 0; xðτðtÞÞ > 0; xðσðtÞÞ > 0 (128)

z′ ðtÞ

> ð<sup>σ</sup>ðt<sup>Þ</sup> T r−1 <sup>α</sup>ðsÞds

0

BBB@

ð<sup>σ</sup>ðt<sup>Þ</sup> T1 r−1 <sup>α</sup>ðsÞds

> ≤ 1 ψðtÞ

ðsÞds

1

α

(123)

(124)

CCCA

1

α

, (125)

dt ¼ ∞ (126)

dt ¼ ∞, (127)

CCCA

≥ ψðtÞ rðtÞ

$$y'(t) = -\frac{\beta \sigma'(t)\psi(t)}{r^{\frac{1}{\pi}}(\sigma(t))} \left( y(t) \right)^{\frac{1+a}{a}} \ddot{\eta}(t), \tag{130}$$

for all t≥T1≥T. Since yðtÞ≥ynðtÞ, Eq. (130) yields

$$y'(t) \!\!\!\sim \frac{\beta \sigma'(t)\psi(t)}{r\_{\pi}^{\frac{1}{\alpha}}(\sigma(t))} y\_n^{\frac{1}{\alpha}}(t) y(t) \!\!\/) \!\!\/ (t) \,. \tag{131}$$

Multiplying the above inequality by the integration factor

$$\exp\left(\int\_{T\_1}^t \frac{\beta \sigma'(s)\psi(s)}{r\_\*^\sharp(\sigma(s))} y\_n^\sharp(s) \, \mathrm{d}s\right),\tag{132}$$

one gets

$$\begin{split} y(t) &\leq \ \exp\left(-\int\_{T\_1}^t \frac{\beta \sigma'(s)\psi(s)}{r^{\frac{1}{\sigma}}(\sigma(s))} y\_n^{\frac{1}{\sigma}}(s) \mathrm{d}s\right) \\ &\quad \left(y(t\_1) - \int\_{T\_1}^t \ddot{q}(s) \exp\left(\int\_{T\_1}^s \frac{\beta \sigma'(u)\psi(u)}{r^{\frac{1}{\sigma}}(\sigma(u))} y\_n^{\frac{1}{\sigma}}(u) \mathrm{d}u\right) \mathrm{d}s\right), \end{split} \tag{133}$$

from which we have that

$$\int\_{T\_1}^t \tilde{q}(s) \exp\left(\int\_{T\_1}^s \frac{\beta \sigma'(u)\psi(u)}{r^\sharp(\sigma(u))} y\_n^\sharp(u) d\mu\right) \text{d}s \le y(T\_1) < \infty. \tag{134}$$

This is a contradiction with Eq. (126).

Now denote

$$u(t) = \int\_{t}^{\infty} \frac{\beta \sigma'(s) \psi(s)}{r\_{\sigma}^{\frac{1}{a}}(\sigma(s))} \left( y(s) \right)^{\frac{1+a}{a}} ds \tag{135}$$

Taking the derivative of uðtÞ, one gets

$$\begin{aligned} u'(t) &= -\frac{\beta \sigma'(t)\psi(t)}{r\_n^{\frac{1}{\sigma}}(\sigma(t))} \left( y(t) \right)^{\frac{1+\omega}{\sigma}} \\ &\leq -\frac{\beta \sigma'(t)\psi(t)}{r\_n^{\frac{1}{\sigma}}(\sigma(t))} y\_n^{\alpha}(t) y(t) \\ &= \frac{\beta \sigma'(t)\psi(t)}{r\_n^{\frac{1}{\sigma}}(\sigma(t))} y\_n^{\alpha}(t) \left( u(t) + y\_0(t) \right) \end{aligned} \tag{136}$$

Proceeding in a similar manner to that above, we conclude that

$$\int\_{T\_1}^{\infty} \frac{\beta \sigma'(t)\psi(t)}{r\_\pi^\dagger(\sigma(t))} y\_n^\frac{1}{\sigma}(t) y\_0(t) \exp\left(\int\_{T\_1}^t \frac{\beta \sigma'(s)\psi(s)}{r\_\pi^\dagger(\sigma(s))} y\_n^\frac{1}{\sigma}(s) ds\right) dt < \infty,\tag{137}$$

which contradicts to Eq. (127). The proof is complete.

### Author details

Said R. Grace1 \* and Irena Jadlovská<sup>2</sup>

\*Address all correspondence to: saidgrace@yahoo.com

1 Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza, Egypt

2 Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Košice, Slovakia

#### References


[6] L. H. Erbe, Q. Kong, and B. G. Zhang. Oscillation Theory for Functional Differential Equations, Marcel Dekker Inc., New York, 1995.

u′

52 Dynamical Systems - Analytical and Computational Techniques

ðtÞ ¼ −

Proceeding in a similar manner to that above, we conclude that

1 α

ð∞ T1

Author details

Said R. Grace1

References

(1936), 106–186.

Clarendon Press, Oxford, 1991.

Egypt

βσ′

r 1 <sup>α</sup>ðσðtÞÞ <sup>y</sup>

ðtÞψðtÞ

which contradicts to Eq. (127). The proof is complete.

\* and Irena Jadlovská<sup>2</sup>

\*Address all correspondence to: saidgrace@yahoo.com

and Informatics, Technical University of Košice, Košice, Slovakia

Equations, Taylor & Francis, London and New York, 2003.

Differential Equations, Marcel Dekker Inc., New York, 2004.

βσ′

r 1 <sup>α</sup>ðσðtÞÞ

≤− βσ′

<sup>¼</sup> βσ′

r 1 <sup>α</sup>ðσðtÞÞ <sup>y</sup>

<sup>n</sup>ðtÞy0ðtÞ exp

r 1 <sup>α</sup>ðσðtÞÞ <sup>y</sup>

ðtÞψðtÞ

ðtÞψðtÞ

ðtÞψðtÞ

� yðtÞ �<sup>1</sup>þ<sup>α</sup> α

1 α <sup>n</sup> ðtÞyðtÞ

1 α <sup>n</sup> ðtÞ �

ðt T1

1 Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza,

2 Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering

[1] C. Sturm. Memoir on linear differential equations of second-order. J. Math. Pures Appl., 1

[2] R. P. Agarwal, S. R. Grace, and D. O'Regan. Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic, Dordrchet, 2002.

[3] R. P. Agarwal, S. R. Grace, and D. O'Regan. Oscillation Theory for Second Order Dynamic

[4] R. P. Agarwal, M. Bohner, and W. T. Li. Nonoscillation and Oscillation: Theory for Functional

[5] I. Györi and G. Ladas. Oscillation Theory of Delay Differential Equations with Applications,

βσ′

r 1 <sup>α</sup>ðσðsÞÞ <sup>y</sup>

ðsÞψðsÞ

!

uðtÞ þ y0ðtÞ

1 α <sup>n</sup>ðsÞds

�

(136)

dt < ∞, (137)


Provisional chapter

## **Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix** Preservation of Synchronization Using a Tracy-Singh Product in the Transformation on Their Linear Matrix

Guillermo Fernadez‐Anaya, Luis Alberto Quezada‐Téllez, Jorge Antonio López‐Rentería, Oscar A. Rosas‐Jaimes, Rodrigo Muñoz‐Vega, Guillermo Manuel Mallen‐Fullerton and José Job Flores‐Godoy Guillermo Fernadez-Anaya, Luis Alberto Quezada-Téllez, Jorge Antonio López-Rentería, Oscar A. Rosas-Jaimes, Rodrigo Muñoz-Vega, Guillermo Manuel Mallen-Fullerton and José Job Flores-Godoy

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/66957

#### Abstract

Preservation is related to local asymptotic stability in nonlinear systems by using dynamical systems tools. It is known that a system, which is stable, asymptotically stable, or unstable at origin, through a transformation can remain stable, asymptotically stable, or unstable. Some systems permit partition of its nonlinear equation in a linear and nonlinear part. Some authors have stated that such systems preserve their local asymptotic stability through the transformations on their linear part. The preservation of synchronization is a typical application of these types of tools and it is considered an interesting topic by scientific community. This chapter is devoted to extend the methodology of the dynamical systems through a partition in the linear part and the nonlinear part, transforming the linear part using the Tracy-Singh product in the Jacobian matrix. This methodology preserves the structure of signs through the real part of eigenvalues of the Jacobian matrix of the dynamical systems in their equilibrium points. The principal part of this methodology is that it permits to extend the fundamental theorems of the dynamical systems, given a linear transformation. The results allow us to infer the hyperbolicity, the stability and the synchronization of transformed systems of higher dimension.

Keywords: preservation, synchronization, Tracy-Singh product, chaotic dynamical system

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons © 2017 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

## 1. Introduction

In nonlinear autonomous dynamical systems, the study of synchronization is not new. We can see several papers about these themes from different approaches. Some examples show the use of change of variables, that is, through a diffeomorphism of the origin. From this, it is possible to say if a system is stable, asymptotically stable, or unstable. Some results are also obtained by the product in a vector field in the nonlinear dynamical system by a continuously differentiable function at the origin [1]. On one hand, there are studies showing the use of statistical properties to characterize the synchronization [2]. The eigenvalues of a system determine a system dynamics, but they are not derivable from the statistical features of such a system. One way to observe the stability is through a linear part of a dynamical system. But the problem to preserve stability by the transformation of its linear part in a nonlinear autonomous system has just been analyzed recently.

In [3], it is presented a methodology under which stability and synchronization of a dynamical master-slave system configuration are preserved under a modification through matrix multiplication. The conservation of stability is important for chaos control. A generalized synchronization can also be derived for different systems by finding a diffeomorphic transformation such as the slave system written as a function of the master system. One example of preservation for asymptotic stability is the use of transformations on rational functions in the frequency domain [4, 5].

This class of transformation can be interpreted as noise in the system or as a simple disturbance on the value of the physical parameters of the model. The chaotic synchronization problem studied in [6] is mainly related to preservation of the stability of the master-slave system presented in it. Results included therein show that stability is preserved by transforming the linear part of system. The same results can also be used in the chaos suppression problem. In [7], the authors show the viability of preserving the hyperbolicity of a masterslave pair of chaotic systems under different types of nonlinear modifications to its Jacobian matrix.

In [8], the developed methodology is used to study the problem of preservation of synchronization in chaotic dynamical systems, in particular the case of dynamical networks. Given a chaotic system, its transformed version is also a chaotic system. By means of a master-slave scheme obtained a controller for the system using a linear-quadratic regulator, preserving the stability even after the master-slave controller is transformed. This chapter is inspired by the same objective, that is, to preserve the stability in a master-slave system even through a transformation is performed over it. One way to achieve it is by extending some of the results in [8], particularly those of the local stable-unstable manifold theorem and extension of the center manifold theorem based in the preservation of the linear part of the vector field in nonlinear dynamical systems. As we will see, these results depart from the hypothesis of the existence of a constant state feedback as anominal synchronization force. In this work, we elaborate another approach to the problem of preservation of synchronization. We focus particularly on autonomous nonlinear dynamical systems, extending the previous results already mentioned.

This chapter is organized as follows: First, in Section 2, we will give basic concepts of dynamical systems. The fundamental theorem for linear systems, the local stable-unstable manifold theorem, the center manifold theorem, the Hartman-Grobman theorem and the concept of group action are introduced. In Section 3, we present some definitions about matrices and Tracy-Singh product of matrices. Also in this section, the main result is presented as a generalization of Proposition 4 in [6]. In Section 4, we will show that it is possible to preserve synchronization under a class of transformations defined under a certain method. Numerical experiments on the stability preservation for chaotic synchronization are shown in Section 5. Finally, a set of concluding remarks is given in Section 6.

## 2. Classical concepts of dynamical systems

1. Introduction

56 Dynamical Systems - Analytical and Computational Techniques

just been analyzed recently.

domain [4, 5].

matrix.

already mentioned.

In nonlinear autonomous dynamical systems, the study of synchronization is not new. We can see several papers about these themes from different approaches. Some examples show the use of change of variables, that is, through a diffeomorphism of the origin. From this, it is possible to say if a system is stable, asymptotically stable, or unstable. Some results are also obtained by the product in a vector field in the nonlinear dynamical system by a continuously differentiable function at the origin [1]. On one hand, there are studies showing the use of statistical properties to characterize the synchronization [2]. The eigenvalues of a system determine a system dynamics, but they are not derivable from the statistical features of such a system. One way to observe the stability is through a linear part of a dynamical system. But the problem to preserve stability by the transformation of its linear part in a nonlinear autonomous system has

In [3], it is presented a methodology under which stability and synchronization of a dynamical master-slave system configuration are preserved under a modification through matrix multiplication. The conservation of stability is important for chaos control. A generalized synchronization can also be derived for different systems by finding a diffeomorphic transformation such as the slave system written as a function of the master system. One example of preservation for asymptotic stability is the use of transformations on rational functions in the frequency

This class of transformation can be interpreted as noise in the system or as a simple disturbance on the value of the physical parameters of the model. The chaotic synchronization problem studied in [6] is mainly related to preservation of the stability of the master-slave system presented in it. Results included therein show that stability is preserved by transforming the linear part of system. The same results can also be used in the chaos suppression problem. In [7], the authors show the viability of preserving the hyperbolicity of a masterslave pair of chaotic systems under different types of nonlinear modifications to its Jacobian

In [8], the developed methodology is used to study the problem of preservation of synchronization in chaotic dynamical systems, in particular the case of dynamical networks. Given a chaotic system, its transformed version is also a chaotic system. By means of a master-slave scheme obtained a controller for the system using a linear-quadratic regulator, preserving the stability even after the master-slave controller is transformed. This chapter is inspired by the same objective, that is, to preserve the stability in a master-slave system even through a transformation is performed over it. One way to achieve it is by extending some of the results in [8], particularly those of the local stable-unstable manifold theorem and extension of the center manifold theorem based in the preservation of the linear part of the vector field in nonlinear dynamical systems. As we will see, these results depart from the hypothesis of the existence of a constant state feedback as anominal synchronization force. In this work, we elaborate another approach to the problem of preservation of synchronization. We focus particularly on autonomous nonlinear dynamical systems, extending the previous results

We introduce theorems and classical definitions on properties of dynamical systems in this section. The fundamental theorem for linear systems, the local stable-unstable manifold theorem and the center manifold theorem are those important propositions mainly needed to develop analyses in this chapter. We will combine them with the Hartman-Grobman theorem in order to achieve a necessary generalization for those particular results of this chapter.

Theorem 2.1. (The local stable-unstable manifold theorem [9]). Let E be an open subset of R<sup>n</sup>

containing the origin. Let f ∈ C<sup>1</sup> ðEÞ and φ<sup>t</sup> be the flow of the nonlinear system of the form x\_ ¼ fðxÞ. Suppose that fð0Þ ¼ 0 and that Dfð0Þ are the Jacobian matrix, which has k eigenvalues with negative real part and n−k eigenvalues with positive real part.


It should be noted that the manifolds S and W mentioned in Theorem 2.1 are unique. We define now the central manifold theorem in the following.

Theorem 2.2. (The center manifold theorem [9]). Let E be an open subset of R<sup>n</sup> containing the origin and r ≥ 1. Let f ∈ C<sup>r</sup> ðEÞ, that is, f is a continuously differentiable function on E of order r. Now we suppose that fð0Þ ¼ 0 and that Dfð0Þ have k eigenvalues with negative real part, j eigenvalues with positive real part and l ¼ n−k−j eigenvalues with zero real part. Therefore, there exists an l


By what it is established in Theorem 2.2, the center manifold <sup>W</sup><sup>C</sup>ð0<sup>Þ</sup> is not unique, which is an important difference for the stable character of the systems to be studied.

Theorem 2.3. (The Hartman-Grobman theorem [9]). Let E be an open subset of R<sup>n</sup> containing the origin, let φ<sup>t</sup> be the flow of the nonlinear system x\_ ¼ fðxÞ. Now, we assume that fð0Þ ¼ 0, that is, the origin is an equilibrium point of the dynamical system; also the Jacobian matrix evaluated at the origin, A ¼ Dfð0Þ. If H is an homeomorphism of an open set W onto an open set V such that for each x0∈W, it exists an open interval I<sup>0</sup> ⊂ R such that for all x<sup>0</sup> ∈ W and t ∈ I<sup>0</sup>

$$H \bullet \phi\_t(\mathbf{x}\_0) = \mathcal{e}^{At} H(\mathbf{x}\_0);\tag{1}$$

that is, H maps trajectories of the nonlinear system x\_ ¼ fðxÞ near the origin onto trajectories of x\_ ¼ Ax near the origin and preserves the parametrization.

From the following argument, it is show that for any matrix <sup>A</sup> <sup>¼</sup> <sup>U</sup>TTAU, there exists an homeomorphism <sup>H</sup>^ <sup>¼</sup> UH such that for an open set <sup>W</sup> containing the origin onto an open set V also containing the origin such that for each x0∈W and there is an open interval I0⊂R containing zero such that for all x0∈W and t∈I<sup>0</sup>

$$
\hat{H} \bullet \phi\_t(\mathbf{x}\_0) = \mathfrak{e}^{T\_A t} \hat{H}(\mathbf{x}\_0);\tag{2}
$$

This last equality is a consequence of the Hartman-Grobman theorem and of the fact of UeAt <sup>¼</sup> <sup>e</sup>TAt <sup>U</sup>, that is, <sup>H</sup>^ maps trajectories of the nonlinear system <sup>x</sup>\_ <sup>¼</sup> <sup>f</sup>ðx<sup>Þ</sup> near the origin onto trajectories of x\_ ¼ TAx near the origin and preserves the parametrization.

On the other hand, some classical definitions are now included. A linear system of the form <sup>x</sup>\_ <sup>¼</sup> Ax where <sup>x</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>, <sup>A</sup> is a <sup>n</sup> · <sup>n</sup> matrix and <sup>x</sup>\_ <sup>¼</sup> dx dt. It is shown that the solution of the linear system together with the initial condition <sup>x</sup>ð0Þ ¼ <sup>x</sup><sup>0</sup> is given by <sup>x</sup>ðtÞ ¼ <sup>e</sup>Atx0. The mapping <sup>e</sup>At : <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup> is called the flow of the linear system.

Definition 2.1. For all eigenvalues of a matrix Að<sup>n</sup> · <sup>n</sup><sup>Þ</sup> have nonzero real part, then the flow eAt is called a hyperbolic flow and therefore, x\_ ¼ Ax is called a hyperbolic linear system [9].

Definition 2.2. A subspace E⊂R<sup>n</sup> is said to be invariant with respect to the flow eAt : <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup> if eAt⊂E for all t∈R [9].

Lemma 2.1. Let A∈R<sup>n</sup> · n. If <sup>R</sup><sup>n</sup> <sup>¼</sup> Es ⊕E<sup>u</sup>⊕E<sup>c</sup> where E<sup>s</sup> , E<sup>u</sup> and E<sup>c</sup> are the stable, unstable and center subspaces of the linear system <sup>x</sup>\_ <sup>¼</sup> Ax. By the above,Es , E<sup>u</sup> and Ec are invariant with respect to the flow eAt, respectively [9].

Definition 2.3. Let E be an open subset of R<sup>n</sup> and let f ∈ C<sup>1</sup> ðEÞ, that is, f is a continuous differentiable function defined on E. For x<sup>0</sup> ∈ E, let φðt, x0Þ be the solution of the initial value problem x\_ ¼ fðxÞ, xð0Þ ¼ x<sup>0</sup> defined on its maximal interval of existence Iðx0Þ. Then for t∈Iðx0Þ, the mapping φ<sup>t</sup> : E ! E defined by φ<sup>t</sup> ðx0Þ ¼ φ<sup>t</sup> ðt, x0Þ is called the flow of the differential equation [9].

Definition 2.4. For any x0∈R<sup>n</sup> , let φ<sup>t</sup> ðx0Þ be the flow of the differential equation through x0.(i) The local stable set S corresponding to a neighborhood V of x<sup>0</sup> is defined by S <sup>¼</sup> <sup>S</sup>ð0Þ ¼ {x<sup>0</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> : φt ðx0Þ ∈ V,t ≥ 0 and φ<sup>t</sup> ðx0Þ ! 0 as t ! ∞}. (ii) The local unstable set W of x<sup>0</sup> corresponding to a neighborhood V of x<sup>0</sup> is defined by W <sup>¼</sup> <sup>W</sup>ð0Þ ¼ {x0∈R<sup>n</sup> : <sup>φ</sup><sup>t</sup> ðx0Þ∈V,t ≤ 0 and φ<sup>t</sup> ðx0Þ ! 0 as t ! ∞}. Then, these stable and unstable local sets are submanifolds of R<sup>n</sup> in a sufficiently small neighborhood V of x0½9�.

Definition 2.5. If G is a group and X is a set, then a (left) group action of G on X is a binary function G · X ! X, denoted by [9]

$$\mathbf{x}(\mathbf{g}, \mathbf{x}) \mapsto \mathbf{g} \cdot \mathbf{x} \tag{3}$$

which satisfies the following two axioms:

Theorem 2.3. (The Hartman-Grobman theorem [9]). Let E be an open subset of R<sup>n</sup> containing the origin, let φ<sup>t</sup> be the flow of the nonlinear system x\_ ¼ fðxÞ. Now, we assume that fð0Þ ¼ 0, that is, the origin is an equilibrium point of the dynamical system; also the Jacobian matrix evaluated at the origin, A ¼ Dfð0Þ. If H is an homeomorphism of an open set W onto an open set V such that for each x0∈W, it

ðx0Þ ¼ e

that is, H maps trajectories of the nonlinear system x\_ ¼ fðxÞ near the origin onto trajectories of x\_ ¼ Ax

From the following argument, it is show that for any matrix <sup>A</sup> <sup>¼</sup> <sup>U</sup>TTAU, there exists an homeomorphism <sup>H</sup>^ <sup>¼</sup> UH such that for an open set <sup>W</sup> containing the origin onto an open set V also containing the origin such that for each x0∈W and there is an open interval I0⊂R

ðx0Þ ¼ e

This last equality is a consequence of the Hartman-Grobman theorem and of the fact of

On the other hand, some classical definitions are now included. A linear system of the form

system together with the initial condition <sup>x</sup>ð0Þ ¼ <sup>x</sup><sup>0</sup> is given by <sup>x</sup>ðtÞ ¼ <sup>e</sup>Atx0. The mapping

Definition 2.1. For all eigenvalues of a matrix Að<sup>n</sup> · <sup>n</sup><sup>Þ</sup> have nonzero real part, then the flow eAt is

Definition 2.2. A subspace E⊂R<sup>n</sup> is said to be invariant with respect to the flow eAt : <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup> if

⊕E<sup>u</sup>⊕E<sup>c</sup> where E<sup>s</sup>

function defined on E. For x<sup>0</sup> ∈ E, let φðt, x0Þ be the solution of the initial value problem x\_ ¼ fðxÞ, xð0Þ ¼ x<sup>0</sup> defined on its maximal interval of existence Iðx0Þ. Then for t∈Iðx0Þ, the mapping

local stable set S corresponding to a neighborhood V of x<sup>0</sup> is defined by S <sup>¼</sup> <sup>S</sup>ð0Þ ¼ {x<sup>0</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> :

ðt, x0Þ is called the flow of the differential equation [9].

ðx0Þ ! 0 as t ! ∞}. (ii) The local unstable set W of x<sup>0</sup> corresponding to a

ðx0Þ be the flow of the differential equation through x0.(i) The

ðx0Þ∈V,t ≤ 0 and φ<sup>t</sup>

TAt

<sup>U</sup>, that is, <sup>H</sup>^ maps trajectories of the nonlinear system <sup>x</sup>\_ <sup>¼</sup> <sup>f</sup>ðx<sup>Þ</sup> near the origin onto

AtHðx0Þ; (1)

<sup>H</sup>^ <sup>ð</sup>x0Þ; (2)

dt. It is shown that the solution of the linear

, E<sup>u</sup> and E<sup>c</sup> are the stable, unstable and

ðEÞ, that is, f is a continuous differentiable

, E<sup>u</sup> and Ec are invariant with

ðx0Þ ! 0 as t ! ∞}.

exists an open interval I<sup>0</sup> ⊂ R such that for all x<sup>0</sup> ∈ W and t ∈ I<sup>0</sup>

near the origin and preserves the parametrization.

58 Dynamical Systems - Analytical and Computational Techniques

containing zero such that for all x0∈W and t∈I<sup>0</sup>

<sup>x</sup>\_ <sup>¼</sup> Ax where <sup>x</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>, <sup>A</sup> is a <sup>n</sup> · <sup>n</sup> matrix and <sup>x</sup>\_ <sup>¼</sup> dx

<sup>e</sup>At : <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup> is called the flow of the linear system.

UeAt <sup>¼</sup> <sup>e</sup>TAt

eAt⊂E for all t∈R [9].

φ<sup>t</sup> : E ! E defined by φ<sup>t</sup>

ðx0Þ ∈ V,t ≥ 0 and φ<sup>t</sup>

φt

Definition 2.4. For any x0∈R<sup>n</sup>

Lemma 2.1. Let A∈R<sup>n</sup> · n. If <sup>R</sup><sup>n</sup> <sup>¼</sup> Es

respect to the flow eAt, respectively [9].

H ∘ φ<sup>t</sup>

<sup>H</sup>^ <sup>∘</sup> <sup>φ</sup><sup>t</sup>

trajectories of x\_ ¼ TAx near the origin and preserves the parametrization.

called a hyperbolic flow and therefore, x\_ ¼ Ax is called a hyperbolic linear system [9].

center subspaces of the linear system <sup>x</sup>\_ <sup>¼</sup> Ax. By the above,Es

, let φ<sup>t</sup>

Definition 2.3. Let E be an open subset of R<sup>n</sup> and let f ∈ C<sup>1</sup>

ðx0Þ ¼ φ<sup>t</sup>

neighborhood V of x<sup>0</sup> is defined by W <sup>¼</sup> <sup>W</sup>ð0Þ ¼ {x0∈R<sup>n</sup> : <sup>φ</sup><sup>t</sup>

1. ðghÞ � x ¼ g � ðh � xÞ for all g, h ∈ G and x ∈ X;

2. e � x ¼ x for every x ∈ X (where e denotes the identity element of G).

The action is faithful (or effective) if for any two different g, h ∈ G, there exists an x ∈ X such that g � x ≠ h � x; or equivalently, if for any g ≠ e in G, there exists an x ∈ X such that g � x ≠ x.

The action is free or semiregular if for any two different g, h ∈ G and all x ∈ X, we have g � x ≠ h � x; or equivalently, if g � x ¼ x for some x implies g ¼ e.

For every x ∈ X, we define the stabilizer subgroup of x (also called the isotropy group or little group) as the set of all elements in G that fix x:

$$\mathcal{G}\_{\mathbf{x}} = \{ \mathbf{g} \in \mathbf{G} : \mathbf{g} \cdot \mathbf{x} = \mathbf{x} \} \tag{4}$$

This is a subgroup of G, though typically not a normal one. The action of G on X is free if and only if all stabilizers are trivial.

#### 3. Tracy-Singh product and other mathematical extensions

In this third section, we show a definition and some properties of the Tracy-Singh product. We also include a simple extension of the local stable-unstable manifold theorem and the center manifold theorem, using the tools presented in Section 2. These extensions are tools that will also be used in Section 4, where we will present the results on preservation of synchronization in nonlinear dynamical systems.

Definition 3.1. Let λ be an eigenvalue of the n · n matrix A of multiplicity m≤n. Then for k ¼ 1, …, m, any nonzero solution w of [9]

$$(A - \lambda I)^k w = 0\tag{5}$$

is called a generalized eigenvector of A.

In this case, let wj ¼ uj þ vj be a generalized eigenvector of the matrix A corresponding to an eigenvalue λ<sup>j</sup> ¼ aj þ ibj (note that if bj ¼ 0 then vj ¼ 0). Then, let B ¼ fu1, v1, …, uk, vk, …, um, vmg

be a basis of <sup>R</sup><sup>n</sup> (with <sup>n</sup> <sup>¼</sup> <sup>2</sup>m−<sup>k</sup> as established by Theorems 1.7.1 and 1.7.2, see [9]). Now, we introduce the definition of Tracy-Singh product and some properties.

Definition 3.2. If taken the matrices A ¼ ðaijÞ and C ¼ ðcijÞ of order m · n and B ¼ ðbklÞ of order p · q. Let A ¼ ðAijÞ be partitioned with Aij of order mi · nj as the ði, jÞ th block submatrix and B ¼ ðBklÞ of order pk · ql as the ðk, lÞ th block submatrix ð∑mi ¼ m, ∑nj ¼ n, ∑pk ¼ p, ∑ql ¼ qÞ. Then, the definitions of the matrix products or sums of A and B are given as follows [10].

Tracy-Singh product

$$A \bullet B = (A\_{i\circ} \bullet B)\_{i\circ} = \left( (A\_{i\circ} \otimes B\_{kl})\_{kl} \right)\_{i\circ} \tag{6}$$

where Aij⊗Bkl is of order mipk · njql , Aij∘B is a Kronecker product of order mip · njq, and A∘B is of order mp · nq.

Tracy-Singh sum

$$A \boxdot{B} = A \circ I\_p + I\_m \circ B \tag{7}$$

where A ¼ ðAijÞ and B ¼ ðBklÞ are square matrices of respective order m · m and p · p with Aij of order mi · mj and Bkl of order pk · pl ; Ip and Im are compatibly partitioned identity matrices.

Theorem 3.1. Let A, B,C, D, E, and F be compatibly partitioned matrices, then [10]


The next proposition presents some extensions to the local stable-unstable manifold theorem and to the center manifold theorem.

Proposition 3.1. Let E be an open subset of R<sup>n</sup> containing the origin, let f ∈ C<sup>1</sup> ðEÞ and φ<sup>t</sup> be the flow of the nonlinear system x\_ ¼ fðxÞ ¼ Ax þ gðxÞ. Suppose that fð0Þ ¼ 0 and that A ¼ Dfð0Þ have k eigenvalues with negative real part and n−k eigenvalues with positive real part, that is, the origin is an hyperbolic fixed point. Then for each matrix M ∈ ΛU, there exists a k


$$\lim\_{t \to \infty} \phi\_{M,t}(\mathbf{x}\_0) = 0,\tag{8}$$

where φM,<sup>t</sup> is the flow of the nonlinear system x\_ ¼ MAx þ gðxÞ and there exists an n−k dimensional differentiable manifold WM tangent to the unstable subspace E<sup>W</sup> <sup>M</sup> of x\_ ¼ MAx at 0 such that for all t≤0, φM,<sup>t</sup> ðWMÞ⊂WM and for all x0∈WM,

$$\lim\_{t \to -\infty} \phi\_{M,t}(\mathbf{x}\_0) = 0.\tag{9}$$

An interesting property is that Proposition 4.1 is valid for each g∈C<sup>1</sup> ðEÞ such that x\_ ¼ fðxÞ ¼ Ax þ gðxÞ and

$$\frac{\|\overline{g}(\mathbf{x})\|\_{2}}{\|\mathbf{x}\|\_{2}} \to \mathbf{0} \text{ as } \mathbb{I}\boldsymbol{\omega}\|\_{2} \to \mathbf{0}.\tag{10}$$

In consequence, the set of matrices Λ<sup>U</sup> generates the action of the group Λ<sup>U</sup> on the set of the hyperbolic nonlinear systems, formally on the set of the hyperbolic vector fields f∈C<sup>1</sup> ðEÞ, <sup>x</sup>\_ <sup>¼</sup> <sup>f</sup>ðxÞ ¼ Ax <sup>þ</sup> <sup>g</sup>ðx<sup>Þ</sup> with <sup>g</sup>∈C<sup>1</sup> ðEÞ and

$$A \in \Omega\_{\mathcal{U}} \triangleq \{ P \in \mathbb{R}^{n \times n} : P = \mathcal{U}^T T\_P \mathcal{U} \text{ with } T\_P \text{ any upper triangular matrix} \} \tag{11}$$

Satisfying the last condition, where U is a fixed unitary matrix, the action is generated by the action of the group Λ<sup>U</sup> on the set ΩU. By that this first action preserves the dimension and a nonlinear systems of the stable and unstable manifolds, that is, an hyperbolic nonlinear system x\_ ¼ Ax þ gðxÞ is mapped in a hyperbolic nonlinear systems x\_ ¼ MAx þ gðxÞ and dimS ¼ dimSM and dimW ¼ dimWM.

The proof of this Proposition 3.1 can be revised in Ref. [8].

Definition 3.2. If taken the matrices A ¼ ðaijÞ and C ¼ ðcijÞ of order m · n and B ¼ ðbklÞ of order p · q. Let A ¼ ðAijÞ be partitioned with Aij of order mi · nj as the ði, jÞ th block submatrix and B ¼ ðBklÞ of order pk · ql as the ðk, lÞ th block submatrix ð∑mi ¼ m, ∑nj ¼ n, ∑pk ¼ p, ∑ql ¼ qÞ. Then, the defini-

where A ¼ ðAijÞ and B ¼ ðBklÞ are square matrices of respective order m · m and p · p with Aij of order

The next proposition presents some extensions to the local stable-unstable manifold theorem

the flow of the nonlinear system x\_ ¼ fðxÞ ¼ Ax þ gðxÞ. Suppose that fð0Þ ¼ 0 and that A ¼ Dfð0Þ have k eigenvalues with negative real part and n−k eigenvalues with positive real part, that is, the origin is an hyperbolic fixed point. Then for each matrix M ∈ ΛU, there

ðSMÞ⊂SM and for all x<sup>0</sup> ∈ SM [8],

Proposition 3.1. Let E be an open subset of R<sup>n</sup> containing the origin, let f ∈ C<sup>1</sup>


lim<sup>t</sup>!<sup>∞</sup> <sup>φ</sup>M,<sup>t</sup>

; Ip and Im are compatibly partitioned identity matrices.

ðAij⊗BklÞkl

, Aij∘B is a Kronecker product of order mip · njq, and A∘B is of order

A⊞B ¼ A∘Ip þ Im∘B (7)

ij (6)

ðEÞ and φ<sup>t</sup> be

<sup>M</sup> of the linear system

ðx0Þ ¼ 0, (8)

tions of the matrix products or sums of A and B are given as follows [10].

A∘B ¼ ðAij∘BÞij ¼

Theorem 3.1. Let A, B,C, D, E, and F be compatibly partitioned matrices, then [10]

Tracy-Singh product

mp · nq.

Tracy-Singh sum

2. A∘B ≠ B∘A.

0 <sup>¼</sup> <sup>A</sup>′ ∘B<sup>0</sup> .

6. ðA∘BÞ∘F ¼ A∘ðB∘FÞ

and to the center manifold theorem.

x\_ ¼ MAx at 0 such that for all t ≥ 0, φM,<sup>t</sup>

4. ðA∘BÞ

exists a k

where Aij⊗Bkl is of order mipk · njql

60 Dynamical Systems - Analytical and Computational Techniques

mi · mj and Bkl of order pk · pl

1. ðA∘BÞðC∘DÞ¼ðACÞ∘ðBDÞ.

3. ðC∘B ¼ B∘CÞ where C ¼ ðcijÞ and cij is a scalar.

5. ðA þ DÞ∘ðB þ EÞ ¼ A∘B þ A∘E þ D∘B þ D∘E.

Given a particular nonlinear system, the stable and unstable manifolds S and W are unique; then for each matrix M∈ΛU, there exists an unique pair of manifolds ðSM, WMÞ in such a way that it is possible to define a pair of functions in the following form

$$\begin{aligned} \Theta: \Lambda\_{\mathrm{II}} \times \mathrm{Man}\_{\mathrm{S}} &\to \mathrm{Man}\_{\mathrm{S}} \\ \Theta(\mathrm{M}, \mathrm{S}) &= \mathrm{S\_{\mathrm{M}}} \\ \Theta: \Lambda\_{\mathrm{II}} \times \mathrm{Man}\_{\mathrm{W}} &\to \mathrm{Man}\_{\mathrm{W}} \end{aligned} \tag{12}$$
 
$$\begin{aligned} \Phi(\mathrm{M}, \mathrm{W}) &= \mathrm{W\_{\mathrm{M}}} \end{aligned}$$

Where ManS is the set of stable manifolds and ManW is the set of unstable manifold for autonomous nonlinear systems.

Therefore, we can say that if A ¼ Dfð0Þ is an stable matrix A has all the n eigenvalues with negative real part, then the origin of the nonlinear system x\_ ¼ M∘Ax þ gðxÞ is asymptotically stable; but if A ¼ Dfð0Þ is an unstable matrix A has n−k (with n > kÞ eigenvalues with positive real part, then the origin of the nonlinear system x\_ ¼ M∘Ax þ gðxÞ is unstable.

As an extension of the local stable-unstable manifold theorem in terms of Tracy-Singh product of matrices in Λ<sup>N</sup> and the matrix A of the vector field fðxÞ, we present the following proposition.

#### Proposition 3.2.

1. Let E be an open subset of R<sup>n</sup> containing the origin, let f∈C<sup>1</sup> ðEÞ and let φ<sup>t</sup> be the flow of the nonlinear system x\_ ¼ fðxÞ ¼ Ax þ gðxÞ. We suppose that fð0Þ ¼ 0 and that A ¼ Dfð0Þ have a k eigenvalues with negative real part and n−k eigenvalues with positive real part; thus, the origin is a hyperbolic fixed point. Now, take a fixed continuously differentiable function

$$F: \mathbb{C}^1(E) \to \mathbb{C}^1(\overline{E}) \tag{13}$$

such that <sup>F</sup>ðgÞ ¼ <sup>g</sup>^ where <sup>g</sup>^ : <sup>E</sup>⊂Rmn ! <sup>R</sup>mn is a fixed continuously differentiable function with domain all C<sup>1</sup> <sup>ð</sup>EÞ; moreover, <sup>g</sup>^∈C<sup>1</sup> <sup>ð</sup>E<sup>Þ</sup> with <sup>E</sup> an open subset of <sup>R</sup><sup>n</sup> containing the origin such that

$$\frac{\|\hat{\mathbf{g}}(\boldsymbol{\chi})\|\_{2}}{\|\boldsymbol{\chi}\|\_{2}} \to 0 \text{ as } \|\boldsymbol{\chi}\|\_{2} \to 0. \tag{14}$$

Then, for each matrix M∈Λ<sup>U</sup> of m · m, there exists a mk− dimensional differentiable manifold SM∘<sup>A</sup> tangent to the stable subspace E<sup>S</sup> <sup>M</sup>∘<sup>A</sup> of the linear system x\_ ¼ ðM∘AÞx at 0 such that for all t≥0, φ<sup>M</sup>∘A,<sup>t</sup> ðSM∘<sup>A</sup>Þ⊂SM∘<sup>A</sup> and for all x0∈SM∘<sup>A</sup>,

$$\lim\_{t \to \infty} \phi\_{M \circ A, t}(\mathbf{x}\_0) = \mathbf{0},\tag{15}$$

where φ<sup>M</sup>∘A,<sup>t</sup> be the flow of the nonlinear system x\_ ¼ ðM∘AÞx þ g^ðxÞ and there exists an mðn−kÞ dimensional differentiable manifold WM∘<sup>A</sup> tangent to the unstable subspace EW <sup>M</sup>∘<sup>A</sup> of x\_ ¼ ðM∘AÞx at 0 such that for all t≤0, φ<sup>M</sup>∘A,<sup>t</sup> ðWM∘<sup>A</sup>Þ⊂WM∘<sup>A</sup> and for all x0∈WM∘<sup>A</sup>,

$$\lim\_{t \to -\infty} \phi\_{M\ast A, t}(\mathbf{x}\_0) = 0. \tag{16}$$

2. Also, there exists a function of the group Λ<sup>N</sup> and the set of all the autonomous hyperbolic nonlinear systems of dimension n (hyperbolic vector fields of dimension n) denoted by Γn, to the set Γmn of all the autonomous hyperbolic nonlinear systems of dimension mn (hyperbolic vector fields of dimension mn); this function (which is similar to an action of the group Λ<sup>N</sup> on the set Γn) is defined as follows

$$\begin{aligned} \mathfrak{G}: \Lambda\_N \times \Gamma\_n &\to \Gamma\_{mn} \\ \mathfrak{G}\left(M, A\mathbf{x} + \mathfrak{g}(\mathbf{x})\right) &= (M \bullet A)\mathbf{x} + \hat{\mathfrak{g}}(\mathbf{x}) \end{aligned} \tag{17}$$

and the new nonlinear system is

Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix http://dx.doi.org/10.5772/66957 63

$$\begin{aligned} \dot{\mathbf{x}} &= \mathfrak{G} \begin{pmatrix} M, Ax + \mathfrak{g}(\mathbf{x}) \end{pmatrix} \\ \dot{\mathbf{x}} &= (M \mathfrak{a} A) \mathbf{x} + \hat{\mathfrak{g}}(\mathbf{x}) \end{aligned} \tag{18}$$

which satisfies the following two axioms:


Where z is associated with Ax þ gðxÞ (denoted by z≗Ax þ gðxÞÞ; h � z means ðMh∘AÞx þ g^ðxÞ (denoted by h � z≗ðMh∘AÞx þ g^ðxÞ); gh is associated with the usual product of matrices Mg, Mh, that is, gh≗MgMh and <sup>e</sup> � <sup>z</sup> means <sup>ð</sup>Im∘AÞ<sup>x</sup> <sup>þ</sup> <sup>g</sup>^ðxÞ, that is, e � z≗ðIm∘AÞx þ g^ðxÞ and g•ðh � zÞ means ðMg∘InÞðMh∘AÞx þ g^ðxÞ (denoted by g•ðh � zÞ≗ðMg∘InÞðMh∘AÞx þ g^ðxÞ).

#### Proof.

As an extension of the local stable-unstable manifold theorem in terms of Tracy-Singh product of matrices in Λ<sup>N</sup> and the matrix A of the vector field fðxÞ, we present the following proposi-

nonlinear system x\_ ¼ fðxÞ ¼ Ax þ gðxÞ. We suppose that fð0Þ ¼ 0 and that A ¼ Dfð0Þ have a k eigenvalues with negative real part and n−k eigenvalues with positive real part; thus, the origin

<sup>ð</sup>EÞ ! <sup>C</sup><sup>1</sup>

such that <sup>F</sup>ðgÞ ¼ <sup>g</sup>^ where <sup>g</sup>^ : <sup>E</sup>⊂Rmn ! <sup>R</sup>mn is a fixed continuously differentiable function with

Then, for each matrix M∈Λ<sup>U</sup> of m · m, there exists a mk− dimensional differentiable manifold

where φ<sup>M</sup>∘A,<sup>t</sup> be the flow of the nonlinear system x\_ ¼ ðM∘AÞx þ g^ðxÞ and there exists an mðn−kÞ dimensional differentiable manifold WM∘<sup>A</sup> tangent to the unstable subspace EW

2. Also, there exists a function of the group Λ<sup>N</sup> and the set of all the autonomous hyperbolic nonlinear systems of dimension n (hyperbolic vector fields of dimension n) denoted by Γn, to the set Γmn of all the autonomous hyperbolic nonlinear systems of dimension mn (hyperbolic vector fields of dimension mn); this function (which is similar to an action of the group Λ<sup>N</sup> on

ϑ : Λ<sup>N</sup> · Γ<sup>n</sup> ! Γmn

lim<sup>t</sup>!<sup>∞</sup> <sup>φ</sup><sup>M</sup>∘A,<sup>t</sup>

lim<sup>t</sup>!−<sup>∞</sup> <sup>φ</sup><sup>M</sup>∘A,<sup>t</sup>

is a hyperbolic fixed point. Now, take a fixed continuously differentiable function

F : C<sup>1</sup>

‖g^ðxÞ‖<sup>2</sup> ‖x‖<sup>2</sup>

ðEÞ and let φ<sup>t</sup> be the flow of the

ðEÞ (13)

<sup>ð</sup>E<sup>Þ</sup> with <sup>E</sup> an open subset of <sup>R</sup><sup>n</sup> containing the origin such

! 0 as ‖x‖<sup>2</sup> ! 0: (14)

ðx0Þ ¼ 0, (15)

ðx0Þ ¼ 0: (16)

¼ ðM∘AÞ<sup>x</sup> <sup>þ</sup> <sup>g</sup>^ðx<sup>Þ</sup> (17)

<sup>M</sup>∘<sup>A</sup> of

<sup>M</sup>∘<sup>A</sup> of the linear system x\_ ¼ ðM∘AÞx at 0 such that for all

ðWM∘<sup>A</sup>Þ⊂WM∘<sup>A</sup> and for all x0∈WM∘<sup>A</sup>,

1. Let E be an open subset of R<sup>n</sup> containing the origin, let f∈C<sup>1</sup>

<sup>ð</sup>EÞ; moreover, <sup>g</sup>^∈C<sup>1</sup>

62 Dynamical Systems - Analytical and Computational Techniques

ðSM∘<sup>A</sup>Þ⊂SM∘<sup>A</sup> and for all x0∈SM∘<sup>A</sup>,

ϑ 

M, Ax þ gðxÞ

SM∘<sup>A</sup> tangent to the stable subspace E<sup>S</sup>

x\_ ¼ ðM∘AÞx at 0 such that for all t≤0, φ<sup>M</sup>∘A,<sup>t</sup>

the set Γn) is defined as follows

and the new nonlinear system is

tion.

Proposition 3.2.

domain all C<sup>1</sup>

t≥0, φ<sup>M</sup>∘A,<sup>t</sup>

that

1. Consider a matrix A with eigenvalues λ<sup>i</sup> for i ¼ 1, 2, …, n and the matrix M with eigenvalues μ<sup>j</sup> for j ¼ 1, 2, …, m. Then, the eigenvalues of the matrix M∘A are the mn numbers λiμ<sup>j</sup> and taking account that μ<sup>j</sup> > 0 for each j ¼ 1, 2, …, m. Therefore, the matrix M∘A has mk eigenvalues with negative real part and mðn−kÞ eigenvalues with positive real part. For this, the result is a consequence of the stable-unstable manifold theorem.

2. The function <sup>ϑ</sup> : <sup>Λ</sup><sup>N</sup> · <sup>Γ</sup><sup>n</sup> ! <sup>Γ</sup>mn is well defined, since <sup>F</sup> : <sup>C</sup><sup>1</sup> <sup>ð</sup>EÞ ! <sup>C</sup><sup>1</sup> ðEÞ is a fixed function; then given gðxÞ, the vector field g^ðxÞ is unique; for a fixed matrix Mh∈ΛN, then Mh<sup>∘</sup> : <sup>R</sup><sup>n</sup> · <sup>n</sup> ! <sup>R</sup>mn · mn is a fixed function and their matrix Mh∘<sup>A</sup> is unique.

Axiom (i): Since Λ<sup>N</sup> is a multiplicative group if Mg, Mh∈ΛN, then MgMh∈ΛN.

Then, by Theorem 3.1, we have that for all g, h∈Λ<sup>N</sup> and z∈Γ<sup>n</sup>

$$(gh) \cdot z \stackrel{\circ}{=} (M\_{\mathcal{S}} M\_{\hbar} \bullet A) \mathbf{x} + \hat{\mathbf{g}}(\mathbf{x}) = (M\_{\mathcal{S}} \bullet I\_{\pi})(M\_{\hbar} \bullet A) \mathbf{x} + \hat{\mathbf{g}}(\mathbf{x}) \stackrel{\circ}{=} \mathbf{g} \bullet (\hbar \cdot z) \tag{19}$$

Axiom (ii): For every z∈Γn, there exists an unique ^z∈Γmn such that e � z≗ðIm∘AÞx þ g^ðxÞ ¼ ^z, then by the Theorem 2.1

$$(h \bullet \hat{\mathbf{z}} \stackrel{\circ}{=} (M\_h \bullet I\_n)(I\_m \bullet A)\mathbf{x} + \hat{\mathbf{g}}(\mathbf{x}) = (M\_h \bullet A)\mathbf{x} + \hat{\mathbf{g}}(\mathbf{x}) \stackrel{\circ}{=} h \cdot \mathbf{z} \tag{20}$$

From what it has been said above, we can note that if A ¼ Dfð0Þ is as stable matrix A, it has all the n eigenvalues with negative real part, then the origin of the nonlinear system x\_ ¼ ðM∘AÞx þ g^ðxÞ is asymptotically stable; if A ¼ Dfð0Þ is an unstable matrix A, it has n−kðn > kÞ eigenvalues with positive real part, then the origin of the nonlinear system x\_ ¼ ðM∘AÞx þ g^ðxÞ is unstable.

Now the following Proposition 3.2 is an extension of the center manifold theorem, similar to Proposition 3.1.

Proposition 3.3. Let be f∈C<sup>r</sup> <sup>ð</sup>E<sup>Þ</sup> where <sup>E</sup> is an open subset of <sup>R</sup><sup>n</sup>

containing the origin and r≥1. Suppose that fð0Þ ¼ 0 and that Dfð0Þ have k eigenvalues with negative real part, j eigenvalues with positive real part and l ¼ n−k−j eigenvalues with zero real part. Then,


$$
\hat{F}: \mathbb{C}'(E) \to \mathbb{C}'(\overline{E})\tag{21}
$$

such that <sup>F</sup>ðgÞ ¼ <sup>g</sup>^ where <sup>g</sup>^ : <sup>E</sup>⊂Rmn ! <sup>R</sup>mn is a fixed continuously differentiable function with domain all Cr <sup>ð</sup>EÞ; moreover, <sup>g</sup>^∈C<sup>r</sup> <sup>ð</sup>E<sup>Þ</sup> with <sup>E</sup> an open subset of <sup>R</sup><sup>n</sup> containing the origin such that

$$\frac{\|\hat{\mathbf{g}}(\mathbf{x})\|\_{2}}{\|\mathbf{x}\|\_{2}} \to 0 \text{ as } \|\mathbf{x}\|\_{2} \to 0. \tag{22}$$

Then for each matrix M∈Λ<sup>N</sup> of m · m, there exists a ml− dimensional differentiable center manifold W<sup>C</sup> <sup>M</sup>∘<sup>A</sup>ð0<sup>Þ</sup> tangent to the center subspace <sup>E</sup><sup>S</sup> <sup>M</sup>∘<sup>A</sup> of the linear system x\_ ¼ ðM∘AÞx at 0 which is invariant under the flow φ<sup>M</sup>∘A,<sup>t</sup> of the nonlinear system x\_ ¼ ðM∘AÞx þ g^ðxÞ.

#### Proof.

The proof is similar to proof of Proposition 3.1 and we make use of the center manifold theorem.

Also, there exists a similar function ϑ^ to ϑ, which satisfies the axiom (i) and axiom (ii) of Proposition 3.2. However, in this case, there does not exist similar functions to Θ and Φ. due to that in general, a center manifold is not unique.

Notice that in this case, if the matrix A has l ¼ n−k−j≠0 eigenvlues with zero real part, then the origin of the nonlinear system x\_ ¼ MAx þ g^ðxÞ and x\_ ¼ ðM∘AÞx þ g^ðxÞ are not asymptotically stable.

Propositions 3.1 and 3.2 generalize Proposition 3 in Ref. [6] and give new tools for preservation of basic properties of dynamical systems and some of these properties are the stability and instability.

## 4. Synchronization in nonlinear dynamical system

In this section, we present that it is possible to preserve synchronization even though the dimension of the systems changes by the action of a class of transformation on the linear part to a chaotic nonlinear system. If we consider the following two n-dimensional chaotic systems, Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix http://dx.doi.org/10.5772/66957 65

$$\begin{array}{l}\dot{\mathbf{x}} = A\mathbf{x} + \mathbf{g}(\mathbf{x})\\\dot{\mathbf{y}} = Ay + f(\mathbf{y}) + \boldsymbol{\mu}(t)\end{array} \tag{23}$$

Where A∈R<sup>n</sup> · <sup>n</sup> is a constant matrix. On the other hand,u∈Rn is the control input and <sup>f</sup> , <sup>g</sup> : <sup>R</sup><sup>n</sup> ! <sup>R</sup><sup>n</sup> are continuous nonlinear functions. Synchronization considered in this section is through the master and the slave system is synchronized by designing an appropriate nonlinear state-feedback control uðtÞ attached to slave system such that lim<sup>t</sup>!<sup>∞</sup> xðtÞ−yðtÞ ! 0, where ‖ � ‖ is the Euclidean norm of a vector [8]. If we consider the error state vector <sup>e</sup> <sup>¼</sup> <sup>y</sup>−x∈Rn, <sup>f</sup>ðyÞ−fðxÞ ¼ <sup>L</sup>ðx, <sup>y</sup><sup>Þ</sup> and an error dynamics equation is <sup>e</sup>\_ <sup>¼</sup> Ae <sup>þ</sup> <sup>L</sup>ðx, <sup>y</sup>Þ þ <sup>u</sup>ðtÞ. Taking the active control approach [5], to eliminate the nonlinear part of the error dynamics and choosing uðtÞ ¼ BvðtÞ−Lðx, yÞ, where B is a constant gain vector which is selected such that ðA, BÞ be controllable, we obtain:

Proposition 3.3. Let be f∈C<sup>r</sup>

64 Dynamical Systems - Analytical and Computational Techniques

with domain all Cr

origin such that

manifold W<sup>C</sup>

Proof.

theorem.

instability.

part. Then,

W<sup>C</sup>

<sup>ð</sup>E<sup>Þ</sup> where <sup>E</sup> is an open subset of <sup>R</sup><sup>n</sup>

containing the origin and r≥1. Suppose that fð0Þ ¼ 0 and that Dfð0Þ have k eigenvalues with negative real part, j eigenvalues with positive real part and l ¼ n−k−j eigenvalues with zero real

1. For each matrix M∈ΛU, there exists a m− dimensional differentiable center manifold

at 0 which is invariant under the flow φM,<sup>t</sup> of the nonlinear system x\_ ¼ MAx þ gðxÞ.

<sup>ð</sup>EÞ ! <sup>C</sup><sup>r</sup>

such that <sup>F</sup>ðgÞ ¼ <sup>g</sup>^ where <sup>g</sup>^ : <sup>E</sup>⊂Rmn ! <sup>R</sup>mn is a fixed continuously differentiable function

Then for each matrix M∈Λ<sup>N</sup> of m · m, there exists a ml− dimensional differentiable center

The proof is similar to proof of Proposition 3.1 and we make use of the center manifold

Also, there exists a similar function ϑ^ to ϑ, which satisfies the axiom (i) and axiom (ii) of Proposition 3.2. However, in this case, there does not exist similar functions to Θ and Φ. due to

Notice that in this case, if the matrix A has l ¼ n−k−j≠0 eigenvlues with zero real part, then the origin of the nonlinear system x\_ ¼ MAx þ g^ðxÞ and x\_ ¼ ðM∘AÞx þ g^ðxÞ are not asymptotically stable. Propositions 3.1 and 3.2 generalize Proposition 3 in Ref. [6] and give new tools for preservation of basic properties of dynamical systems and some of these properties are the stability and

In this section, we present that it is possible to preserve synchronization even though the dimension of the systems changes by the action of a class of transformation on the linear part to a chaotic nonlinear system. If we consider the following two n-dimensional chaotic systems,

which is invariant under the flow φ<sup>M</sup>∘A,<sup>t</sup> of the nonlinear system x\_ ¼ ðM∘AÞx þ g^ðxÞ.

F^ : Cr

<sup>M</sup> of the linear system x\_ ¼ MAx þ gðxÞ

ðEÞ (21)

<sup>M</sup>∘<sup>A</sup> of the linear system x\_ ¼ ðM∘AÞx at 0

<sup>ð</sup>E<sup>Þ</sup> with <sup>E</sup> an open subset of <sup>R</sup><sup>n</sup> containing the

! 0 as ‖x‖<sup>2</sup> ! 0: (22)

<sup>M</sup>ð0<sup>Þ</sup> of class Cr tangent to the center subspace EC

<sup>ð</sup>EÞ; moreover, <sup>g</sup>^∈C<sup>r</sup>

<sup>M</sup>∘<sup>A</sup>ð0<sup>Þ</sup> tangent to the center subspace <sup>E</sup><sup>S</sup>

4. Synchronization in nonlinear dynamical system

that in general, a center manifold is not unique.

‖g^ðxÞ‖<sup>2</sup> ‖x‖<sup>2</sup>

2. If taken a fixed continuously differentiable function

$$
\dot{e} = Ae + Bv(t) \tag{24}
$$

We can see that the original synchronization problem is equivalent to stabilize the zero-input solution of the slave system through a suitable choice of the state-feedback control [8]. If the pair ðA, BÞ is controllable, then one such suitable choice for state feedback is a linear-quadratic regulator [5], which minimizes the quadratic cost function in the next expression,

$$J\left(\mu(t)\right) = \bigcap\_{0}^{\bullet} (e(t)^{\mathsf{I}} Qe(t) + \upsilon(t) R\upsilon(t))dt \tag{25}$$

Where Q and R are positive semi-definite and positive definite weighting matrices, respectively. The state-feedback law is given by <sup>v</sup> <sup>¼</sup> <sup>−</sup>Ke with <sup>K</sup> <sup>¼</sup> <sup>R</sup><sup>−</sup><sup>1</sup> B⊺ S and S the solution to the Riccati equation

$$A^\text{I'}S + SA \text{--} SBR^{-1}B^\text{I} + Q = 0\tag{26}$$

This state-feedback law makes the error equation to be e\_ ¼ ðA−BKÞe, with ðA−BKÞ a Hurwitz matrix.1 The linear-quadratic regulator is a technique to obtain feedback gains [5]. It is an interesting property of (LQR) which is robustness. On the other hand, if we consider T∈Rm · <sup>m</sup> be a matrix with strictly positive eigenvalues, supposing that the following two nm-dimensional systems are chaotic:

$$\begin{aligned} \dot{\mathbf{x}} &= (T \bullet A)\mathbf{x} + \hat{\mathbf{g}}(\mathbf{x}) \\ \dot{\mathbf{y}} &= (T \bullet A)\mathbf{y} + \hat{f}(\mathbf{y}) + \hat{u}(t) \end{aligned} \tag{27}$$

for some ^<sup>f</sup> , <sup>g</sup>^ : <sup>R</sup>nm ! Rnm continuous nonlinear functions and <sup>u</sup>^∈Rnm is the control input. Then, for the Proposition 4.1 and the former procedure, we have that <sup>u</sup>^ðtÞ ¼ <sup>θ</sup>^ðtÞ−L^ðx, <sup>y</sup><sup>Þ</sup> stabilizes the zero solution of the error dynamics system, where <sup>θ</sup>^ðtÞ ¼ <sup>−</sup>ðBK∘TÞe, that is, the resultant system

<sup>1</sup> A Hurwitz matrix is a matrix for which all its eigenvalues are such that their real part is strictly less than zero.

$$
\dot{e} = (T \bullet A)e + \hat{\theta} \left( t \right) \dot{e} = (T \bullet A - T \bullet BK)e \tag{28}
$$

is asymptotically stable. Then, by using Lemma 2.1 and <sup>K</sup> <sup>¼</sup> <sup>−</sup>R<sup>−</sup><sup>1</sup> B⊺ S, we obtain that:

$$\begin{aligned} \dot{\varepsilon} &= \left( T \bullet (A + BK) \right) \varepsilon \\ \dot{e} &= \left( T \bullet (A - BR^{-1}B^{\dagger}S) \right) e \end{aligned} \tag{29}$$

Now, the original control uðtÞ ¼ BKe−Lðx, yÞ is preserved in its linear part by the Tracy-Singh product <sup>T</sup>∘ð�Þ and the new control is given by <sup>u</sup>^ðtÞ ¼ <sup>−</sup>ðT∘BKÞe−L^ðx, <sup>y</sup>Þ. Therefore, we can interpreted the last procedure as one in which the controller uðtÞ that achieves the synchronization in the two systems is preserved by the transformation T∘ð�Þ so that u^ðtÞ achieves the synchronization in the two resultant systems after the transformation. For that, a similar procedure is possible if we consider the transformation ð�Þ∘T.

In general, under the transformation ðA, gÞ!ðMA, gÞ or ðA, gÞ!ðM∘A, gÞ and under the hypothesis of existence of a constant state feedback U ¼ −Kx, which achieves synchronization of the original chaotic systems and also that the transformed system is chaotic, then synchronization can be preserved [8]. The major contribution does not refer a better synchronization methodology; it deals that synchronization is preserved when a chaotic system changes from a lower dimension to a higher dimension.

## 5. Synchronization of the classical Lü system

In this section, we present the synchronization of a chaotic system. First, we propose a master and slave system. Then, from these systems, we will apply a linear transformation that allows us to preserve the synchronization. We will use the well-known Lü and Chen [11] model to show the possibility to preserve synchronization, described by

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= a(\mathbf{x}\_2 - \mathbf{x}\_1) \\ \dot{\mathbf{x}}\_2 &= c\mathbf{x}\_2 - \mathbf{x}\_1 \mathbf{x}\_3 \\ \dot{\mathbf{x}}\_3 &= \mathbf{x}\_1 \mathbf{x}\_2 - b\mathbf{x}\_3 \end{aligned} \tag{30}$$

which has a chaotic attractor when the parameters are a ¼ 35, b ¼ 3 and c ¼ 14:5. In order to observe synchronization behavior, we have a modified Lü attractor arranged as a master-slave configuration. The master and the slave systems are almost identical and the only difference is that the slave system includes an extra term which is used for the purpose of synchronization with the master system. The master system is defined by the following equations,

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= 35(\mathbf{x}\_2 - \mathbf{x}\_1) \\ \dot{\mathbf{x}}\_2 &= 28\mathbf{x}\_2 - \mathbf{x}\_1 \mathbf{x}\_3 \\ \dot{\mathbf{x}}\_3 &= \mathbf{x}\_1 \mathbf{x}\_2 - 3\mathbf{x}\_3 \end{aligned} \tag{31}$$

and the slave system is a copy of the master system with a control function uðtÞ to be determined in order to synchronize the two systems.

Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix http://dx.doi.org/10.5772/66957 67

$$\begin{array}{l} \dot{y}\_1 = 35(y\_2 - y\_1) + \nu\_1(t) \\ \dot{y}\_2 = 28y\_2 - y\_1y\_3 + \nu\_2(t) \\ \dot{y}\_3 = y\_1y\_2 - 3y\_3 + \nu\_3(t) \end{array} \tag{32}$$

Now, we consider the errors e<sup>1</sup> ¼ x1−y1,e<sup>2</sup> ¼ x2−y<sup>2</sup> and e<sup>3</sup> ¼ x3−y3,; then, the error dynamics can be written as:

$$\begin{aligned} \dot{e}\_1 &= 35(e\_2 - e\_1) + \mu\_1(t) \\ \dot{e}\_2 &= 28e\_2 - y\_1 y\_3 + \mathbf{x}\_1 \mathbf{x}\_3 + \mu\_2(t) \\ \dot{e}\_3 &= y\_1 y\_2 - \mathbf{x}\_1 \mathbf{x}\_2 - 3e\_3 + \mu\_3(t) \end{aligned} \tag{33}$$

If we introduce the matrices

<sup>e</sup>\_ ¼ ðT∘AÞ<sup>e</sup> <sup>þ</sup> <sup>θ</sup>^ <sup>ð</sup>tÞe\_ ¼ ðT∘A−T∘BKÞ<sup>e</sup> (28)

B⊺

S, we obtain that:

(29)

(30)

(31)

is asymptotically stable. Then, by using Lemma 2.1 and <sup>K</sup> <sup>¼</sup> <sup>−</sup>R<sup>−</sup><sup>1</sup>

66 Dynamical Systems - Analytical and Computational Techniques

procedure is possible if we consider the transformation ð�Þ∘T.

5. Synchronization of the classical Lü system

show the possibility to preserve synchronization, described by

determined in order to synchronize the two systems.

lower dimension to a higher dimension.

e\_ ¼ 

e\_ ¼  T∘ðA þ BKÞ

<sup>T</sup>∘ðA−BR<sup>−</sup><sup>1</sup>

Now, the original control uðtÞ ¼ BKe−Lðx, yÞ is preserved in its linear part by the Tracy-Singh product <sup>T</sup>∘ð�Þ and the new control is given by <sup>u</sup>^ðtÞ ¼ <sup>−</sup>ðT∘BKÞe−L^ðx, <sup>y</sup>Þ. Therefore, we can interpreted the last procedure as one in which the controller uðtÞ that achieves the synchronization in the two systems is preserved by the transformation T∘ð�Þ so that u^ðtÞ achieves the synchronization in the two resultant systems after the transformation. For that, a similar

In general, under the transformation ðA, gÞ!ðMA, gÞ or ðA, gÞ!ðM∘A, gÞ and under the hypothesis of existence of a constant state feedback U ¼ −Kx, which achieves synchronization of the original chaotic systems and also that the transformed system is chaotic, then synchronization can be preserved [8]. The major contribution does not refer a better synchronization methodology; it deals that synchronization is preserved when a chaotic system changes from a

In this section, we present the synchronization of a chaotic system. First, we propose a master and slave system. Then, from these systems, we will apply a linear transformation that allows us to preserve the synchronization. We will use the well-known Lü and Chen [11] model to

> x\_ <sup>1</sup> ¼ aðx2−x1Þ x\_ <sup>2</sup> ¼ cx2−x1x<sup>3</sup> x\_ <sup>3</sup> ¼ x1x2−bx<sup>3</sup>

which has a chaotic attractor when the parameters are a ¼ 35, b ¼ 3 and c ¼ 14:5. In order to observe synchronization behavior, we have a modified Lü attractor arranged as a master-slave configuration. The master and the slave systems are almost identical and the only difference is that the slave system includes an extra term which is used for the purpose of synchronization

> x\_ <sup>1</sup> ¼ 35ðx2−x1Þ x\_ <sup>2</sup> ¼ 28x2−x1x<sup>3</sup> x\_ <sup>3</sup> ¼ x1x2−3x<sup>3</sup>

and the slave system is a copy of the master system with a control function uðtÞ to be

with the master system. The master system is defined by the following equations,

 e

B⊺ SÞ e

$$A = \begin{pmatrix} -35 & 35 & 0 \\ 0 & 14.5 & 0 \\ 0 & 0 & -3 \end{pmatrix}, L(x, y) = \begin{pmatrix} 0 \\ -y\_1 y\_3 + \mathbf{x}\_1 \mathbf{x}\_3 \\ y\_1 y\_2 \mathbf{-} \mathbf{x}\_1 \mathbf{x}\_2 \end{pmatrix}, \mathbf{u} = \begin{pmatrix} u\_1(t) \\ u\_2(t) \\ u\_3(t) \end{pmatrix}. \tag{34}$$

and selecting the matrix B such that ðA, BÞ is controllable: B ¼ I, the LQR controller is obtained by using weighting matrices <sup>Q</sup> <sup>¼</sup> <sup>I</sup> and <sup>R</sup> <sup>¼</sup> <sup>B</sup><sup>⊺</sup> B ¼ I. Then, state-feedback matrix is given by

$$K = \begin{pmatrix} 0.0143 & 0.0101 & 0 \\ 0.0101 & 29.0587 & 0 \\ 0 & 0 & 0.1623 \end{pmatrix} \tag{35}$$

From the formerly said, we now present simulations made for the synchronized system of Lü and for the system also synchronized, but after the transformation of its linear part. All simulations here presented were made in Matlab software. In Figure 1, we show the trajectories of the master system of Lü. Each line represents one trajectory of the system along the time, taking an initial condition of ð1, 1, 1Þ.

For the case of Figure 3, we show the trajectories of the slave system of Lü. As it was in the first case, each line represents one trajectory of the system along the time, taking a initial condition as ð3, 3, 3Þ. Figures 2 and 4 are phase space mappings of each system while maintaining the same initial condition.

On the other hand, in Figure 5, we can see the error magnitude between master and slave systems. Phase space of synchronization of the master and slave systems in Figure 6 is presented. Now, we shall present a system showing modifications performed on the Lü attractor. The modified Lü master and slave systems linear and nonlinear parts may be defined as follows:

$$\begin{array}{l}\dot{\mathbf{x}} = (T \bullet A)\mathbf{x} + \begin{bmatrix} 0 & -\mathbf{x}\_1 \mathbf{x}\_3 & \mathbf{x}\_1 \mathbf{x}\_2 \mathbf{0} & -\mathbf{x}\_4 \mathbf{x}\_6 & \mathbf{x}\_4 \mathbf{x}\_5 \end{bmatrix}\mathbf{I} \\\dot{y} = (T \bullet A)y + \begin{bmatrix} 0 & -y\_1 y\_3 & y\_1 y\_2 \mathbf{0} & -y\_4 y\_6 & y\_4 y\_5 \end{bmatrix}\mathbf{I} + \mu\end{array} \tag{36}$$

Considering the error vector e ¼ y−x, then the error dynamics can be written as:

$$
\dot{e} = (T \bullet A)e + L(\mathbf{x}, y) + u \tag{37}
$$

with u ¼ −Lðx, yÞ þ v and v ¼ −ðT∘BKÞe and

Figure 1. Master system of Lü.

Figure 2. Master system of Lü.

Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix http://dx.doi.org/10.5772/66957 69

Figure 3. Slave system of Lü.

Figure 1. Master system of Lü.

68 Dynamical Systems - Analytical and Computational Techniques

Figure 2. Master system of Lü.

Figure 4. Slave system of Lü.

Figure 5. Magnitude of the error between the master and the slave systems.

Figure 6. Synchronization of master and slave system of Lü.

Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix http://dx.doi.org/10.5772/66957 71

$$\begin{aligned} A &= \begin{pmatrix} -35 & 35 & 0 \\ 0 & 14.5 & 0 \\ 0 & 0 & -3 \end{pmatrix}, T = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, B = [111111]^\top, \\ L(\mathbf{x}, y) &= \begin{bmatrix} 0 & -y\_1 y\_3 + \mathbf{x}\_1 \mathbf{x}\_3 & y\_1 y\_2 - \mathbf{x}\_1 \mathbf{x}\_2 \ 0 & -y\_4 y\_6 + \mathbf{x}\_4 \mathbf{x}\_6 & y\_4 y\_5 - \mathbf{x}\_4 \mathbf{x}\_5 \end{bmatrix}^\top \end{aligned} \tag{38}$$

Now, the LQR controller is obtained by using weighting matrices, <sup>B</sup> <sup>¼</sup> IQ <sup>¼</sup> <sup>I</sup> and <sup>R</sup> <sup>¼</sup> <sup>B</sup><sup>⊺</sup> B ¼ I. So the vector Lðx, yÞ takes these values because T is an upper triangular matrix and the value one on the diagonal is repeated.

$$T \circ A = \begin{pmatrix} -35 & 35 & -35 & 35 & 0 & 0 \\ 0 & 14.5 & 0 & 14.5 & 0 & 0 \\ 0 & 0 & -35 & 35 & 0 & 0 \\ 0 & 0 & 0 & 14.5 & 0 & 0 \\ 0 & 0 & 0 & 0 & -3 & -3 \\ 0 & 0 & 0 & 0 & 0 & -3 \end{pmatrix} \tag{39}$$

$$K = \begin{pmatrix} 0.0143 & 0.0101 & 0 & -0.0071 & 0.0050 & 0 \\ 0.0101 & 23.3051 & 0 & -0.0151 & 11.5941 & 0 \\ 0 & 0 & 0.1614 & 0 & 0 & -0.0757 \\ -0.0071 & -0.0151 & 0 & 0.0214 & 0.0050 & 0 \\ 0.0050 & 11.5941 & 0 & 0.0050 & 34.8411 & 0 \\ 0 & 0 & -0.0757 & 0 & 0 & 0.2324 \end{pmatrix} \tag{40}$$

Figure 7. Transformation of the master system of Lü.

Figure 5. Magnitude of the error between the master and the slave systems.

70 Dynamical Systems - Analytical and Computational Techniques

Figure 6. Synchronization of master and slave system of Lü.

Figure 8. Phase space of the transformation of the master system of Lü.

Figure 9. Transformation of the slave system of Lü.

Preservation of Synchronization Using a Tracy‐Singh Product in the Transformation on Their Linear Matrix http://dx.doi.org/10.5772/66957 73

Figure 10. Phase space of the transformation of the slave system of Lü.

Figure 8. Phase space of the transformation of the master system of Lü.

72 Dynamical Systems - Analytical and Computational Techniques

Figure 9. Transformation of the slave system of Lü.

Figure 11. Magnitude of the error between the transformation of master and slave systems.

Figure 12. Synchronization of the transformation of the master and slave systems of Lü.

After the transformation in its linear part of Lü attractor, we also have several simulations allowing us to analyze the dynamics of the transformed system. In Figure 7, we present the trajectories of the transformation of the master system of Lü. Each line represents one trajectory of the system along with the time taking an initial condition of ð0:5, 0:5, 0:5, 0:5, 0:5, 0:5Þ. For the case of Figure 9, we show the trajectories of the transformation of the slave system of Lü. Each line represents one trajectory of the system also, along the time, taking an initial condition of ð3, 3, 3, 3, 3, 3Þ. Figures 8 and 10 are the phase space mappings of each transformed system while maintaining the same initial condition. By last, in Figure 11, we can see the error magnitude of the transformation of synchronized system. A phase space mapping of the transformation of synchronized system is presented in Figure 12.

#### 6. Conclusion

We have studied the preservation of stability of a chaotic dynamic system, from an extension of the stable-unstable manifold theorem and an extension of the center manifold theorem based on the preservation of the linear part in nonlinear dynamical systems. However, we can check that given a chaotic system, its transformed version is also chaotic. A scheme consisting of a master-slave system for which a controller gain is obtained using a linearquadratic regulator has been presented and synchronization is achieved and preserved even after the master-slave controller is transformed, obtaining as a consequence that the chaotic system changes to an higher dimension. It is important to note the transformation of the linear part of the chaotic system from Tracy-Singh product in which it was used to modify a Lü system, showing the effectiveness of the proposed method. The results can be extended to other techniques for feedback design, for example, adaptive control, sliding mode regulator and etcetera.

## Author details

Guillermo Fernadez-Anaya<sup>1</sup> \*, Luis Alberto Quezada-Téllez<sup>1</sup> , Jorge Antonio López-Rentería<sup>1</sup> , Oscar A. Rosas-Jaimes<sup>2</sup> , Rodrigo Muñoz-Vega<sup>3</sup> , Guillermo Manuel Mallen-Fullerton<sup>1</sup> and José Job Flores-Godoy<sup>4</sup>

\*Address all correspondence to: guillermo.fernandez@ibero.mx

1 Departamento de Física y Matemáticas, Universidad Iberoamericana, Ciudad de México, México

2 Facultad de Ingeniería, Universidad Autónoma del Estado de México, Toluca, Estado de, México

3 Universidad Autónoma de la Ciudad de México, Ciudad de México, México

4 Departamento de Matemática, Facultad de Ingeniería y Tecnologías, Universidad Católica del Uruguay, Uruguay

## References

After the transformation in its linear part of Lü attractor, we also have several simulations allowing us to analyze the dynamics of the transformed system. In Figure 7, we present the trajectories of the transformation of the master system of Lü. Each line represents one trajectory of the system along with the time taking an initial condition of ð0:5, 0:5, 0:5, 0:5, 0:5, 0:5Þ. For the case of Figure 9, we show the trajectories of the transformation of the slave system of Lü. Each line represents one trajectory of the system also, along the time, taking an initial condition of ð3, 3, 3, 3, 3, 3Þ. Figures 8 and 10 are the phase space mappings of each transformed system while maintaining the same initial condition. By last, in Figure 11, we can see the error magnitude of the transformation of synchronized system. A phase space mapping of the

We have studied the preservation of stability of a chaotic dynamic system, from an extension of the stable-unstable manifold theorem and an extension of the center manifold theorem based on the preservation of the linear part in nonlinear dynamical systems. However, we can check that given a chaotic system, its transformed version is also chaotic. A scheme consisting of a master-slave system for which a controller gain is obtained using a linearquadratic regulator has been presented and synchronization is achieved and preserved even

transformation of synchronized system is presented in Figure 12.

Figure 12. Synchronization of the transformation of the master and slave systems of Lü.

74 Dynamical Systems - Analytical and Computational Techniques

6. Conclusion


Provisional chapter
