Oscillation Criteria of Two-Dimensional Time-Scale Systems

Ozkan Ozturk

#### Abstract

Oscillation and nonoscillation theories have recently gotten too much attention and play a very important role in the theory of time-scale systems to have enough information about the long-time behavior of nonlinear systems. Some applications of such systems in discrete and continuous cases arise in control and stability theories for the unmanned aerial and ground vehicles (UAVs and UGVs). We deal with a two-dimensional nonlinear system to investigate the oscillatory behaviors of solutions. This helps us understand the limiting behavior of such solutions and contributes several theoretical results to the literature.

Keywords: oscillation, nonoscillation, two-dimensional systems, time scale, nonlinear system, fixed point theorems

### 1. Introduction

This chapter analyses the oscillatory behavior of solutions of two-dimensional (2D) nonlinear time-scale systems of first-order dynamic equations. We also investigate the existence and asymptotic properties of such solutions. The tools that we use are the most well-known fixed point theorems to consider the sign of the component functions of solutions of our system. A time scale, denoted by T, is an arbitrary nonempty closed subset of the real numbers R, which is introduced by a German mathematician, Stefan Hilger, in his PhD thesis in 1988 [1]. His primary purpose was to unify continuous and discrete analysis and extend the results to one comprehensive theory. For example, the results hold for differential equations when T ¼ R, while the results hold for difference equations when T ¼ Z. Therefore, there might happen to be two different proofs and maybe similar in most cases. In other words, our essential desire is to combine continuous and discrete cases in one comprehensive theory and remove the obscurity from both. For more details in the theory of differential and difference equations, we refer the books [2–4] to interested readers. As for the time-scale theory, we assume most of the readers are not familiar with the time-scale calculus, and thus we give a concise introduction to the theory of time scales from the books [5, 6] written by Bohner and Peterson in 2001 and 2003, respectively.

Two-dimensional dynamical systems have recently gotten too much attention because of their potential in applications in engineering, biology, and physics (see, e.g., [7–11]). For example, Bartolini and Pvdvnowski [12] consider a nonlinear

system and propose a new method for the asymptotic linearization by means of continuous control law. Also Bartolini et al. [13, 14] consider an uncertain secondorder nonlinear system and propose a new approximate linearization and sliding mode to control such systems. In addition to the nonoscillation for two-dimensional systems of first-order equations, periodic and subharmonic solutions are also investigated in [15–17], and significant contributions have been made. Another type of two-dimensional systems of dynamic equations is the Emden-Fowler type equation, named after E. Fowler after he did the mathematical foundation of a second-order differential equation in a series of four papers during 1914–1931 (see [18–21]). This system has several fascinating applications such as in gas dynamics and fluid mechanics, astrophysics, nuclear physics, relativistic mechanics, and chemically reacting systems (see [9, 22–24]).

This chapter is organized as follows: In Section 2, we give the calculus of the time-scale theory for those who are not familiar with the time scale (see [5]). In Section 3, referred to [25, 26], we show the existence and asymptotic behaviors of nonoscillatory solutions of a two-dimensional homogeneous dynamical system on time scales by using improper integrals and some inequalities. We also give enough examples for readers to see our results work nicely. Section 4, referred to [27], provides us oscillation criteria for two-dimensional nonhomogeneous time-scale systems by using famous inequalities and rules such as comparison theorem and chain rules on time scales. Finally, we give a conclusion and provide some exercises to the readers to have them comprehend the main results in the last two sections.

#### 2. Preliminaries

The examples of the time scales are not restricted with the set of real numbers R and the set of integers Z. There are several other time scales which are used in many ˜ ° application areas such as <sup>q</sup>N<sup>0</sup> <sup>¼</sup> <sup>1</sup>; <sup>q</sup>; <sup>q</sup><sup>2</sup>; <sup>⋯</sup>; , q . <sup>1</sup> (called <sup>q</sup>-difference equations <sup>2</sup> [28]), <sup>T</sup> <sup>¼</sup> <sup>h</sup>Z, h . 0, <sup>T</sup> <sup>¼</sup> <sup>N</sup><sup>2</sup> <sup>0</sup> <sup>¼</sup> ˜ n : n∈ N<sup>0</sup> ° , etc. On the other hand, the set of rational numbers Q, the set of irrational numbers R\Q, and the open interval ða; bÞ are not time scales since they are not closed subsets of R. For the following definitions and theorems in this section, we refer [5], (Chapter 1), and [29] to the readers.

Definition 2.1 Let T be a time scale. Then, the forward jump operator σ : T ! T is defined by

σð Þt ≔ inffs∈T : s . tg for all t∈ T

while the backward jump operator ρ : T ! T is given by

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

Finally, the graininess function μ : T ! ½0; ∞Þ is defined by μð Þt ≔ σðÞ�t t for all t ∈T:

For a better explanation, the operator σ is the first next point, while the operator ρ is the first back point on a time scale. And μ is the length between the next point and the current point. So it is always nonnegative. Table 1 shows some examples of the forward/backward jump operators and the graininess function for most known time scales.

If t , supT and σðÞ¼ t t, then t is said to be right-dense, and if t . infT and ρðÞ¼ t t, we say t is left-dense. Also, if t is right- and left-dense at the same time, then t is said to be dense. In addition to left and right-dense points, it is said to be right-scattered when σð Þt . t, and t is called left-scattered when ρð Þt , t. Also, if t is

Oscillation Criteria of Two-Dimensional Time-Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.83375

right-and left-scattered at the same time, then t is called isolated. Figure 1 shows the classification of points on time scales, clarifying the operators σ, ρ and μ (see [5]).

Next, we introduce the definition of derivative on any time scale. Note that if sup<sup>T</sup> , <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 f : T ! R is a function. Then f <sup>σ</sup> : <sup>T</sup> ! <sup>R</sup> is defined by f σ ðÞ¼ t fðσð Þt Þ for all t∈T:

Definition 2.2 If there does exist a δ . 0 such that

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

for any ε, then g is called delta differentiable on T<sup>κ</sup> and g<sup>Δ</sup> is said to be delta derivative of g. Sometimes, delta derivative is referred as Hilger derivative in the literature (see [5]).

Theorem 2.3 Suppose that f, <sup>g</sup> : <sup>T</sup> ! <sup>R</sup> is <sup>a</sup> function with <sup>t</sup> <sup>∈</sup>T<sup>κ</sup> . Then.

i. g is said to be continuous at t if g is differentiable at t.

ii. g is differentiable at t and

$$\mathbf{g}^{\Delta}(t) = \frac{\mathbf{g}(\sigma(t)) - \mathbf{g}(t)}{\mu(t)},$$

provided g is continuous at t and t is right-scattered.

iii. Let t be right-dense, then g is differentiable at t if and only if

$$\mathbf{g}^{\Delta}(t) = \lim\_{s \to t} \frac{\mathbf{g}(t) - \mathbf{g}(s)}{t - s}.$$

is equal to a finite number.


Table 1.

Examples of most known time scales.

Figure 1. Classification of points.

iv. If g tð Þgðσð Þ<sup>t</sup> Þ ¼6 0, then <sup>f</sup> is differentiable at <sup>t</sup> with g

$$\left(\frac{f}{\mathfrak{g}}\right)^{\Delta}(t) = \frac{f^{\Delta}(t)\mathfrak{g}(t) - f(t)\mathfrak{g}^{\Delta}(t)}{\mathfrak{g}(t)\mathfrak{g}(\sigma(t))}.$$

If T ¼ R, then f <sup>Δ</sup> turns out to be the usual derivative f 0 on continuous case, while f <sup>Δ</sup> is reduced to forward difference operator <sup>Δ</sup>f, defined by <sup>Δ</sup>f tðÞ¼ <sup>f</sup>ð<sup>t</sup> <sup>þ</sup> <sup>1</sup>Þ � f tð Þ if T ¼ Z: The following example is a good example of time scale applications in electrical engineering (see [5], Example 1.39–1.40).

Example 2.4 Consider a simple electric circuit, shown in Figure 2 with resistor R, inductor L, capacitor C and the current I.

Suppose, we discharge the capacitor periodically every time unit and assume that the discharging small δ . 0 time units. Then we can model it as

$$\mathbb{P}\_{\mathbf{1}-\delta,\delta} = \bigcup\_{k \in \mathbb{M}\_0} [k, k+\mathbf{1} - \delta]$$

by using the time scale. Suppose that Q tð Þ is the total charge on the capacitor at time t and I tð ) is the current with respect to time t. Then the total charge Q can be defined by

$$Q^\Delta(t) = \begin{cases} bQ(t) & \text{if } \quad t \in \bigcup\_{k \in \mathbb{N}} \{k - \delta\}, \\\\ I & \text{otherwise} \end{cases}$$

and

$$I^{\Delta}(t) = \begin{cases} 0 & \text{if } \quad t \in \bigcup\_{k \in \mathbb{N}} \{k - \delta\}, \\\\ -\frac{1}{LC}Q(T) - \frac{R}{L}I(t) & \text{otherwise,} \end{cases}$$

where �1 , bδ , 0.

Finally, we introduce the integrals on time scales, but before that, we must give the following definition to define delta integrable functions (see [5]).

Definition 2.5 g : T ! R is said to be right-dense continuous (rd-continuous) if its left-sided limits exist at left-dense points in T and it is continuous at right-dense points in T. We denote rd-continuous functions by CrdðT; RÞ. The set of functions g that are differentiable and whose derivative is rd-continuous is denoted by <sup>C</sup><sup>1</sup> <sup>ð</sup>T; <sup>R</sup>Þ. rd Finally, we denote continuous functions by C throughout this chapter.

Figure 2. Electric circuit.

Theorem 2.6 ([5], Theorem 1.60) For σ : T ! T and f : T ! R, we have the following:

i. The jump operator σ is rd-continuous.

ii. If f is continuous, then it is rd-continuous.

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 Every rd-continuous function has an antiderivative. Moreover, F given by

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

is an antiderivative of f.

Similar to the continuous analysis, we have integral properties and some of them are presented as follows ([5] or [29]):

Theorem 2.8 Suppose that h<sup>1</sup> and h<sup>2</sup> are rd-continuous functions, c, d,e∈T and β ∈ R.


$$\text{v. } \int\_{\mathcal{c}}^{d} h\_1(t) h\_2^{\Delta}(t) \Delta t = (h\_1 h\_2)(d) - (h\_1 h\_2)(\mathcal{c}) - \int\_{\mathcal{c}}^{d} h\_1^{\Delta}(t) h\_2(\sigma(t)) \Delta t$$

<sup>R</sup> <sup>a</sup> vi. <sup>f</sup>ð Þ<sup>t</sup> <sup>Δ</sup><sup>t</sup> <sup>¼</sup> 0. <sup>a</sup>

Table 2 shows how the derivative and integral are defined for some time scales for a, b∈T.


Table 2.

Derivative and integrals for most common time scales.

We finish the section by Schauder's fixed point theorem, proved by Juliusz Schauder in 1930, and Knaster fixed point theorem, proved by Knaster in 1928 (see [30], Theorem 2.A and [31], respectively).

Theorem 2.9 Schauder's fixed point theorem. Suppose that S is a nonempty, bounded, closed, and convex subset of a Banach space Y and that F : S ! S is a compact operator. Then, we conclude that F has a fixed point such that y ¼ Fy.

Theorem 2.10 The Knaster fixed point theorem. Suppose that ðS; ≤ Þ is a complete lattice and that F : S ! S is order preserving, then F has a fixed point such that y ¼ Fy. In fact, we say that the set of fixed points of F is a complete lattice.

Finally, we note that throughout this paper, we assume that T is unbounded above and whenever we write t≥ t1, we mean t∈ ½t1; ∞Þ<sup>T</sup> ≔ ½t1; ∞Þ ∩T.

#### 3. Nonoscillation on a two-dimensional time-scale systems

This section focuses on the nonoscillatory solutions of a two-dimensional dynamical system on time scales. To do this, we consider the system

$$\begin{cases} \varkappa^{\Delta}(t) = p(t)f(\mathbf{y}(t)) \\\\ \mathbf{y}^{\Delta}(t) = r(t)\mathbf{g}(\mathbf{x}(t)), \end{cases} \tag{1}$$

� � where p,r∈Crd ½t0; ∞ÞT; R<sup>þ</sup> and f and g are nondecreasing functions such that ufð Þ u . 0 and ugð Þ u . 0 for u 6¼ 0.

By a solution of (1), we mean a collection of functions, where x, <sup>y</sup>∈C<sup>1</sup> ð½t0, <sup>∞</sup>ÞT; RÞ, T <sup>≥</sup>t<sup>0</sup> and <sup>ð</sup>x; <sup>y</sup><sup>Þ</sup> satisfies system (1) for all large <sup>t</sup>≥T: rd

Note that system (1) is reduced to the system of differential equations when the time scale is the set of real numbers R, i.e., f <sup>Δ</sup> <sup>¼</sup> <sup>f</sup> <sup>0</sup> (see [32]). And when T ¼ Z, system (1) turns out to be a system of difference equations, i.e., f <sup>Δ</sup> <sup>¼</sup> <sup>Δ</sup><sup>f</sup> (see [33]). Other versions of system (1), the case T ¼ Z, are investigated by Li et al. [34], Cheng et al. [35], and Marini et al. [36]. More details about the continuous and discrete versions of system (1) are given in the conclusion section.

Definition 3.1 A solution ðx; yÞ of system (1) is said to be proper if

supf∣x sð Þ∣, ∣y sð Þ∣, ∣z sð Þ∣ : s∈½t, ∞ÞTg . 0

holds for t≥t0.

Definition 3.2 A proper solution ðx; yÞ of (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.

Suppose that N is the set of all nonoscillatory solutions of system (1). It can easily be shown that any nonoscillatory solution ðx; yÞ of system (1) belongs to one of the following classes:

> � � � � N<sup>þ</sup> ≔ ðx; yÞ∈ N : xy . 0 eventually N� ≔ ðx; yÞ∈ N : xy , 0 eventually :

Let ðx; yÞ be a solution of system (1). Then one can show that the component functions x and y are themselves nonoscillatory (see, e.g., [37]). Throughout this section, we assume that the first component function x of the nonoscillatory

solution ðx; yÞ is eventually positive. The results can be obtained similarly for the case x , 0 eventually.

We obtain the existence criteria for nonoscillatory solutions of system (1) in N<sup>þ</sup> and N� by using the fixed point theorems and the following improper integrals:

$$\begin{array}{ll} I\_{1} = \int\_{t\_{0}}^{\infty} p(t) f\Big(\mathbb{k}\_{1} \int\_{t\_{0}}^{t} r(s) \Delta s\Big) \Delta t, & I\_{2} = \int\_{t\_{0}}^{\infty} r(t) \Big\lg\left(\mathbb{k}\_{2} \int\_{t\_{0}}^{t} p(s) \Delta s\right) \Delta t, \\\\ I\_{3} = \int\_{t\_{0}}^{\infty} p(t) f\Big(\mathbb{k}\_{3} - \mathbb{k}\_{4} \int\_{t}^{\infty} r(s) \Delta s\Big) \Delta t, & I\_{4} = \int\_{t\_{0}}^{\infty} r(t) \lg\left(\mathbb{k}\_{5} \int\_{t}^{\infty} p(s) \Delta s\right) \Delta t, \\\\ P(t\_{0}, t) = \int\_{t\_{0}}^{t} p(s) \Delta s, & R(t\_{0}, t) = \int\_{t\_{0}}^{t} r(s) \Delta s, \end{array}$$

where ki, i ¼ 1 � 5 are some constants.

#### 3.1 Existence of nonoscillatory solutions of (1) in N<sup>þ</sup>

Suppose that ðx; yÞ is a nonoscillatory solution of (1) such that x . 0. Then system (1) implies that x<sup>Δ</sup> . 0 and y<sup>Δ</sup> . 0 eventually. Therefore, as a result of this, we have that x converges to a positive finite number or x ! ∞ and similarly y tends to a positive finite number or y ! ∞. One can have very similar asymptotic behaviors when x , 0. Hence, as a result of this information, the following subclasses of N<sup>þ</sup> are obtained:

$$\begin{aligned} N\_{F,F}^{+} &= \left\{ (\varkappa, \jmath) \in \mathcal{N}^{+} : \lim\_{t \to \infty} |\varkappa(t)| = c, \quad \lim\_{t \to \infty} |\jmath(t)| = d \right\}, \\\\ N\_{F,\infty}^{+} &= \left\{ (\varkappa, \jmath) \in \mathcal{N}^{+} : \lim\_{t \to \infty} |\varkappa(t)| = c, \quad \lim\_{t \to \infty} |\jmath(t)| = \infty \right\}, \\\\ N\_{\infty,F}^{+} &= \left\{ (\varkappa, \jmath) \in \mathcal{N}^{+} : \lim\_{t \to \infty} |\varkappa(t)| = \infty, \quad \lim\_{t \to \infty} |\jmath(t)| = d \right\}, \\\\ N\_{\infty,\infty}^{+} &= \left\{ (\varkappa, \jmath) \in \mathcal{N}^{+} : \lim\_{t \to \infty} |\varkappa(t)| = \infty, \quad \lim\_{t \to \infty} |\jmath(t)| = \infty \right\}. \end{aligned}$$

To focus on Nþ, first consider the following four cases for t<sup>0</sup> ∈ T :


Suppose P tð <sup>0</sup>; ∞Þ ¼ ∞ and R tð <sup>0</sup>; ∞Þ ¼ ∞ and that ðx; yÞ is a nonoscillatory solution in Nþ: Integrating the equations of system (1) from t<sup>0</sup> to t separately gives us

$$\varkappa(t) \ge \varkappa(t\_0) + f(\varkappa(t\_0)) \int\_{t\_0}^t p(s) \Delta s$$

and

$$\mathbf{y}(t) \ge \mathbf{y}(t\_0) + \mathbf{g}(\mathbf{x}(t\_0)) \int\_{t\_0}^t r(s) \Delta s, \quad t \ge t\_0.s$$

Thus, we get x tðÞ! ∞ and y tðÞ! ∞ as t ! ∞. In view of this information, the following theorem is given without any proof.

Theorem 3.3 Let Pðt0; ∞Þ ¼ ∞ and Rðt0; ∞Þ ¼ ∞. Then any nonoscillatory solution of system (1) belongs to Nþ <sup>∞</sup>,∞.

Next, we consider the other three cases to obtain the nonoscillation criteria for system (1).

## 3.1.1 The case Pðt0; ∞Þ ¼ ∞ and Rðt0; ∞Þ , ∞

Suppose that ðx; yÞ is a nonoscillatory solution of system (1) such that x . 0 and y . 0 eventually. Then by the integration of the first equation of system (1) from t<sup>0</sup> to t, we have that there exists k . 0

$$
\varkappa(t) \ge \varkappa(t\_0) + k \int\_{t\_0}^t p(s) \Delta s, \quad t\_0 \in \mathbb{T}. \tag{2}
$$

Then by taking the limit of (2) as t ! ∞, we have that x diverges. Therefore, we have the following lemma in the light of this information.

Lemma 3.4 Any nonoscillatory solution in N<sup>þ</sup> belongs to N<sup>þ</sup> <sup>∞</sup>,F, or N<sup>þ</sup> <sup>∞</sup>,<sup>∞</sup> for 0 , c, d , ∞.

It is not easy to give the sufficient conditions for the existence of nonoscillatory solutions in N<sup>þ</sup> <sup>∞</sup>,∞. So, we only provide the existence of nonoscillatory solutions in Nþ <sup>∞</sup>,F.

Theorem 3.5 There exists a nonoscillatory solution in N<sup>þ</sup> <sup>∞</sup>,F if and only if I<sup>2</sup> , ∞ for all k<sup>2</sup> . 0.

Proof. Suppose that there exists a solution in N<sup>þ</sup> <sup>∞</sup>,F such that x tð Þ . 0, y tð Þ . 0 for t≥t0, x tðÞ! ∞ and y tðÞ! d as t ! ∞ for d . 0. Since y is eventually increasing, there exist k<sup>2</sup> . 0 and t<sup>1</sup> ≥ t<sup>0</sup> such that fð y tð ÞÞ≥k<sup>2</sup> for t≥ t1. Integrating the first equation from t<sup>1</sup> to t, the monotonicity of f yields us

$$\varkappa(t) = \varkappa(t\_1) + \int\_{t\_1}^t p(s)f(\wp(s))\Delta s \ge k\_2 \int\_{t\_1}^t p(s)\Delta s, \quad t \ge t\_1. \tag{3}$$

Integrating the second equation from t<sup>1</sup> to t, the monotonicity of g and (3) gives us

$$y(\mathbf{t}) = y(\mathbf{t}\_1) + \int\_{t\_1}^{t} r(s)\mathbf{g}(\mathbf{x}(s))\Delta s \geq \int\_{t\_1}^{t} r(s)\mathbf{g}\left(k\_2 \int\_{t\_1}^{s} p(u)\Delta u\right)\Delta s, \quad t \geq t\_1. \tag{4}$$

So as t ! ∞, we have that I<sup>2</sup> , ∞ holds.

Conversely, suppose that I<sup>2</sup> , ∞ for all k<sup>2</sup> . 0. Then, there exists a large t<sup>1</sup> ≥t<sup>0</sup> such that

$$\int\_{t\_1}^{\infty} r(t) \mathbf{g}\left(k\_2 \int\_{t\_1}^{t} p(s) \Delta s\right) \Delta t \le \frac{c}{2},\tag{5}$$

Oscillation Criteria of Two-Dimensional Time-Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.83375

where k<sup>2</sup> ¼ f cð Þ. Let Y be the set of all bounded and continuous real-valued functions y tð Þ on ½t1; ∞Þ<sup>T</sup> with the supremum norm supt≥t<sup>1</sup> ∣y tð Þ∣. Then Y is a Banach space (see [38]). Let us define a subset Ω of Y such that

$$\mathfrak{Q} := \left\{ \mathcal{Y}(t) \in Y : \frac{c}{2} \le \mathcal{Y}(t) \le c, \quad t \ge t\_1 \right\}.$$

One can prove that Ω is bounded, closed, and also convex subset of Y. Suppose that T : Ω ! Y is an operator given by

$$(Ty)(t) = c - \int\_{t}^{\infty} r(s)g\left(\int\_{t\_1}^{s} p(u)f(y(u))\Delta u\right)d\sigma. \tag{6}$$

The very first thing we do is to show that T is mapping into itself, i.e., T : Ω ! Ω.

$$\frac{c}{2} \le c - \int\_t^{\infty} r(s) \lg\left(\int\_{t\_1}^s p(u)f(c)\Delta u\right) \Delta s \le (T\chi)(t) \le c$$

� � by using (5) for y∈ Ω. The second thing we show that T must be continuous on Ω: Hence, for y ∈ Ω, suppose that y is a sequence in Ω so that �y � y� ! 0: Then <sup>n</sup> <sup>n</sup>

$$\begin{aligned} &|(T\mathfrak{y}\_n)(t) - (T\mathfrak{y})(t)| \\ &\leq \int\_t^\infty r(s) \Big| \Big\mathfrak{g} \left( \int\_{t\_1}^s p(u) f\left(\mathfrak{y}\_n(u)\right) \Delta u \right) - \mathfrak{g} \left( \int\_{t\_1}^s p(u) f(\mathfrak{y}(u)) \Delta u \right) \Big| \Delta s \end{aligned}$$

� � Then by the Lebesgue dominated convergence theorem and by the continuity of f and g, we have that �Ty � Ty� ! 0 as n ! ∞, i.e., T, is continuous. Finally, we <sup>n</sup> show that TΩ is relatively compact, i.e., equibounded and equicontinuous. Since

$$0 \le (T\mathfrak{y})^{\Delta}(t) = r(t)\mathfrak{g}\left(\int\_{t\_1}^t p(u)f(\mathfrak{y}(u))\Delta u\right) \le r(t)\mathfrak{g}\left(k\_2\int\_{t\_1}^t p(u)\Delta u\right) \le \infty,$$

we have that Ty is relatively compact by the Arzelá-Ascoli and mean value theorems. Therefore, Theorem 2.9 implies that there exists y∈ Ω such that y ¼ Ty: Then we have

$$\left(\overline{\mathbf{y}}^{\Delta}(t) = \left(T\overline{\mathbf{y}}\right)^{\Delta}(t) = r(t)\mathbf{g}\left(\int\_{t\_1}^{t} p(u)f(\overline{\mathbf{y}}(u))\Delta u\right) \quad t \ge t\_1. \tag{7}$$

6 <sup>R</sup> <sup>t</sup> <sup>Δ</sup> Setting <sup>x</sup>ðtÞ ¼ <sup>t</sup><sup>1</sup> p uð Þfðyð Þ <sup>u</sup> <sup>Þ</sup>Δ<sup>u</sup> gives us <sup>x</sup> ðÞ¼ <sup>t</sup> p tð Þfðyð Þ<sup>t</sup> <sup>Þ</sup>: Hence, we have that ðx; yÞ is a nonoscillatory solution of system (1) such that xðtÞ ! ∞ and yðtÞ ! c as t ! ∞, i.e., N<sup>þ</sup> <sup>∞</sup>,F ¼ ø.

## 3.1.2 The case Pðt0; ∞Þ , ∞ and Rðt0; ∞Þ , ∞

In this subsection, we show that the existence of nonoscillatory solutions of (1) is only possible in N<sup>þ</sup> F,F and N<sup>∞</sup> þ ,<sup>∞</sup> for P tð <sup>0</sup>; ∞Þ , ∞ and R tð <sup>0</sup>; ∞Þ , ∞, i.e., N<sup>þ</sup> F,<sup>∞</sup> ¼ N<sup>þ</sup> <sup>∞</sup>,F ¼ ø:

Lemma 3.6 Suppose Pðt0; ∞Þ , ∞ and Rðt0; ∞Þ , ∞ and that ðx; yÞ is a nonoscillatory solution of system (1). Then xð Þt tends to a finite nonzero number c if and only if yð Þt tends to a finite nonzero number d as t ! ∞.

#### Oscillators - Recent Developments

Proof. We prove the theorem by assuming x . 0 without loss of generality. Therefore by the definition of Nþ, y is also a positive component function of the solution ðx; yÞ. By taking the integral of the second equation of system (1) from t<sup>0</sup> to t and by the monotonicity of g and x, we have that there exists a positive constant k such that

$$
\mathcal{Y}(t) \le \mathcal{Y}(t\_0) + k \int\_{t\_0}^t r(s) \Delta s,
$$

where k ¼ g cð Þ: Then we have that y is convergent because P tð <sup>0</sup>; ∞Þ , ∞ as t ! ∞. The sufficiency can be shown similarly.

Theorem 3.7 N<sup>þ</sup> 6¼ ø if and only if I<sup>1</sup> , ∞ for all k<sup>1</sup> . 0. F,F

Proof. The necessity part can be shown similar to Theorem 3.5. So for sufficiency, suppose I<sup>1</sup> , ∞ holds for all k<sup>1</sup> . 0. Then choose t<sup>1</sup> ≥t<sup>0</sup> such that

$$\int\_{t\_1}^{\infty} p(t) f\left(k\_1 \int\_{t\_1}^{t} r(s) \Delta s\right) \Delta t \le \frac{c}{2},\tag{8}$$

where k<sup>1</sup> ¼ g cð Þ and t ≥t1: Let X be the Banach space of all bounded real-valued and continuous functions on ½t0; ∞Þ<sup>T</sup> with usual pointwise ordering ≤ and the norm sup ∣x tð Þ∣. Let Y be a subset of X such that <sup>t</sup>≥t<sup>1</sup>

$$Y \coloneqq \left\{ \mathbf{x} \in X : \frac{c}{2} \le \mathbf{x}(t) \le c \quad t \ge t\_1 \right\}.$$

and F : Ω ! X be an operator such that

$$(F\mathfrak{x})(t) = \frac{c}{2} + \int\_{t\_1}^t p(s)f\left(\int\_{t\_1}^s r(u)\mathfrak{g}\left(\mathfrak{x}(u)\right)\Delta u\right)d\mathfrak{x}, \quad t \ge t\_1.$$

One can easily have that inf B ∈Y and sup B∈ Y for any subset B of Y, which implies that ðY; ≤ Þ is a complete lattice. First, let us show that F : Y ! Y is an increasing mapping.

$$\frac{c}{2} \le (F\infty)(t) \le \frac{c}{2} + \int\_{t\_1}^t p(s)f\left(g(c)\int\_{t\_1}^s r(u)\Delta u\right)\Delta t \le c, \quad t \ge t\_1.$$

that is F : Y ! Y: Note also that for x<sup>1</sup> ≤ x2, x1, x<sup>2</sup> ∈Y, we have Fx<sup>1</sup> ≤ Fx2, i.e., F, which is an increasing mapping. Then by Theorem 2.10, there exists a function x∈Y such that x ¼ Fx: By taking the derivative of Fx, we have

$$\left(F\overline{\boldsymbol{x}}\right)^{\Delta}(t) = p(t)f\left(\int\_{t\_1}^{t} r(u)\mathbf{g}\left(\overline{\boldsymbol{x}}(u)\right)\Delta u\right), \quad t \ge t\_1.s$$

By letting

$$
\overline{\mathcal{Y}}(t) = \int\_{t\_1}^{t} r(u) \underline{\mathcal{g}}\left(\overline{\mathcal{X}}(u)\right) \Delta u,
$$

<sup>Δ</sup> we have <sup>y</sup> ðÞ¼ <sup>t</sup> r tð Þgðxð Þ<sup>t</sup> <sup>Þ</sup>, and <sup>ð</sup>x; <sup>y</sup><sup>Þ</sup> is <sup>a</sup> nonoscillatory solution of system (1) such that x and y have finite limits as t ! ∞. This completes the assertion.

Remark 3.8 Suppose that P tð <sup>0</sup>; ∞Þ , ∞ and R tð <sup>0</sup>; ∞Þ , ∞. Then, as a result of this, we have I<sup>1</sup> , ∞. So Theorem 3.7 also holds for P tð <sup>0</sup>; ∞Þ , ∞ and R tð <sup>0</sup>; ∞Þ , ∞. Exercise 3.9 Prove Remark 3.8.

3.1.3 The case Pðt0; ∞Þ , ∞ and Rðt0; ∞Þ ¼ ∞

We present the nonoscillation criteria in N<sup>þ</sup> under the case P tð <sup>0</sup>; ∞Þ , ∞ and R tð <sup>0</sup>; ∞Þ ¼ ∞ in this subsection. Therefore, we have the following lemma.

Lemma 3.10 Suppose that Rðt0; ∞Þ ¼ ∞. Then any nonoscillatory solution in N<sup>þ</sup> belongs to Nþ F,<sup>∞</sup> or N<sup>þ</sup> <sup>∞</sup>,∞, i.e., NF þ ,F ¼ N<sup>þ</sup> <sup>∞</sup>,F ¼ ø.

Exercise 3.11 Prove Lemma 3.10.

The following theorem shows us the nonexistence of nonoscillatory solutions in Nþ F,∞: We skip the proof of the following theorem, since it is very similar to the proof of Theorem 3.5.

Theorem 3.12 N<sup>þ</sup> F,<sup>∞</sup> 6¼ ø if and only if I<sup>1</sup> , ∞ for all k<sup>1</sup> . 0.

#### 3.1.4 Examples

Examples are great ways to see that theoretical claims actually work. Therefore, we provide two examples about the existence of nonoscillatory solutions of system (1). But before the examples, we need the following proposition because our examples consist of scattered points.

Proposition 1 ([5], Theorem 1.79) Let a, b∈T and h∈ Crd: If ½a; b� consists of only isolated points, then

$$\int\_{a}^{b} h(t)\Delta t = \sum\_{t \in \left[a,b\right)\_{\mathbb{T}}} \mu(t)h(t)\,.$$

Example 3.13 Let <sup>T</sup> <sup>¼</sup> <sup>2</sup><sup>N</sup><sup>0</sup> . Consider

$$\begin{cases} \Delta\_q \varkappa(t) = \left(\frac{t}{2t-1}\right)^{\frac{1}{61}} (\varkappa(t))^{\frac{1}{61}} \\\\ \Delta\_q \varkappa(t) = \frac{1}{2t^{\frac{13}{5}}} (\varkappa(t))^{\frac{3}{5}}, \end{cases} \tag{9}$$

<sup>h</sup>ðσð Þ<sup>t</sup> Þ�h tð Þ where <sup>Δ</sup><sup>q</sup> is known as <sup>a</sup> <sup>q</sup>-derivative and defined as <sup>Δ</sup>qh tðÞ¼ , where <sup>μ</sup>ð Þ<sup>t</sup> <sup>μ</sup> <sup>t</sup> ðÞ¼ <sup>2</sup>t, and <sup>t</sup> <sup>¼</sup> <sup>2</sup><sup>n</sup> ðÞ¼ <sup>t</sup>, <sup>σ</sup> <sup>t</sup> , (see [5]). In this example, it is shown that we have <sup>a</sup> nonoscillatory solution in N<sup>þ</sup> <sup>∞</sup>,F to highlight Theorem 3.5. Therefore, we need that P tð <sup>0</sup>; ∞Þ is divergent and R tð <sup>0</sup>; ∞Þ is convergent. Indeed, by Proposition 1, we have

$$P(\mathbf{1}, T) = \int\_{1}^{T} p(t) \Delta t = \sum\_{t \in \left[1, T\right)\_{2} \mathbb{N}\_{0}} \left(\frac{t}{2t - 1}\right)^{\frac{1}{64}} \cdot t.$$

Hence, we have Pð1; ∞Þ ¼ ∞ as T tends to infinity. Note that we use the limit divergence test to show the divergence of Pð1; ∞Þ. Next, we continue with the convergence of Rð1; ∞Þ. To do that, we note

$$R(\mathbf{1}, T) = \int\_{1}^{T} r(t) \Delta t = \sum\_{t \in \left[1, T\right)\_{2} \mathbb{K}\_{0}} \frac{\mathbf{1}}{2t^{\frac{13}{5}}} \cdot t.$$

As T ! ∞, we have

$$\sum\_{n=0}^{\infty} \left(\frac{1}{2 \cdot 2^n}\right)^{\frac{n}{5}} < \infty$$

by the geometric series, i.e., Rð1; ∞Þ , ∞. Finally, we have to show I<sup>2</sup> , ∞. Let k<sup>2</sup> ¼ 1: Then we get

$$\begin{split} &\int\_{1}^{T} r(t) \mathfrak{g} \left( k\_{2} \iint\_{\mathbf{t}} p(\mathbf{s}) \Delta \mathbf{s} \right) \Delta t = \iint\_{1}^{T} \frac{1}{2t^{\frac{31}{5}}} \sum\_{\mathbf{s} \in [1,t]\_{2^{\mathbb{N}}}} \left( \frac{\mathbf{s}}{2\mathfrak{s}-1} \right)^{\frac{1}{64}} \cdot \mathfrak{s} \right) \Big| \Delta t \\ &\leq \int\_{1}^{T} \frac{1}{2t^{\frac{31}{5}}} \sum\_{\mathbf{s} \in [\mathbf{1}^{\mathbb{J}}]\_{2^{\mathbb{N}}}} t^{\frac{62}{56}} \Big)^{\frac{2}{5}} \Delta t \leq \int\_{1}^{T} \frac{1}{2t^{\frac{31}{5}}} \cdot t^{\frac{62}{105}} \Delta t = \sum\_{\mathbf{t} \in [1,T]\_{2^{\mathbb{N}}}} \frac{1}{t^{\frac{208}{105}}}. \end{split}$$

So as t ! ∞, we have

$$\sum\_{n=0}^{\infty} \left(\frac{1}{\mathcal{Z}^n}\right)^{\frac{208}{105}} < \infty$$

� � by the ratio test. Therefore, I<sup>2</sup> , ∞ by the comparison test. One can also show that <sup>t</sup>; <sup>2</sup> � <sup>1</sup> is <sup>a</sup> solution of system (9) such that x tðÞ! <sup>∞</sup> and y tðÞ! <sup>2</sup> as <sup>t</sup> ! <sup>∞</sup>, <sup>t</sup> i.e., N<sup>þ</sup> <sup>∞</sup>,F 6¼ ø by Theorem 3.5

<sup>p</sup>ffiffi <sup>p</sup>ffiffi � � <sup>1</sup> <sup>1</sup> 2 2�<sup>1</sup><sup>Þ</sup> <sup>n</sup> <sup>ð</sup> Example 3.14 Let <sup>T</sup> <sup>¼</sup> <sup>2</sup> : <sup>n</sup> <sup>∈</sup> <sup>N</sup><sup>0</sup> , f <sup>z</sup> <sup>3</sup>, g <sup>z</sup> <sup>5</sup>, p <sup>t</sup> <sup>1</sup> <sup>ð</sup> Þ ¼ <sup>z</sup> <sup>ð</sup> Þ ¼ <sup>z</sup> ðÞ¼ <sup>2</sup><sup>t</sup> , <sup>2</sup> <sup>3</sup> <sup>ð</sup>3�2<sup>t</sup> �1Þ<sup>3</sup> <sup>p</sup>ffiffi <sup>p</sup>ffiffi <sup>2</sup><sup>ð</sup> <sup>2</sup>�<sup>1</sup><sup>Þ</sup> r tðÞ¼ <sup>1</sup>, and <sup>t</sup> <sup>¼</sup> <sup>n</sup> <sup>4</sup><sup>t</sup> in system (1). We show that there exists <sup>a</sup> nonoscillatory <sup>2</sup> <sup>2</sup> <sup>5</sup> <sup>ð</sup>2�2<sup>t</sup> �1Þ<sup>5</sup> solution in Nþ F,F. So by Theorem 3.7, we need to show Pðt0; ∞Þ , ∞ and Rðt0; ∞Þ , ∞ and I<sup>1</sup> , ∞. Proposition 1 gives us

$$\iint\_{\mathbf{t}}^{T} p(\mathbf{t}) \Delta \mathbf{t} = \sum\_{\mathbf{t} \in [0,T)\_{\mathbb{T}}} \frac{\sqrt{2} (\sqrt{2} \not\!\! - \mathbf{1})}{2^{\frac{2t}{3}} (\mathbf{3} \cdot \mathbf{2}^{t} \not\!\!\! - \mathbf{1})^{\frac{1}{3}}} \cdot \frac{\mathbf{1}}{\mathbf{2}} \le \sum\_{\mathbf{t} \in [0,T)\_{\mathbb{T}}} \frac{1}{2^{\frac{2t}{3}}} \cdot \mathbf{1}$$

So as T ! ∞, we have

$$\sum\_{n=0}^{\infty} \frac{1}{2^{\frac{n}{5}}} < \infty$$

by the geometric series, i.e., P tð <sup>0</sup>; ∞Þ , ∞. Also

$$\iint\_{\mathbf{t}\in\mathbb{T}^{3}}^{T} r(t)\Delta t = \sum\_{\mathbf{t}\in[0,T)\_{\mathbb{T}}} \frac{\sqrt{2}(\sqrt{2}\langle -\mathbf{1} \rangle\_{\mathbb{T}} \cdot \mathbf{1})}{2^{\#}(2\cdot 2^{\sharp} \langle -\mathbf{1} \rangle\_{\mathbb{T}}^{\sharp})} \cdot \frac{\mathbf{1}}{2} \leq \sum\_{\mathbf{t}\in[0,T)\_{\mathbb{T}}} \frac{\mathbf{1}}{2^{\#}}.$$

Hence, we have

$$\sum\_{n=0}^{\infty} \frac{1}{2^{\#}} < \infty$$

as T ! ∞. Note also that I � <sup>1</sup> , ∞ if Pðt0; � ∞Þ , ∞ and Rðt0; ∞Þ , ∞ (see Remark (8)). It can be confirmed that 2 � <sup>2</sup> 1 <sup>t</sup> ; <sup>3</sup> � <sup>1</sup> is <sup>a</sup> nonoscillatory solution of <sup>2</sup><sup>t</sup>

Oscillation Criteria of Two-Dimensional Time-Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.83375

$$\begin{cases} \mathbf{x}^{\Delta}(t) = \frac{\sqrt{2}\left(\sqrt{2}-1\right)}{2^{\frac{4}{5}}\left(3\cdot 2^{t}-1\right)^{\frac{1}{5}}}(\mathbf{y}(t))^{\frac{1}{5}} \\\\ \mathbf{y}^{\Delta}(t) = \frac{\sqrt{2}\left(\sqrt{2}-1\right)}{2^{\frac{4}{5}}\left(2\cdot 2^{t}-1\right)^{\frac{1}{5}}}(\mathbf{x}(t))^{\frac{1}{5}} \end{cases}$$

such that x tðÞ! 2 and y tðÞ! 3 as t ! ∞, i.e., NF þ ,F ¼6 ø by Theorem 3.7.

#### 3.2 Existence of nonoscillatory solutions of (1) in N�

Suppose that ðx; yÞ is a nonoscillatory solution of system (1) such that x . 0 eventually. Then by the first and second equations of system (1) and the similar discussion as in Section 3.1, we obtain the following subclasses of N�.

$$\begin{aligned} N\_{F,F}^- &= \left\{ (\mathbf{x}, \boldsymbol{\mathcal{y}}) \in \boldsymbol{N}^- : \lim\_{t \to \infty} \boldsymbol{x}(t) = c, \quad \lim\_{t \to \infty} \boldsymbol{y}(t) = -d \right\}, \\\\ N\_{F,0}^- &= \left\{ (\mathbf{x}, \boldsymbol{\mathcal{y}}) \in \boldsymbol{N}^- : \lim\_{t \to \infty} \boldsymbol{x}(t) = c, \quad \lim\_{t \to \infty} \boldsymbol{y}(t) = \mathbf{0} \right\}, \\\\ N\_{0,F}^- &= \left\{ (\mathbf{x}, \boldsymbol{\mathcal{y}}) \in \boldsymbol{N}^- : \lim\_{t \to \infty} \boldsymbol{x}(t) = \mathbf{0}, \quad \lim\_{t \to \infty} \boldsymbol{y}(t) = -d \right\}, \\\\ N\_{0,0}^- &= \left\{ (\mathbf{x}, \boldsymbol{\mathcal{y}}) \in \boldsymbol{N}^- : \lim\_{t \to \mathbf{0}} \boldsymbol{x}(t) = \mathbf{0}, \quad \lim\_{t \to \mathbf{0}} \boldsymbol{y}(t) = \mathbf{0} \right\}. \end{aligned}$$

This section presents us the existence and nonexistence of nonoscillatory solutions of system (1) under the monotonicity condition on f and g.

Theorem 3.15 Let Rðt0; ∞Þ , ∞. Then there exists a nonoscillatory solution in N� F,F 6¼ ø if and only if I<sup>3</sup> , ∞ for all k<sup>3</sup> , 0 and k<sup>4</sup> . 0.

Proof. Suppose NF � ,F ¼6 ø. Then there exists a solution ðx; yÞ∈ N� F,F such that x . 0, y , 0, x tðÞ! c1, and y tðÞ!�d<sup>1</sup> as t ! ∞ for 0 , c<sup>1</sup> , ∞ and 0 , d<sup>1</sup> , ∞. By integrating the second equation of system (1) from t to ∞, we obtain

$$\begin{split} \mathcal{Y}(t) &= \mathcal{Y}(\infty) - \int\_{t}^{\infty} r(s) \mathcal{g}(\mathbf{x}(s)) \Delta s \\ &\leq -d\_{1} - k\_{4} \int\_{t}^{\infty} r(s) \Delta s, \quad \text{where} \quad k\_{4} = \mathcal{g}(c\_{1}). \end{split} \tag{10}$$

Integrating the first equation from t<sup>1</sup> to t, using (10) and the fact that x is bounded yield us

$$\begin{aligned} c\_1 \le \mathbf{x}(t) &= \mathbf{x}(t\_1) + \int\_{t\_1}^t p(s) f(\mathbf{y}(s)) \, \Delta s \\ &\le \mathbf{x}(t\_1) + \int\_{t\_1}^t p(s) f\left(-d\_1 - k\_4 \int\_s^\infty r(u) \Delta u\right) \Delta s \le \mathbf{x}(t\_1), \quad t \ge t\_1. \end{aligned}$$

Therefore, it implies I<sup>3</sup> , ∞ as t ! ∞, where �d<sup>1</sup> ¼ k3.

Conversely, suppose that I<sup>3</sup> , ∞: Then there exist t<sup>1</sup> ≥ t<sup>0</sup> and k<sup>3</sup> , 0, k<sup>4</sup> . 0 such that

$$\int\_{t\_0}^{\infty} p(t)f\left(k\_3 - k\_4 \int\_t^{\infty} r(s)\Delta s\right) \Delta t > \frac{-1}{2} \tag{11}$$

� � where <sup>3</sup> <sup>k</sup><sup>4</sup> <sup>¼</sup> <sup>g</sup> . Let CB be the set of all continuous and bounded real-valued <sup>2</sup> functions x tð Þ on ½t1; ∞Þ<sup>T</sup> with the supremum norm supt≥t<sup>1</sup> ∣x tð Þ∣. Observe that CB is a Banach space (see [38]). Suppose that B is a subset of CB such that

$$B \coloneqq \left\{ \varkappa(t) \in \mathcal{C}\_B \,:\, 1 \le \varkappa(t) \le \frac{3}{2}, \quad t \ge t\_1 \right\}.$$

We have that B meets the assumptions of Theorem 2.9. Suppose also that F : B ! B is an operator such that

$$(F\mathbf{x})(t) = \mathbf{1} - \int\_{t}^{\infty} a(s)f\left(k\_{\beta} - \int\_{s}^{\infty} b(u)\mathbf{g}(\mathbf{x}(u))\Delta u\right)d\mathbf{s} \tag{12}$$

First, we need to show F is a mapping into itself, i.e., F : B ! B. Indeed,

$$\mathbf{1} \le (F\mathbf{x})(t) \le \mathbf{1} - \int\_t^\infty a(s) f\left(k\_3 - \mathbf{g}\left(\frac{\mathfrak{J}}{2}\right) \int\_{t\_1}^s b(u) \Delta u\right) d\mathbf{s} \le \frac{\mathfrak{J}}{2}$$

because x∈B and (5) hold. Next, let us verify that F is continuous on B: In order to do that, let xn be a sequence in B such that xn ! x, where x∈ B ¼ B: Then

$$\begin{aligned} &|(\mathrm{Fx}\_{\mathfrak{n}})(t) - (\mathrm{Fx})(t)| \\ &\leq \int\_{\mathfrak{t}}^{\infty} p(\mathfrak{s}) \Big| f\left(\mathbb{k}\_{3} - \int\_{\mathfrak{s}}^{\infty} r(\mathfrak{u}) \mathbf{g}(\mathbf{x}\_{\mathfrak{n}}(\mathfrak{u})) \Delta \mathbf{u}\right) - f\left(\mathbb{k}\_{3} - \int\_{\mathfrak{s}}^{\infty} r(\mathfrak{u}) \mathbf{g}(\mathbf{x}(\mathfrak{u})) \Delta \mathbf{u}\right) \Big| \Delta \mathbf{s}. \end{aligned}$$

Therefore, the continuity of f and g and the Lebesgue dominated convergence theorem gives us Fxn ! Fx as n ! ∞, which implies F is continuous on B. Finally, we prove that FY is equibounded and equicontinuous, i.e., relatively compact. Because

$$\begin{aligned} 0 &< - (F\mathfrak{x})^\Delta(t) = -p(t)f\left(k\_3 - \int\_t^\infty r(u)\mathfrak{g}(\mathfrak{x}(u))\Delta u\right) \\ &\leq -p(t)f\left(k\_3 - k\_4 \int\_{t\_1}^t r(u)\Delta u\right) < \infty, \end{aligned}$$

we have that Fx is relatively compact. Hence, Theorem 2.9 implies that there exists x∈B such that x ¼ Fx: Thus, we have x . 0 eventually and xðtÞ ! 1 as t ! ∞. Also

$$\overline{\mathfrak{X}}^{\Delta}(t) = \left(F\overline{\mathfrak{X}}\right)^{\Delta}(t) = p(t)f\left(k\_3 - \int\_t^{\infty} r(u)\overline{\mathfrak{g}}(\overline{\mathfrak{x}}(u))\Delta u\right) \quad t \ge t\_1.$$

Letting

$$\overline{\mathfrak{F}}(t) = k\_3 - \int\_t^{\infty} r(u)\overline{\mathfrak{g}}(\overline{\mathfrak{X}}(u))\Delta u \le 0, \quad t \ge t\_1 \tag{13}$$

6 and taking the derivative of (13) give <sup>y</sup><sup>Δ</sup>ðÞ¼ <sup>t</sup> <sup>b</sup>ð Þt gðxð Þ<sup>t</sup> <sup>Þ</sup>: So, we conclude that ðx; yÞ is a nonoscillatory solution of system (1). Finally, taking the limit of Eq. (13) results in yðtÞ ! k<sup>3</sup> , 0. Therefore, we get N� F,F ¼ ø.

Theorem 3.16 Suppose Pðt0; ∞Þ , ∞. N� <sup>0</sup>,F 6¼ ø if and only if I<sup>4</sup> , ∞ for k<sup>5</sup> . 0. Exercise 3.17 Prove Theorem 3.16.

Theorem 3.18 Suppose Pðt0; ∞Þ , ∞: N� <sup>0</sup>,<sup>0</sup> 6¼ ø if I<sup>3</sup> , ∞ and I<sup>4</sup> ¼ ∞ for all k<sup>3</sup> ¼ 0, k<sup>4</sup> , 0 and k<sup>5</sup> . 0, provided f is odd.

Proof. Suppose that I<sup>3</sup> , ∞, and I<sup>4</sup> ¼ ∞. Then there exists t<sup>1</sup> ≥t<sup>0</sup> such that

$$\int\_{t\_1}^{\infty} p(s)f\left(-k\_4 \int\_s^{\infty} r(u)\Delta u\right) \Delta s \le 1$$

and

$$\int\_{t\_1}^{\infty} r(s) \mathbf{g} \left( k\_5 \int\_s^{\infty} p(u) \Delta u \right) \Delta s \geq \frac{1}{2}$$

for t≥t1, k<sup>4</sup> ¼ �gð Þ1 . Let X be the space that is claimed as in the proof of Theorem 3.7. Let Y be a subset of X and given by

$$Y := \left\{ \mathbf{x} \in X : c\_1 \int\_t^{\infty} a(s) \Delta s \le \mathbf{x}(t) \le 1 \quad t \ge t\_1 \right\},$$

� � where <sup>1</sup> <sup>c</sup><sup>1</sup> <sup>¼</sup> <sup>f</sup> . Define an operator <sup>T</sup> : <sup>Y</sup> ! <sup>X</sup> such that <sup>2</sup>

$$(T\mathbf{x})(t) = \int\_{t}^{\infty} p(s)f\left(\int\_{s}^{\infty} r(u)\mathbf{g}(\mathbf{x}(u))\Delta u\right)dt, \quad t \ge t\_1.$$

One can show that ðY; ≤ Þ is a complete lattice and T is an increasing mapping such that T : Y ! Y. As a matter of fact,

$$(T\infty)(t) \le \int\_{t}^{\infty} p(s)f\left(\mathbf{g}(\mathbf{1})\int\_{s}^{\infty} r(u)\Delta u\right)\Delta t \le \mathbf{1}, \quad t \ge t\_1$$

and

$$\begin{aligned} (T\mathbf{x})(t) &\geq \int\_t^{\infty} p(s) f\left(\int\_s^{\infty} r(u) \mathbf{g}\left(c\_1 \int\_u^{\infty} p(v) \Delta v\right) \Delta u\right) d\mathbf{s}, \\ &\geq f\left(\frac{1}{2}\right) \int\_t^{\infty} p(s) \Delta s, \end{aligned}$$

where c<sup>1</sup> ¼ k5, i.e., T : Y ! Y: Then by Theorem 2.10, there exists a function x∈Y such that x ¼ Tx: By taking the derivative of Tx and using the fact that f is odd, we have

$$(T\overline{\mathfrak{X}})^{\Delta}(t) = p(t)f\left(-\int\_{t}^{\infty} r(u)\overline{g}(\overline{\mathfrak{X}}(u))\Delta u\right), \quad t \ge t\_1.$$

Setting

$$\overline{\mathfrak{J}}(t) = -\int\_{t}^{\infty} r(u)\mathfrak{g}\left(\overline{\mathfrak{x}}(u)\right)\Delta u$$

<sup>Δ</sup> yields <sup>y</sup> <sup>t</sup> ð Þ <sup>ð</sup> ð ÞÞ <sup>ð</sup>x; <sup>y</sup> <sup>0</sup>,<sup>0</sup> ðÞ¼ <sup>b</sup> t g <sup>x</sup> <sup>t</sup> , and <sup>Þ</sup> is <sup>a</sup> solution of system (1) in <sup>N</sup> , i.e., <sup>x</sup> and � y both tend to zero.

Theorem 3.19 Suppose Rðt0; ∞Þ , ∞. NF � ,<sup>0</sup> ¼6 ø if and only if I<sup>3</sup> , ∞, where k<sup>3</sup> ¼ 0 and k<sup>4</sup> . 0.

Exercise 3.20 Prove Theorem 3.19. Hint: Use Theorem 2.10 with the operator

$$(F\infty)(t) = \frac{1}{2} - \int\_t^{\infty} a(s)f\left(-\int\_s^{\infty} b(u)g(\infty(u))\Delta u\right)d\mathbf{t}, \quad t \ge t\_1.$$

Examples make results clearer and give more information to readers. Therefore, we give the following example to validate our claims. The beauty of our example is that we do not only show the theorem holds but also find the explicit solutions, which might be very hard for some nonlinear systems.

<sup>2</sup> Example 3.21 Consider <sup>T</sup> <sup>¼</sup> <sup>N</sup><sup>2</sup> <sup>0</sup> <sup>¼</sup> � n : n∈ N<sup>0</sup> � with the system

$$\begin{cases} \varkappa^{\Delta}(t) = \frac{1}{t^{\frac{1}{2}} (\sqrt{t} + 1)^{2} (t^{2} + 1)^{\frac{1}{5}}} (\varkappa(t))^{\frac{1}{5}} \\\\ \varkappa^{\Delta}(t) = \frac{\left(\sqrt{t} + 1\right)^{4} - t^{2}}{t^{\frac{3}{2}} (\sqrt{t} + 1)^{4} (1 + 2\sqrt{t})} (\varkappa(t))^{\frac{1}{5}}, \end{cases} \tag{14}$$

�pffiffi �<sup>2</sup> <sup>p</sup>ffiffi <sup>f</sup>ðσð Þ<sup>t</sup> Þ�f tð Þ where <sup>f</sup> <sup>Δ</sup>ðÞ¼ <sup>t</sup> for <sup>σ</sup>ðtÞ ¼ <sup>t</sup> <sup>þ</sup> <sup>1</sup> and <sup>μ</sup>ðÞ¼ t t <sup>1</sup> <sup>þ</sup> <sup>2</sup> (see [5]). First, let <sup>μ</sup>ð Þ<sup>t</sup> us show Pðt0; ∞Þ , ∞, where t<sup>0</sup> ≥1.

$$\int\_{1}^{T} p(t)\Delta t = \sum\_{t \in [1,T)\_{\mathbb{N}\_{0}}} \frac{1}{t^{\frac{1}{2}}(\sqrt{t}+1)^{2}(t^{2}+1)^{\frac{1}{2}}} \cdot (1+2\sqrt{t}) \leq \sum\_{t \in [1,T)\_{\mathbb{N}\_{0}}} \frac{1+2\sqrt{t}}{t^{2}}.$$

Since <sup>t</sup> <sup>¼</sup> <sup>n</sup><sup>2</sup>, as T ! <sup>∞</sup>, we have

$$\sum\_{n=1}^{\infty} \frac{1+2n}{n^4} < \infty$$

by the geometric series. Therefore, Pð1; ∞Þ , ∞ by the comparison test. Next, <sup>R</sup> <sup>∞</sup> we show <sup>I</sup><sup>4</sup> , <sup>∞</sup>. Since <sup>P</sup>ð1; <sup>∞</sup><sup>Þ</sup> , <sup>∞</sup>, we have p sð ÞΔ<sup>s</sup> , <sup>α</sup> for <sup>t</sup>≥<sup>1</sup> and <sup>0</sup> , <sup>α</sup> , <sup>∞</sup>. <sup>t</sup> Hence,

$$\begin{aligned} \int\_{1}^{T} r(t)g\left(\int\_{t}^{\infty} p(s)\Delta t\right) \Delta t &\leq a \int\_{1}^{T} r(t)\Delta t \\ &= a \sum\_{t \in \left[1, T\right)\_{\mathbb{N}\_{0}}} \frac{\left(\sqrt{t} + 1\right)^{4} - t^{2}}{t^{2} \left(\sqrt{t} + 1\right)^{4} \left(1 + 2\sqrt{t}\right)} \cdot \left(1 + 2\sqrt{t}\right)^{2} \\ &\leq a \sum\_{t \in \left[1, T\right)\_{\mathbb{N}\_{0}}} \frac{1}{t^{5}} .\end{aligned}$$

So as T tends to infinity, we get

$$\sum\_{n=1}^{\infty} \frac{1}{n^{\frac{n}{5}}} < \infty,$$

<sup>1</sup> i.e., <sup>I</sup><sup>2</sup> , <sup>∞</sup>. Also, note that � ; �<sup>1</sup> � <sup>1</sup> � is a solution of system (14) in N� such that x <sup>t</sup> <sup>t</sup><sup>2</sup> tends to zero, while y tends to �1, i.e., N� <sup>0</sup>,F ¼6 ø.

#### 4. Oscillation of a two-dimensional time-scale systems

Motivated by [39], this section deals with the system

$$\begin{cases} \varkappa^{\Delta}(t) = a(t)f(\wp(t)) \\ \jmath^{\Delta}(t) = -b(t)\lg(\varkappa(t)) + c(t), \end{cases} \tag{15}$$

� � � � where a, b∈Crd ½t0; ∞ÞT; R<sup>þ</sup> , c∈Crd ½t0; ∞ÞT; R and functions f g have the same characteristics as in system (1) and g is continuously differentiable. Note that we can rewrite system (15) as a non-homogenous dynamic equations on time scales and putting σ on x inside the function g. Therefore, we have the following dynamic equation

$$\left(a(t)\mathfrak{x}^{\Delta}(t)\right)^{\Delta} + b(t)\mathfrak{g}\left(\mathfrak{x}^{\sigma}(t)\right) = c(t) \tag{16}$$

and systems of dynamical equations

$$\begin{cases} \mathbf{x}^{\Delta}(t) = a(t)\mathbf{f}(\mathbf{y}(t)) \\ \mathbf{y}^{\Delta}(t) = -b(t)\mathbf{g}(\mathbf{x}^{\sigma}(t)) + c(t). \end{cases} \tag{17}$$

Oscillation criteria for Eq. (16), system (17), and other similar versions of (15) and (17) are investigated in [39–42]. A solution ðx; yÞ of system (15) is called oscillatory if x and y have arbitrarily large zeros. System (15) is called oscillatory if all solutions are oscillatory.

Before giving the main results, we present some propositions so that we can use them in our theoretical claims (see [43], Theorem 4.2 (comparison theorem) and [5], Theorem 1.90).

Proposition 2 Let z<sup>1</sup> be a function from T to R and v be a nondecreasing function from R to R such that v ∘ z<sup>1</sup> is rd-continuous. Suppose also that p≥ 0 is rd-continuous and α∈ R: Then

$$z\_1(t) \le a + \int\_{t\_0}^t p(\tau)v(z\_1(\tau))\Delta\tau, \quad t \ge t\_0$$

implies z1ð Þt ≤ z2ð Þt , where z<sup>2</sup> solves the initial value problem

$$z\_2^{\Delta}(t) = p(t)v(z\_2(t)), \quad z\_2(t\_0) = z\_{20} > a.$$

Proposition 3 (chain rule). ([5], Theorem 1.90) Let h<sup>1</sup> : R ! R be continuously differentiable and suppose h<sup>2</sup> : T ! R is delta differentiable. Then h<sup>1</sup> ∘ h<sup>2</sup> : T ! R is delta differentiable, and the formula

$$(h\_1 \circ h\_2)^{\Delta}(t) = \left\{ \int\_0^1 h\_{1'} \left( h\_2(t) + h\mu(t) h\_2^{\Delta}(t) \right) dh \right\} h\_2^{\Delta}(t)$$

holds.

For simplicity, set

$$\begin{aligned} A(t,s) &= \int\_t^s a(u)\Delta u, \quad B(t,s) = \int\_t^s b(u)\Delta u, \\ C(t,s) &= \int\_t^s |c(u)|\Delta u, \quad D(t,s) = \int\_t^s \left(b(u) - \frac{c(u)}{\mathcal{g}(\varkappa(u))}\right) \Delta u, \\ Y(t,s) &= \int\_t^s \frac{\mathcal{y}^\sigma(u)\varkappa^\Delta(u)\int\_0^1 \left[\mathcal{g}'(\varkappa(u) + h\mu(u)x^\Delta(u)\right]dh\right]}{\mathcal{g}(\varkappa(u))\mathcal{g}(\varkappa^\sigma(u))} \Delta u. \end{aligned}$$

Next, note that if ðx; yÞ is a nonoscillatory solution of system (15), then one can easily prove that x is also nonoscillatory. This result was shown by Anderson in [37] when c tðÞ� 0: Because the proof when c tð Þ�= 0 is very similar to the proof of the case c tðÞ� 0, we leave it to the readers.

Lemma 4.1 Suppose that ðx; yÞ is a nonoscillatory solution of system (15) and t1, t<sup>2</sup> ∈T. If there exists a constant K . 0 such that

$$H(t) \ge K, \quad t \ge t\_2,\tag{18}$$

where H is defined as

$$H(t) = -\frac{\mathcal{y}(t\_1)}{\mathcal{g}(\mathcal{x}(t\_1))} + D(t\_1, t) + Y(t\_1, t\_2),\tag{19}$$

then yð Þt ≤ � Kgðxð Þ t<sup>2</sup> Þ, t≥t2.

Proof. Suppose that ðx; yÞ is a nonoscillatory solution of system (15). Then, we have that x is also nonoscillatory. Without loss of generality, assume that x tð Þ . 0 for t≥ t<sup>1</sup> ≥t0, where t1, t<sup>0</sup> ∈ T. Integrating the second equation of system (15) from t<sup>1</sup> to t and Theorem 2.8 (v.) gives us

$$\int\_{t\_1}^{t} b(\boldsymbol{\varsigma}) \Delta \boldsymbol{\varsigma} = \frac{\boldsymbol{y}(t\_1)}{\boldsymbol{g}(\boldsymbol{\varsigma}(t\_1))} - \frac{\boldsymbol{y}(t)}{\boldsymbol{g}(\boldsymbol{\varsigma}(t))} + \int\_{t\_1}^{t} \left(\frac{1}{\boldsymbol{g}(\boldsymbol{\varsigma}(s))}\right)^{\Delta} \boldsymbol{\jmath}^{\sigma}(s) \Delta \boldsymbol{\varsigma} + \int\_{t\_1}^{t} \frac{\boldsymbol{c}(s)}{\boldsymbol{g}(\boldsymbol{\varsigma}(s))} \Delta \boldsymbol{\varsigma}.\tag{20}$$

By applying Theorem 2.3 (iv) and Proposition 3 to Eq. (20), we have

$$\int\_{t\_1}^{t} b(s) \Delta s = \frac{\mathbf{y}(t\_1)}{\mathbf{g}(\mathbf{x}(t\_1))} - \frac{\mathbf{y}(t)}{\mathbf{g}(\mathbf{x}(t))} + \int\_{t\_1}^{t} \frac{c(s)}{\mathbf{g}(\mathbf{x}(s))} \Delta s - Y(t\_1, t), \quad t \ge t\_1. \tag{21}$$

Rewriting Eq. (21) gives us

$$-\frac{\mathcal{y}(t)}{\mathcal{g}(\mathbf{x}(t))} = D(t\_1, t) - \frac{\mathcal{y}(t\_1)}{\mathcal{g}(\mathbf{x}(t\_1))} + Y(t\_1, t), \quad t \ge t\_1. \tag{22}$$

Now by using (18) and (19), we get

$$-\frac{y(t)}{g(\varkappa(t))} \ge K + Y(t\_2, t), \quad t \ge t\_2 \ge t\_1. \tag{23}$$

<sup>Δ</sup> <sup>t</sup> <sup>Δ</sup> Note that y tð Þ , <sup>0</sup> and <sup>x</sup> ð Þ , <sup>0</sup> for <sup>t</sup>≥t<sup>2</sup> since y tð Þ<sup>x</sup> ðÞ¼ <sup>t</sup> a sð Þy sð Þfðy sð ÞÞ . <sup>0</sup>: y tð Þ Otherwise, we would have � <sup>Þ</sup> . 0, which is <sup>a</sup> contradiction. Let gðx tð Þ

$$\frac{-\nu(t)}{g(\varkappa(t))} = K + Y(t\_2, t), \quad t \ge t\_2. \tag{24}$$

Oscillation Criteria of Two-Dimensional Time-Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.83375

So one can obtain

$$\left(\frac{-\nu(t)}{\mathfrak{g}(\mathfrak{x}(t))}\right)^{\Delta} = \frac{\mathcal{Y}^{\sigma}(t)\mathfrak{x}^{\Delta}(t)\int\_{0}^{1} [\mathfrak{g}'(\mathfrak{x}(t) + h\mu(t)\mathfrak{x}^{\Delta}(t))dh]}{\mathfrak{g}(\mathfrak{x}(t))\mathfrak{g}(\mathfrak{x}^{\sigma}(t))} > 0, \quad t \ge t\_{2}. \tag{25}$$

Because x tð Þ is a positive and v tð Þ is a negative function for t≥t2, we have �y tð Þ �v tð Þ <sup>≥</sup> gxt <sup>Þ</sup> , i.e., y tð Þ<sup>≤</sup> v tð Þ , <sup>0</sup> for <sup>t</sup> <sup>≥</sup>t2: Therefore, we have by (25) that gxt <sup>ð</sup> ð ÞÞ <sup>ð</sup> ð Þ

$$\left(\frac{-v(t)}{\mathbf{g}(\mathbf{x}(t))}\right)^{\Delta} \ge \frac{v''(t)\mathbf{x}^{\Delta}(t)\int\_0^1 \left[\mathbf{g}'(\mathbf{x}(t) + h\mu(t)\mathbf{x}^{\Delta}(t))dh\right]}{\mathbf{g}(\mathbf{x}(t))\mathbf{g}(\mathbf{x}^{\sigma}(t))} > 0, \quad t \ge t\_{22}$$

since v tð Þ , 0 and x<sup>Δ</sup>ð Þt , 0 for t ≥t2: By setting

$$\frac{w(t)}{\log(\varkappa(t))} = K - \int\_{t\_1}^{t} \frac{w''(s)\varkappa^{\Delta}(s) \int\_0^1 \left[ \mathbf{g}'(\varkappa(s) + h\mu(s)\varkappa^{\Delta}(s)) dh \right]}{\mathbf{g}(\varkappa(s))\mathbf{g}(\varkappa^{\sigma}(s))} d\varkappa \tag{26}$$

v tð Þ<sup>2</sup> <sup>¼</sup> <sup>K</sup> <sup>¼</sup> w tð Þ<sup>2</sup> and using (24), we have � gxt <sup>Þ</sup> : Then, setting gxt <sup>ð</sup> ð Þ<sup>2</sup> <sup>Þ</sup> <sup>ð</sup> ð Þ<sup>2</sup> <sup>1</sup> <sup>¼</sup> v tð Þ w tð Þ ð Þ<sup>t</sup> <sup>z</sup> , z<sup>2</sup> <sup>¼</sup> � ,h u<sup>ð</sup> Þ ¼ <sup>u</sup><sup>σ</sup> in Proposition 2, it follows v tð Þ<sup>≤</sup> � w tð Þ, which gxt <sup>ð</sup> ð ÞÞ gxt <sup>ð</sup> ð ÞÞ gxt <sup>ð</sup> ð ÞÞ implies y tð Þ≤ � w tð Þ, t ≥t2: Note also by Theorem 2.3 (iv) and Proposition 3 that

$$\left(\frac{w(t)}{g(\mathbf{x}(t))}\right)^{\Delta} = \frac{w^{\Delta}(t)}{g(\mathbf{x}^{\sigma}(t))} - \frac{w^{\sigma}(t)\mathbf{x}^{\Delta}(t)\int\_{0}^{1} \mathbf{g}'(\mathbf{x}(t) + h\mu(t)\mathbf{x}^{\Delta}(t))dh}{g(\mathbf{x}(t))\mathbf{g}(\mathbf{x}^{\sigma}(t))}, \quad t \ge t\_{2}. \tag{27}$$

Taking the derivative of (26) and comparing the resulting equation with (27) yield us

$$\frac{w^{\Delta}(t)}{\lg(\varkappa^{\sigma}(t))} = 0, \quad i.e., w^{\Delta}(t) = 0, \quad t \ge t\_2.$$

Therefore, we have

$$
\mu w(t\_2) = K \cdot \mathbf{g}(\mathbf{x}(t\_2)) = w(t), \quad i.e., \mathbf{y}(t) \le -w(t) = -K \cdot \mathbf{g}(\mathbf{x}(t\_2)).
$$

So the proof is completed.

#### 4.1 Results for oscillation

After giving the preliminaries in the previous section, it is presented the conditions for oscillatory solutions in this section.

Theorem 4.2 Let A tð <sup>0</sup>; ∞Þ ¼ ∞,B tð <sup>0</sup>; ∞Þ , ∞, and C tð <sup>0</sup>; ∞Þ , ∞: Assume

$$f(u)f(v) \le f(uv) \le -f(u)f(-v) \tag{28}$$

and

$$\int\_{t\_0}^{\infty} \frac{\varkappa^{\Delta}(s)}{f(\lg(\varkappa(s)))} \Delta s \le \infty. \tag{29}$$

Then system (15) is oscillatory if

$$\int\_{t\_0}^{\infty} a(t)f(B(t,\infty) - k \cdot C(t,\infty))\Delta t = \infty \tag{30}$$

6 for k ¼ 0.

Proof. Suppose that system (15) has a nonoscillatory solution ðx; yÞ such that x . 0 eventually. Then there exist t<sup>1</sup> ≥ t<sup>0</sup> and a constant k<sup>6</sup> such that gðx tð ÞÞ≥k<sup>6</sup> for t≥t<sup>1</sup> by the monotonicity of g. Then by Eq. (22), we have

$$\frac{\mathbf{y}(t)}{\mathbf{g}(\mathbf{x}(t))} = \frac{\mathbf{y}(t\_1)}{\mathbf{g}(\mathbf{x}(t\_1))} - D(t\_1, t) - Y(t\_1, t), \quad t \ge t\_1. \tag{31}$$

Note that Y tð <sup>1</sup>; tÞ , ∞. Otherwise, we have a contradiction to the fact that x tð Þ . 0 for t≥t<sup>1</sup> since A tð <sup>0</sup>; ∞Þ ¼ ∞. Equality (31) can be rewritten as

$$\frac{y(t)}{\mathbf{g}(\mathbf{x}(t))} = \chi + D(t, \infty) + Y(t, \infty),\tag{32}$$

y t<sup>ð</sup> <sup>1</sup><sup>Þ</sup> where <sup>γ</sup> ¼ - D t<sup>ð</sup> <sup>1</sup>; <sup>∞</sup>Þ - Y t<sup>ð</sup> <sup>1</sup>; <sup>∞</sup>Þ, t≥t1: It can be shown that <sup>γ</sup> <sup>≥</sup>0. Oth- <sup>g</sup>ðx t<sup>ð</sup> <sup>1</sup>ÞÞ <sup>γ</sup> erwise, we can choose <sup>a</sup> large <sup>t</sup><sup>2</sup> such that B t<sup>ð</sup> ; <sup>∞</sup>Þ<sup>≤</sup> - <sup>γ</sup>, Y t<sup>ð</sup> <sup>2</sup>; <sup>∞</sup>Þ<sup>≤</sup> - , and <sup>≤</sup> -<sup>γ</sup> 4 <sup>R</sup> <sup>∞</sup> c sð Þ for <sup>t</sup>≥t2. Then H tð Þ<sup>≥</sup> -<sup>γ</sup> . <sup>0</sup> for <sup>t</sup>≥t2. Then by setting <sup>K</sup> <sup>¼</sup> -<sup>γ</sup> <sup>Δ</sup> <sup>4</sup> <sup>4</sup> <sup>s</sup> in <sup>t</sup> <sup>g</sup>ðx sð ÞÞ <sup>4</sup> Lemma 4.1 found, we have y tð Þ≤ - Kgðxð Þ t<sup>2</sup> Þ for t≥t2. Integrating the first equation of system (15) from t<sup>2</sup> to ∞ and the monotonicity of f yields us

$$
\varkappa(t) \le \varkappa(t\_2) + f(-\mathsf{Kg}(\varkappa(t\_2))) \int\_{t\_2}^{t} a(s) \Delta s, \quad t \ge t\_2.
$$

So as t ! ∞, we have a contradiction to x . 0 eventually. Therefore γ ≥0. Then by Eq. (32), we have

$$y(t) \ge \mathbf{g}\left(\mathbf{x}(t)\right) \left[ \int\_t^{\infty} b(s) \Delta s - \frac{\mathbf{1}}{k\_6} \int\_t^{\infty} |c(s)| \Delta s \right], \quad t \ge t\_2. 4$$

By the first equation of system (15), the monotonicity of f and Eq. (28), we have

$$\alpha^{\Delta}(t) \ge a(t)f(\mathbf{g}(\mathbf{x}(t)))f\left(\int\_{t}^{\infty}b(s)\Delta s - \frac{1}{k\_{\delta}}\int\_{t}^{\infty}|c(s)|\Delta s\right), \quad t \ge t\_{2}.\tag{33}$$

Then by Eqs. (33) and (29), we have

$$\int\_{t\_2}^t a(s)f\left(\int\_s^{\infty} b(u)\Delta u - k \int\_s^{\infty} |c(u)|\Delta u\right) \le \int\_{t\_2}^t \frac{x^{\Delta}(s)}{f(\mathsf{g}(\varkappa(s)))} \Delta s \le \infty$$

where <sup>k</sup> <sup>¼</sup> <sup>1</sup> <sup>k</sup><sup>6</sup> : But as t ! ∞, this contradicts to Eq. (30). The proof is completed. Theorem 4.3 System (15) is oscillatory if Aðt0; ∞Þ ¼ B tð <sup>0</sup>; ∞Þ ¼ ∞ and Cðt0; ∞Þ , ∞.

Proof. We use the method of contradiction to prove the theorem. Thus, assume there is a nonoscillatory solution ðx; yÞ of system (15) such that the component function x is eventually positive. Because g is nondecreasing, we have that there exist t<sup>1</sup> ≥t<sup>0</sup> and k<sup>7</sup> . 0 such that gðx tð ÞÞ≥k<sup>7</sup> for t≥ t1. Then since C tð <sup>0</sup>; ∞Þ , ∞, we have that there exists 0 , k<sup>8</sup> , ∞ such that

$$\left| \int\_{t\_1}^{t} \frac{c(s)}{\mathfrak{g}(\varkappa(s))} \Delta s \right| \le \frac{1}{k\_7} \int\_{t\_1}^{t} |c(s)| \Delta s \le k\_8, \quad t \ge t\_1. \tag{34}$$

The first equation of system (15), and the monotonicity of g give us that there exist K . 0 and t<sup>2</sup> ≥ t<sup>1</sup> so large that

$$\mathbf{x}^{\Delta}(t) \le a(t)f(-\mathbf{Kg}(\mathbf{x}(t\_2))), \quad t \ge t\_2. \tag{35}$$

Integrating (35) from t<sup>2</sup> to t yields

$$\varkappa(t) \le \varkappa(t\_2) + k\_{\mathcal{\theta}} \int\_{t\_2}^t a(s) \Delta s, \quad \text{where} \quad k\_{\mathcal{\theta}} = f(-\text{Kg}(\varkappa(t\_2))) \le 0, \quad t \ge t\_2.$$

As t ! ∞, we have a contradiction to x tð Þ . 0 for t≥t2: This proves the assertion.

Finally, an example is provided to highlight Theorem 4.3 by finding the explicit solution of the dynamical system.

1 <sup>ð</sup> Þ ð2tþ7<sup>Þ</sup> Example 4.4 Consider the time scale <sup>T</sup> <sup>¼</sup> <sup>5</sup>Z<sup>þ</sup> with a <sup>t</sup> <sup>3</sup> ðÞ¼ , <sup>t</sup>þ<sup>4</sup> <sup>2</sup> 5ðtþ1Þ3ðtþ6Þ <sup>3</sup><sup>t</sup> <sup>1</sup> ð�1<sup>Þ</sup> ð�<sup>3</sup><sup>t</sup> <sup>5</sup>�27<sup>t</sup> <sup>4</sup>�125<sup>t</sup> <sup>3</sup>�237<sup>t</sup> <sup>2</sup>�195t�<sup>59</sup>Þ <sup>t</sup> <sup>5</sup>þ<sup>t</sup> <sup>4</sup>þ<sup>t</sup> <sup>3</sup>þ<sup>t</sup> <sup>2</sup>þtþ<sup>1</sup> <sup>3</sup> b tðÞ¼ , fðzÞ ¼ <sup>z</sup>3,g z<sup>ð</sup> Þ ¼ <sup>z</sup> ,c tðÞ¼ <sup>4</sup> , <sup>5</sup>ðtþ1Þðtþ4Þðtþ6Þðtþ9<sup>Þ</sup> <sup>5</sup>ðtþ1Þ ðtþ4Þðtþ6Þðtþ9<sup>Þ</sup> and t ¼ 5n, where n∈ N in system (15). We show that Aðt0; ∞Þ ¼ ∞, Bðt0; ∞Þ ¼ ∞, and Cðt0; ∞Þ , ∞. Indeed,

$$A(\mathfrak{5},T) = \int\_{\mathfrak{5}}^{T} \frac{(t+4)^{\frac{1}{3}}(2t+7)}{\mathfrak{5}(t+1)^{\frac{2}{3}}(t+6)} \Delta t = \sum\_{t \in [\mathfrak{5},T)\_{\mathfrak{5}\mathbb{Z}^+}} \frac{(t+4)^{\frac{1}{3}}(2t+7)}{(t+1)^{\frac{2}{3}}(t+6)}.$$

So as T ! ∞, we have

<sup>∞</sup> ð5n þ 4Þ<sup>3</sup> 1 <sup>ð</sup>10<sup>n</sup> <sup>þ</sup> <sup>7</sup><sup>Þ</sup> <sup>∑</sup> <sup>2</sup> <sup>¼</sup> <sup>∞</sup> by the limit comparison test: Therefore, <sup>A</sup>ð5; <sup>∞</sup>Þ ¼ <sup>∞</sup>: <sup>n</sup><sup>¼</sup> <sup>3</sup> <sup>1</sup> <sup>ð</sup>5<sup>n</sup> <sup>þ</sup> <sup>1</sup>Þ ð5<sup>n</sup> <sup>þ</sup> <sup>6</sup><sup>Þ</sup>

Similarly,

$$B(5,T) \quad = \int\_{5}^{T} \frac{t^{5} + t^{4} + t^{3} + t^{2} + t + 1}{5(t+1)(t+4)(t+6)(t+9)} dt = \sum\_{t \in [5,T]\_{5t^{+}}} \frac{t^{5} + t^{4} + t^{3} + t^{2} + t + 1}{(t+1)(t+4)(t+6)(t+9)}$$

$$\geq \sum\_{t \in [5,T]\_{5t^{+}}} \frac{t^{5}}{(t+1)(t+4)(t+6)(t+9)}.$$

Taking the limit as T ! ∞ gives us

$$B(5,\infty) \ge 625 \cdot \sum\_{n=1}^{\infty} \frac{n^5}{(5n+1)(5n+4)(5n+6)(5n+9)} = \infty$$

by the limit divergence test. Therefore, Bð5; ∞Þ ¼ ∞ by the comparison test. Finally, we show Cðt0; ∞Þ , ∞.

$$C(5,T) \quad = \sum\_{t \in [5,T)\_{\mathbb{S}\mathbb{Z}^+}} \frac{3t^5 + 27t^4 + 125t^3 + 237t^2 + 195t + 59}{(t+1)^4(t+4)(t+6)(t+9)}$$

$$\leq \sum\_{t \in [5,T)\_{\mathbb{S}\mathbb{Z}^+}} \frac{3}{t^2} + \frac{27}{t^3} + \frac{125}{t^5} + \frac{195}{t^6} + \frac{59}{t^7}.$$

So as T ! ∞, we have

Oscillators - Recent Developments

$$C(5,\infty) \le \sum\_{n=1}^{\infty} \frac{3}{n^2} + \frac{27}{n^3} + \frac{125}{n^5} + \frac{195}{n^6} + \frac{59}{n^7} < \infty$$

� � by the geometric series. One can also show that <sup>t</sup>þ<sup>1</sup> <sup>3</sup><sup>t</sup> ð�1<sup>Þ</sup> ð�1<sup>Þ</sup> ; <sup>t</sup>þ<sup>1</sup> <sup>ð</sup>tþ1Þðtþ4<sup>Þ</sup> is an oscillatory solution of system

$$\begin{cases} \mathbf{x}^{\Delta}(t) = \frac{(t+4)^{\frac{1}{2}}(2t+7)}{5(t+1)^{\frac{3}{2}}(t+6)} y^{\frac{1}{2}}(t) \\\\ y^{\Delta}(t) = -\frac{t^{5}+t^{4}+t^{3}+t^{2}+t+1}{5(t+1)(t+4)(t+6)(t+9)} \mathbf{x}^{3}(t) + \frac{(-1)^{3}(-3t^{5}-27t^{4}-125t^{3}-237t^{2}-195t-59)}{5(t+1)^{4}(t+4)(t+6)(t+9)} \end{cases}$$

<sup>h</sup>ðσð Þ<sup>t</sup> Þ�hð Þ<sup>t</sup> where we define <sup>h</sup><sup>Δ</sup>ðÞ¼ <sup>t</sup> for <sup>σ</sup>ðtÞ ¼ <sup>t</sup> <sup>þ</sup> <sup>5</sup> and <sup>μ</sup>ðÞ¼ <sup>t</sup> <sup>5</sup> (see [5]). <sup>μ</sup>ð Þ<sup>t</sup>

#### 5. Conclusion

This chapter focuses on the oscillation/nonoscillation criteria of twodimensional dynamical systems on time scales. We do not only show the oscillatory behaviors of such solutions but also guarantee the existence of such solutions, which might be challenging most of the time for nonlinear systems. In the first and second sections, we present some introductory parts to dynamical systems and basic calculus of the time-scale theory for the readers to comprehend the idea behind the time scales. In Section 3, we consider

$$\begin{cases} \mathcal{X}^\Delta(t) = p(t)f(\mathcal{y}(t)) \\ \mathcal{Y}^\Delta(t) = r(t)\mathcal{g}(\mathcal{x}(t)) \end{cases}$$

and investigate the nonoscillatory behavior of solutions under some certain circumstances. Recall that system (1) turns out to be a differential equation system

$$\begin{cases} \mathbf{x}'(t) = p(t)f(\mathbf{y}(t)) \\ \mathbf{y}'(t) = r(t)\mathbf{g}(\mathbf{x}(t)) \end{cases}$$

when T ¼ R. And the asymptotic behaviors of nonoscillatory solutions were presented by Li in [32]. Also when T ¼ Z, system (1) is reduced to the difference equation system,

$$\begin{cases} \Delta \mathbf{x}\_n = p\_n f(\mathbf{y}\_n) \\ \Delta \mathbf{y}\_n = r\_n \mathbf{g}(\mathbf{x}\_n), \end{cases}$$

and the existence of nonoscillatory solutions were investigated in [33]. Therefore, we unify the results for oscillation and nonoscillation theory, which was shown in R and Z and extends them in one comprehensive theory, which is called time-scale theory. These results were inspired from the book chapter written by Elvan Akın and Özkan Öztürk (see [29]). In that book chapter, it was considered a second-order dynamical system

$$\begin{cases} \varkappa^{\Delta}(t) = p(t)f(\mathbf{y}(t)) \\ \mathbf{y}^{\Delta}(t) = -r(t)\mathbf{g}(\varkappa(t)) \end{cases} \tag{36}$$

Oscillation Criteria of Two-Dimensional Time-Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.83375

and delay system

$$\begin{cases} \varkappa^{\Delta}(t) = p(t)f(\mathbf{y}(t)) \\\\ \jmath^{\Delta}(t) = -r(t)\lg(\varkappa((\tau(t)))), \end{cases} \tag{37}$$

where τ is rd-continuous function such that τð Þt ≤ t and τðÞ!t ∞ as t ! ∞. When the latter systems were considered, because of the negative sign of the second equation of systems, the subclasses for N<sup>þ</sup> an N� would be totally different. So in [29], the existence of nonoscillatory solutions in different subclasses was shown. Another crucial thing on the results is that it is assumed that f must be an odd function for some main results. However, we do not have these strict conditions on our results. Another interesting observation for system (37) is that we lose some subclasses when we consider the delay in system (37). It is because of the setup fixed point theorem and the delay function τ. Therefore, this is a big disadvantage of delayed systems on time scales.

Akın and Öztürk also considered the system

$$\begin{cases} \varkappa^{\Delta}(t) = p(t)|\boldsymbol{y}(t)|^{a} \operatorname{sgn} \boldsymbol{y}(t) \\\\ \boldsymbol{y}^{\Delta}(t) = -r(t)|\boldsymbol{x}(\sigma(t))|^{\beta} \operatorname{sgn} \boldsymbol{x}^{\sigma}(t), \end{cases} \tag{38}$$

where α, β . 0. System (38) is known as Emden-Fowler dynamical systems on time scales in the literature that has been mentioned in Section 1 with applications. Akın et al. [44, 45] showed the asymptotic behavior of nonoscillatory solutions by using α and β relations.

For example, system (38) turns out to be a system of first-order differential equation

$$\begin{cases} \varkappa'(t) = p(t)|\wp(t)|^a \operatorname{sgn} \wp(t) \\\\ \jmath'(t) = -r(t)|\varkappa(t)|^\beta \operatorname{sgn} \varkappa(t), \end{cases}$$

when the time scale T ¼ R. On the other hand, system (38) ends up with the system of difference equations

$$\begin{cases} \Delta \mathbf{x}\_n = p\_n |\mathbf{y}\_n|^a \operatorname{sgn} \mathbf{y}\_n \\\\ \Delta \mathbf{y}\_n = -r\_n |\mathbf{x}\_{n+1}|^\beta \operatorname{sgn} \mathbf{x}\_{n+1}. \end{cases}$$

when the time scale T ¼ Z. For both cases, several contributions have been made by Zuzana et al. in [46] and [47], respectively.

� Finally, we finish this section with the following tables, showing summaries about the existence of nonoscillatory solutions of system (1) in N<sup>þ</sup> and N (Tables 3 and 4).


Table 3. <sup>þ</sup> Existence for (1) in N .


Table 4. Existence for (1) in N�.

#### A. Appendix

We give the following exercises to the interested readers that help them practicing the theoretical results. The examples are in q-calculus which takes too much attention recently. Recall from Example 3.13 that Δ<sup>q</sup> is defined as

$$
\Delta\_q f(t) = \frac{f(tq) - f(t)}{(q-1)t}.\tag{39}
$$

With the help of Eq. (39), we provide the following exercises. Exercise 6.1 Let <sup>T</sup> <sup>¼</sup> <sup>2</sup><sup>N</sup><sup>0</sup> : Consider the following system:

$$\begin{cases} \Delta\_q \boldsymbol{\chi}(t) = \frac{1}{4t^2(1+t)^{\frac{1}{\boldsymbol{\gamma}}}} (\boldsymbol{\chi}(t))^{\frac{1}{\boldsymbol{\gamma}}} \\\\ \Delta\_q \boldsymbol{\chi}(t) = \frac{2t}{4t-1} \boldsymbol{\chi}(t) \end{cases} \tag{40}$$

� � <sup>2</sup> � <sup>1</sup> <sup>2</sup><sup>t</sup> is a nonoscillatory solution of Eq. (40) in N<sup>þ</sup> and show that ; t þ 1 F,<sup>∞</sup> 6¼ ø by checking the conditions given in Theorem 3.12 for k<sup>1</sup> ¼ 1.

Exercise 6.2 Let T ¼ qN<sup>0</sup> , q . 1. Consider the following system:

$$\begin{cases} \Delta \mathbf{x}\_q(t) = \frac{1}{qt^{\frac{8}{7}}(2t^2 + 1)^{\frac{1}{7}}} (\boldsymbol{y}(t))^{\frac{1}{7}} \\ \Delta \mathbf{y}\_q(t) = \frac{q+1}{q^2 t^2 (t+1)} \boldsymbol{x}(t), \end{cases} \tag{41}$$

� � and show that there exists <sup>a</sup> nonoscillatory solution of <sup>h</sup>ðσð Þ<sup>t</sup> Þ�h tð Þ where <sup>Δ</sup>hqðtÞ ¼ <sup>μ</sup>ð Þ<sup>t</sup> <sup>1</sup> <sup>þ</sup> <sup>1</sup> ; �<sup>2</sup> � <sup>1</sup> t t<sup>2</sup> , in N� F,F system (41), given by 6¼ ø by Theorem 3.15 for k<sup>3</sup> ¼ �1 and k<sup>4</sup> ¼ 1.

Oscillation Criteria of Two-Dimensional Time-Scale Systems DOI: http://dx.doi.org/10.5772/intechopen.83375

## Author details

Ozkan Ozturk

Department of Mathematics, College of Engineering, American University of the Middle East, Eqaila, Kuwait

\*Address all correspondence to: ozkan.ozturk@aum.edu.kw

© 2019 The Author(s). Licensee IntechOpen. This chapteris distributed underthe terms oftheCreative 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.

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Section 2
