1. Introduction

Arbitrary-order differential equations are the excellent tools in the description of many phenomena and process in different fields of science, technology, and engineering (see [1, 2]). Therefore, considerable attention has been paid to the subject of differential equations of arbitrary order (see [3–5] and the references therein). The area devoted to the existence of positive solutions to fractional differential equations and their system especially coupled systems was greatly studied by many authors (for details see [6–9]). In all these articles, the concerned results were obtained by using classical fixed point theorems like Banach contraction principle, Leray-Schauder fixed point theorem, and fixed point theorems of cone type. The aforesaid area has been very well explored for both ordinary- and arbitrary-order differential equations. Existence and uniqueness

© 2016 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. © 2018 The Author(s). Licensee IntechOpen. 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.

results for nonlinear and linear, classical, as well as arbitrary-order differential equations have been investigated in many papers (see few of them as [10–13]). results for nonlinear and linear, classical, as well as arbitrary-order differential equations have been investigated in many papers (see few of them as [10–13]).

where j ¼ 0; 1; 2, ⋯m � 2, m ≥ 3, I ¼ ½ � 0; 1 , η, ξ∈ð Þ 0; 1 , H1, H<sup>2</sup> : ½ �� 0; 1 f g0 ∪R<sup>þ</sup> � f g0 ∪R<sup>þ</sup> ! f g<sup>0</sup> <sup>∪</sup>R<sup>þ</sup> are continuous functions, and <sup>D</sup><sup>θ</sup><sup>1</sup> , <sup>D</sup><sup>θ</sup><sup>2</sup> stand for Riemann-Liouville fractional derivative of order θ1, θ<sup>2</sup> in sequel. We obtain necessary and sufficient conditions for the existence of solution to system (1) by using another type of fixed point result based on a concave-type operator with increasing or decreasing property. The idea then extends to form some conditions which ensure multiplicity of solutions to the considered problem. Also, we discuss some results about the Hyers-Ulam stability for the considered problem. Further by providing examples, we illustrate the

where j ¼ 0;1; 2, ⋯m � 2, m ≥ 3, I ¼ ½ � 0; 1 , η, ξ∈ð Þ 0; 1 , H1, H<sup>2</sup> :½ �� 0; 1 f g0 ∪R<sup>þ</sup> � f g0 ∪R<sup>þ</sup> ! f g<sup>0</sup> <sup>∪</sup>R<sup>þ</sup> are continuous functions, and <sup>D</sup><sup>θ</sup><sup>1</sup> , <sup>D</sup><sup>θ</sup><sup>2</sup> stand for Riemann-Liouville fractional derivative of order θ1, θ<sup>2</sup> in sequel. We obtain necessary and sufficient conditions for the existence of solution to system (1) by using another type of fixed point result based on a concave-type operator with increasing or decreasing property. The idea then extends to form some conditions which ensure multiplicity of solutions to the considered problem. Also, we discuss some results about the Hyers-Ulam stability for the considered problem. Further by providing examples, we illustrate the

In the current section, we review few fundamental lemmas and results found in [2, 4, 6, 28, 29].

In the current section, we review few fundamental lemmas and results found in [2, 4, 6, 28, 29].

ðt 0

1 Γð Þ θ<sup>1</sup>

Definition 2.2. Arbitrary-order derivative in Riemann-Liouville sense for a function ψ∈ ð Þ ð Þ 0; ∞ ; R

Definition 2.2. Arbitrary-order derivative in Riemann-Liouville sense for a functionψ∈ ð Þ ð Þ 0; ∞ ; R

Lemma 2.3. [16] Let θ<sup>1</sup> > 0, then for arbitrary Cj ∈ R, j ¼ 1, 2, …, m, m ¼ ½ �þ θ<sup>1</sup> 1, and the

Lemma 2.3. [16] Let θ<sup>1</sup> > 0, then for arbitrary Cj ∈ R, j ¼ 1, 2, …, m, m ¼ ½ �þ θ<sup>1</sup> 1, and the

<sup>D</sup><sup>θ</sup>1ψðÞ¼ <sup>t</sup> f tð Þ

<sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> <sup>C</sup>2<sup>t</sup>

Definition 2.4. [17, 28] Consider a Banach space E with a closed set C ⊂ E. Then, C is said to be partially ordered if p⪯q such that q � p ∈ C: Further, C is said to be a cone if it holds the given

<sup>D</sup><sup>θ</sup>1ψðÞ¼ <sup>t</sup> f tð Þ

<sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> <sup>C</sup>2<sup>t</sup>

Definition 2.4. [17, 28] Consider a Banach space E with a closed set C ⊂ E. Then, C is said to be partially ordered if p⪯q such that q � p ∈ C: Further, C is said to be a cone if it holds the given

ð Þ <sup>t</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

ð Þ <sup>t</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

ðt 0

ψð Þs ds,

ψð Þs ds,

Existence Theory of Differential Equations of Arbitrary Order

http://dx.doi.org/10.5772/intechopen.75523

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

37

37

ψð Þs ds, θ<sup>1</sup> > 0, where m ¼ ½ �þ θ<sup>1</sup> 1:

ψð Þs ds,θ<sup>1</sup> > 0, wherem ¼ ½ �þ θ<sup>1</sup> 1:

<sup>θ</sup>1�<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> Cmt

<sup>θ</sup>1�<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> Cmt

<sup>θ</sup>1�<sup>m</sup>:

<sup>θ</sup>1�<sup>m</sup>:

Definition 2.1. Arbitrary-order integral of function ψ : ð Þ! 0; ∞ R is recalled as

<sup>I</sup><sup>θ</sup>1ψðÞ¼ <sup>t</sup>

where θ<sup>1</sup> > 0 is a real number and also the integral is pointwise defined on R<sup>þ</sup>

1 Γð Þ θ<sup>1</sup>

Definition 2.1. Arbitrary-order integral of functionψ : ð Þ! 0; ∞ R is recalled as

<sup>I</sup><sup>θ</sup>1ψðÞ¼ <sup>t</sup>

where θ<sup>1</sup> > 0 is a real number and also the integral is pointwise defined on R<sup>þ</sup>

ð Þ <sup>t</sup> � <sup>s</sup> <sup>m</sup>�θ1�<sup>1</sup> Γð Þ m � θ<sup>1</sup>

ð Þ <sup>t</sup> � <sup>s</sup> <sup>m</sup>�θ1�<sup>1</sup> Γð Þ m � θ<sup>1</sup>

established results.

established results.

2. Preliminaries

2. Preliminaries

is given by

is given by

solution of

solution of

is provided by

is provided by

conditions:

conditions:

<sup>D</sup><sup>θ</sup>1ψðÞ¼ <sup>t</sup>

<sup>D</sup><sup>θ</sup>1ψðÞ¼ <sup>t</sup>

d dt � �<sup>m</sup> <sup>ð</sup><sup>t</sup> 0

d dt � �<sup>m</sup> <sup>ð</sup><sup>t</sup> 0

<sup>ψ</sup>ðÞ¼ <sup>t</sup> <sup>I</sup><sup>θ</sup><sup>1</sup> f tð Þþ <sup>C</sup>1<sup>t</sup>

<sup>ψ</sup>ðÞ¼ <sup>t</sup> <sup>I</sup><sup>θ</sup><sup>1</sup> f tð Þþ <sup>C</sup>1<sup>t</sup>

1. p ∈ C and for a real constant κ ≥ 0 the relation κp∈ C holds.

2. p and �p∈ C yield that 0 ∈ C, where 0 is zero element of Banach space E

2. p and �p∈ C yield that 0 ∈ C, where 0 is zero element of Banach space E

1. p ∈ C and for a real constant κ ≥ 0 the relation κp∈ C holds.

Another warm area of research in the theory of fractional-order differential equations (FDEs) is devoted to the multiplicity of solutions. Plenty of research articles are available on this topic in literature. In [14], the author studied the given boundary value problem (BVP) for existence of multiple solutions: Another warm area of research in the theory of fractional-order differential equations (FDEs) is devoted to the multiplicity of solutions. Plenty of research articles are available on this topic in literature. In [14], the author studied the given boundary value problem (BVP) for existence of multiple solutions:

$$\begin{cases} \mathcal{B}^{\theta\_1} p(t) + \mathcal{H}(t, p(t)) = 0, & t \in \mathbf{I}, \ \theta\_1 \in (1, 2], \\\\ p(t)|\_{t=0} = p(t)|\_{t=1} = 0. \end{cases}$$

where D is the Riemann-Liouville derivative of non-integer order and I ¼ ½ � 0; 1 . In same line, Kaufmann and Mboumi [15] studied the given boundary value problem of fractional differential equations for multiplicity of positive solutions: where D is the Riemann-Liouville derivative of non-integer order and I ¼½ � 0; 1 . In same line, Kaufmann and Mboumi [15] studied the given boundary value problem of fractional differential equations for multiplicity of positive solutions:

$$\begin{cases} \mathcal{B}^{\theta\_1} p(t) + \phi(t) \mathcal{H}(t, p(t)) = 0, & t \in \mathcal{I}, \ \theta\_1 \in (1, 2], \\ p(t)|\_{t=0} = p'(t)|\_{t=1} = 0, \end{cases}$$

where D is the Riemann-Liouville derivative and ϕ∈ Cð Þ I; R , H ∈ Cð Þ I � R; R : where D is the Riemann-Liouville derivative and ϕ∈ Cð Þ I; R , H ∈ Cð Þ I � R; R :

In the last few decades, the theory devoted to the multiplicity of solutions is very well extended to coupled systems of nonlinear FDEs, and we refer to few papers in [16–18]. Wang et al. [19] established some conditions under which the given system of three point BVP In the last few decades, the theory devoted to the multiplicity of solutions is very well extended to coupled systems of nonlinear FDEs, and we refer to few papers in [16–18]. Wang et al. [19] established some conditions under which the given system of three point BVP

$$\begin{cases} \mathcal{B}^{\theta\_1} p(t) = \mathcal{H}\_1(t, q(t)); & t \in \mathcal{I}, \\\\ \mathcal{B}^{\theta\_2} q(t) = \mathcal{H}\_2(t, p(t)); & t \in \mathcal{I}, \\\\ p(t)\_{t=0} = 0, \ p(t)\_{t=1} = \mu p(t)|\_{t=\xi}, & q(t)|\_{t=0} = 0, \ \end{cases}$$

� t¼ξ,

(1)

has a solution, where θ1, θ<sup>2</sup> ∈ ð � 1; 2 and μ, ν∈I, ξ∈ ð Þ 0; 1 , H<sup>i</sup> : ½ �� 0; 1 R ! R for i ¼ 1, 2 are nonlinear functions. has a solution, whereθ1, θ<sup>2</sup> ∈ ð � 1; 2 and μ, ν∈I, ξ∈ ð Þ 0; 1 , H<sup>i</sup> :½ �� 0; 1 R ! Rfor i ¼ 1, 2 are nonlinear functions.

In the last few decades, another important aspect devoted to stability analysis of FDEs with initial/boundary conditions has been given much attention. This is because stability is very important from the numerical and optimization point of view. Various forms of stabilities were studied for the aforesaid FDEs including exponential, Mittag-Leffler, and Lyapunov stability. Recently, Hyers-Ulam stability has given more attention. This concept was initially introduced by Ulam and then by Hyers (for details see [20–22]). Now, many articles have been written on this concept (see [23–27]). So far, the aforementioned stability has not yet well studied for multipoint BVPs of FDEs. Motivated by the aforesaid discussion, we propose the following coupled system of four-point BVP provided as In the last few decades, another important aspect devoted to stability analysis of FDEs with initial/boundary conditions has been given much attention. This is because stability is very important from the numerical and optimization point of view. Various forms of stabilities were studied for the aforesaid FDEs including exponential, Mittag-Leffler, and Lyapunov stability. Recently, Hyers-Ulam stability has given more attention. This concept was initially introduced by Ulam and then by Hyers (for details see [20–22]). Now, many articles have been written on this concept (see [23–27]). So far, the aforementioned stability has not yet well studied for multipoint BVPs of FDEs. Motivated by the aforesaid discussion, we propose the following coupled system of four-point BVP provided as

$$\begin{cases} \mathcal{B}^{\theta\_1} p(t) = \mathcal{H}\_1(t, p(t), q(t)); \quad t \in \mathcal{I}; \ \mathcal{O}\_1 \in (m-1, m]\_{\prime} \\\\ \mathcal{B}^{\theta\_2} q(t) = \mathcal{H}\_2(t, p(t), q(t)); \quad t \in \mathcal{I}; \ \mathcal{O}\_2 \in (m-1, m]\_{\prime} \\\\ p^{(j)}(t)\_{t=0} = q^{(j)}(t)|\_{t=0} = 0, \ \left. p(t)|\_{t=1} = p(t)|\_{t=\eta^{\prime}} \right. \end{cases} \tag{1}$$

where j ¼ 0; 1; 2, ⋯m � 2, m ≥ 3, I ¼ ½ � 0; 1 , η, ξ∈ð Þ 0; 1 , H1, H<sup>2</sup> : ½ �� 0; 1 f g0 ∪R<sup>þ</sup> � f g0 ∪R<sup>þ</sup> ! f g<sup>0</sup> <sup>∪</sup>R<sup>þ</sup> are continuous functions, and <sup>D</sup><sup>θ</sup><sup>1</sup> , <sup>D</sup><sup>θ</sup><sup>2</sup> stand for Riemann-Liouville fractional derivative of order θ1, θ<sup>2</sup> in sequel. We obtain necessary and sufficient conditions for the existence of solution to system (1) by using another type of fixed point result based on a concave-type operator with increasing or decreasing property. The idea then extends to form some conditions which ensure multiplicity of solutions to the considered problem. Also, we discuss some results about the Hyers-Ulam stability for the considered problem. Further by providing examples, we illustrate the established results. where j ¼ 0; 1; 2, ⋯m � 2, m ≥ 3, I ¼½ �0; 1 , η, ξ∈ð Þ 0; 1 , H1, H<sup>2</sup> :½ �� 0; 1 f g0 ∪R<sup>þ</sup> � f g0 ∪R<sup>þ</sup> ! f g<sup>0</sup> <sup>∪</sup>R<sup>þ</sup> are continuous functions, and <sup>D</sup><sup>θ</sup><sup>1</sup> , <sup>D</sup><sup>θ</sup><sup>2</sup> stand for Riemann-Liouville fractional derivative of order θ1, θ<sup>2</sup> in sequel. We obtain necessary and sufficient conditions for the existence of solution to system (1) by using another type of fixed point result based on a concave-type operator with increasing or decreasing property. The idea then extends to form some conditions which ensure multiplicity of solutions to the considered problem. Also, we discuss some results about the Hyers-Ulam stability for the considered problem. Further by providing examples, we illustrate the established results.

#### 2. Preliminaries 2. Preliminaries

results for nonlinear and linear, classical, as well as arbitrary-order differential equations have

results for nonlinear and linear, classical, as well as arbitrary-order differential equations have

Another warm area of research in the theory of fractional-order differential equations (FDEs) is devoted to the multiplicity of solutions. Plenty of research articles are available on this topic in literature. In [14], the author studied the given boundary value problem (BVP) for existence of

<sup>D</sup><sup>θ</sup><sup>1</sup> p tð Þþ <sup>H</sup>ð Þ¼ <sup>t</sup>; p tð Þ <sup>0</sup>, t<sup>∈</sup> <sup>I</sup>, <sup>θ</sup><sup>1</sup> <sup>∈</sup>ð � <sup>1</sup>; <sup>2</sup> ,

where D is the Riemann-Liouville derivative of non-integer order and I ¼½ � 0; 1 . In same line, Kaufmann and Mboumi [15] studied the given boundary value problem of fractional differen-

<sup>D</sup><sup>θ</sup><sup>1</sup> p tð Þþ <sup>ϕ</sup>ð Þ<sup>t</sup> <sup>H</sup>ð Þ¼ <sup>t</sup>; p tð Þ <sup>0</sup>, t <sup>∈</sup>I, <sup>θ</sup><sup>1</sup> <sup>∈</sup> ð � <sup>1</sup>; <sup>2</sup> ,

In the last few decades, the theory devoted to the multiplicity of solutions is very well extended to coupled systems of nonlinear FDEs, and we refer to few papers in [16–18]. Wang et al. [19] established some conditions under which the given system of three point BVP

Another warm area of research in the theory of fractional-order differential equations (FDEs) is devoted to the multiplicity of solutions. Plenty of research articles are available on this topic in literature. In [14], the author studied the given boundary value problem (BVP) for existence of

<sup>D</sup><sup>θ</sup><sup>1</sup> p tð Þþ <sup>H</sup>ð Þ¼ <sup>t</sup>; p tð Þ <sup>0</sup>, t<sup>∈</sup> <sup>I</sup>, <sup>θ</sup><sup>1</sup> <sup>∈</sup>ð � <sup>1</sup>; <sup>2</sup> ,

where D is the Riemann-Liouville derivative of non-integer order and I ¼ ½ � 0; 1 . In same line, Kaufmann and Mboumi [15] studied the given boundary value problem of fractional differen-

<sup>D</sup><sup>θ</sup><sup>1</sup> p tð Þþ <sup>ϕ</sup>ð Þ<sup>t</sup> <sup>H</sup>ð Þ¼ <sup>t</sup>; p tð Þ <sup>0</sup>, t <sup>∈</sup>I, <sup>θ</sup><sup>1</sup> <sup>∈</sup> ð � <sup>1</sup>; <sup>2</sup> ,

In the last few decades, the theory devoted to the multiplicity of solutions is very well extended to coupled systems of nonlinear FDEs, and we refer to few papers in [16–18]. Wang et al. [19] established some conditions under which the given system of three point BVP

�

<sup>D</sup><sup>θ</sup><sup>1</sup> p tðÞ¼ <sup>H</sup>1ð Þ <sup>t</sup>; p tð Þ; q tð Þ ; t <sup>∈</sup>I; <sup>θ</sup><sup>1</sup> <sup>∈</sup>ð � <sup>m</sup> � <sup>1</sup>; <sup>m</sup> , <sup>D</sup><sup>θ</sup><sup>2</sup> q tðÞ¼ <sup>H</sup>2ð Þ <sup>t</sup>; p tð Þ; q tð Þ ; t∈I; <sup>θ</sup><sup>2</sup> <sup>∈</sup> ð � <sup>m</sup> � <sup>1</sup>; <sup>m</sup> ,

<sup>D</sup><sup>θ</sup><sup>1</sup> p tðÞ¼ <sup>H</sup>1ð Þt; p tð Þ; q tð Þ ; t∈I; <sup>θ</sup><sup>1</sup> <sup>∈</sup>ð � <sup>m</sup> � <sup>1</sup>; <sup>m</sup> , <sup>D</sup><sup>θ</sup><sup>2</sup> q tðÞ¼ <sup>H</sup>2ð Þ <sup>t</sup>; p tð Þ; q tð Þ ; t∈I; <sup>θ</sup><sup>2</sup> <sup>∈</sup> ð � <sup>m</sup> � <sup>1</sup>; <sup>m</sup> ,

has a solution, where θ1, θ<sup>2</sup> ∈ ð � 1; 2 and μ, ν∈I, ξ∈ ð Þ 0; 1 , H<sup>i</sup> : ½ �� 0; 1 R ! R for i ¼ 1, 2 are

has a solution, whereθ1, θ<sup>2</sup> ∈ ð � 1; 2 and μ, ν∈I, ξ∈ ð Þ 0; 1 , H<sup>i</sup> :½ �� 0; 1 R ! R for i ¼ 1, 2 are

In the last few decades, another important aspect devoted to stability analysis of FDEs with initial/boundary conditions has been given much attention. This is because stability is very important from the numerical and optimization point of view. Various forms of stabilities were studied for the aforesaid FDEs including exponential, Mittag-Leffler, and Lyapunov stability. Recently, Hyers-Ulam stability has given more attention. This concept was initially introduced by Ulam and then by Hyers (for details see [20–22]). Now, many articles have been written on this concept (see [23–27]). So far, the aforementioned stability has not yet well studied for multipoint BVPs of FDEs. Motivated by the aforesaid discussion, we propose the following

�

In the last few decades, another important aspect devoted to stability analysis of FDEs with initial/boundary conditions has been given much attention. This is because stability is very important from the numerical and optimization point of view. Various forms of stabilities were studied for the aforesaid FDEs including exponential, Mittag-Leffler, and Lyapunov stability. Recently, Hyers-Ulam stability has given more attention. This concept was initially introduced by Ulam and then by Hyers (for details see [20–22]). Now, many articles have been written on this concept (see [23–27]). So far, the aforementioned stability has not yet well studied for multipoint BVPs of FDEs. Motivated by the aforesaid discussion, we propose the following

<sup>t</sup>¼<sup>ξ</sup>; q tð Þ <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>; q tð Þ<sup>t</sup>¼<sup>1</sup> <sup>¼</sup> <sup>ν</sup>q tð Þ � � �

<sup>t</sup>¼<sup>ξ</sup>; q tð Þ <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>; q tð Þ<sup>t</sup>¼<sup>1</sup> <sup>¼</sup> <sup>ν</sup>q tð Þ � � �

<sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>, ptð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>η</sup>, qtð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> q tð Þ <sup>t</sup>¼<sup>ξ</sup>:

<sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>, ptð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>η</sup>, qtð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> q tð Þ <sup>t</sup>¼<sup>ξ</sup>:

� t¼ξ,

� t¼ξ,

(1)

(1)

p tð Þj j <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> <sup>0</sup>:

p tð Þj j<sup>t</sup>¼<sup>0</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> <sup>0</sup>:

0 ð Þ<sup>t</sup> � � �

p tð Þ <sup>t</sup>¼<sup>0</sup> ¼ p

� <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> <sup>0</sup>,

0 ð Þ<sup>t</sup> � � �

� <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> <sup>0</sup>,

where D is the Riemann-Liouville derivative and ϕ∈ Cð Þ I; R , H ∈ Cð Þ I � R; R :

where D is the Riemann-Liouville derivative and ϕ∈ Cð Þ I; R , H ∈ Cð Þ I � R; R :

been investigated in many papers (see few of them as [10–13]).

been investigated in many papers (see few of them as [10–13]).

(

(

36 Differential Equations - Theory and Current Research

36 Differential Equations - Theory and Current Research

tial equations for multiplicity of positive solutions:

(

coupled system of four-point BVP provided as

coupled system of four-point BVP provided as

8 >>>< 8 >>><

>>>:

>>>:

<sup>p</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>q</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup>

<sup>p</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>q</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup>

� � � �

8 >>>< 8 >>><

>>>:

>>>:

nonlinear functions.

nonlinear functions.

p tð Þ <sup>t</sup>¼<sup>0</sup> ¼ p

tial equations for multiplicity of positive solutions:

(

<sup>D</sup><sup>θ</sup><sup>1</sup> p tðÞ¼ <sup>H</sup>1ð Þ <sup>t</sup>; q tð Þ ; t <sup>∈</sup>I, <sup>D</sup><sup>θ</sup><sup>2</sup> q tðÞ¼ <sup>H</sup>2ð Þ <sup>t</sup>; p tð Þ ; t <sup>∈</sup>I, p tð Þ<sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>; p tð Þ<sup>t</sup>¼<sup>1</sup> <sup>¼</sup> <sup>μ</sup>p tð Þ�

<sup>D</sup><sup>θ</sup><sup>1</sup> p tðÞ¼ <sup>H</sup>1ð Þ <sup>t</sup>; q tð Þ ; t <sup>∈</sup>I, <sup>D</sup><sup>θ</sup><sup>2</sup> q tðÞ¼ <sup>H</sup>2ð Þ <sup>t</sup>; p tð Þ ; t <sup>∈</sup>I, p tð Þ<sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>; p tð Þ<sup>t</sup>¼<sup>1</sup> <sup>¼</sup> <sup>μ</sup>p tð Þ�

multiple solutions:

multiple solutions:

In the current section, we review few fundamental lemmas and results found in [2, 4, 6, 28, 29]. In the current section, we review few fundamental lemmas and results found in [2, 4, 6, 28, 29].

Definition 2.1. Arbitrary-order integral of function ψ : ð Þ! 0; ∞ R is recalled as Definition 2.1. Arbitrary-order integral of functionψ : ð Þ! 0; ∞ R is recalled as

$$\mathcal{T}^{\theta\_1}\psi(t) = \frac{1}{\Gamma(\theta\_1)} \int\_0^t (t-s)^{\theta\_1-1} \psi(s)ds,$$

where θ<sup>1</sup> > 0 is a real number and also the integral is pointwise defined on R<sup>þ</sup> where θ<sup>1</sup> > 0 is a real number and also the integral is pointwise defined on R<sup>þ</sup>

Definition 2.2. Arbitrary-order derivative in Riemann-Liouville sense for a function ψ∈ ð Þ ð Þ 0; ∞ ; R is given by Definition 2.2. Arbitrary-order derivative in Riemann-Liouville sense for a functionψ∈ ð Þ ð Þ 0; ∞ ; R is given by

$$\mathcal{B}^{\theta\_1}\psi(t) = \left(\frac{d}{dt}\right)^m \int\_0^t \frac{(t-s)^{m-\theta\_1-1}}{\Gamma(m-\theta\_1)}\psi(s)ds,\\ \theta\_1 > 0, \text{where } m = [\theta\_1] + 1.$$

Lemma 2.3. [16] Let θ<sup>1</sup> > 0, then for arbitrary Cj ∈ R, j ¼ 1, 2, …, m, m ¼ ½ �þ θ<sup>1</sup> 1, and the solution of Lemma 2.3. [16]Let θ<sup>1</sup> > 0, then for arbitrary Cj ∈ R, j ¼ 1, 2, …, m, m ¼ ½ �þ θ<sup>1</sup> 1, and the solution of

$$
\mathcal{B}^{\theta\_1} \psi(t) = f(t).
$$

is provided by is provided by

$$
\psi(t) = \mathcal{T}^{\theta\_1} f(t) + \mathsf{C}\_1 t^{\theta\_1 - 1} + \mathsf{C}\_2 t^{\theta\_1 - 2} + \dots + \mathsf{C}\_m t^{\theta\_1 - m}.
$$

Definition 2.4. [17, 28] Consider a Banach space E with a closed set C ⊂ E. Then, C is said to be partially ordered if p⪯q such that q � p ∈ C: Further, C is said to be a cone if it holds the given conditions: Definition 2.4. [17, 28] Consider a Banach space E with a closed set C ⊂ E. Then, C is said to be partially ordered if p⪯q such that q � p ∈ C: Further, C is said to be a cone if it holds the given conditions:


Definition 2.5. [17, 28] A closed and convex set C of E is said to be a normal cone if it obeys the given properties: Definition 2.5. [17, 28] A closed and convex set C of E is said to be a normal cone if it obeys the given properties:

where Gð Þ t;s is the Green's function defined by

8

>>>>>>>>>>>>><

1 Γð Þ θ<sup>1</sup>

>>>>>>>>>>>>>:

1 λ1

8

>>>>>>>>>>>>><

1 λ1

1 λ1

� 1 λ1

� 1 λ1

Proof. In view of Lemma 2.3, we may write Eq. (2) as

where Gð Þ t;s is the Green's function defined by

1 λ1

� 1 λ1

� 1 λ1

>>>>>>>>>>>>>:

Proof. In view of Lemma 2.3, we may write Eq. (2) as

�½ � <sup>t</sup>ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> ½ � <sup>t</sup>ð Þ <sup>η</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup> h i <sup>þ</sup> ð Þ <sup>t</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

�½ � <sup>t</sup>ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> ½ � <sup>t</sup>ð Þ <sup>η</sup>�<sup>s</sup> <sup>θ</sup>1�<sup>1</sup> h i <sup>þ</sup> ð Þ <sup>t</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

�½ � <sup>t</sup>ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> ½ � <sup>t</sup>ð Þ <sup>η</sup>�<sup>s</sup> <sup>θ</sup>1�<sup>1</sup> h i, <sup>0</sup>≤<sup>t</sup> <sup>≤</sup><sup>s</sup> <sup>≤</sup> <sup>η</sup> <sup>≤</sup> <sup>1</sup>,

�½ � <sup>t</sup>ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> ½ � <sup>t</sup>ð Þ <sup>η</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup> h i, <sup>0</sup> <sup>≤</sup> <sup>t</sup> <sup>≤</sup> <sup>s</sup> <sup>≤</sup> <sup>η</sup> <sup>≤</sup> <sup>1</sup>,

, 0 ≤ η ≤ t ≤ s ≤ 1:

, 0 ≤ η≤t≤ s ≤ 1:

<sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> <sup>C</sup>2<sup>t</sup>

<sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> <sup>C</sup>2<sup>t</sup>

In view of conditions <sup>p</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, …<sup>m</sup> � <sup>2</sup>, m <sup>≥</sup> <sup>3</sup>, , Eq. (5) suffers from singularity;

In view of conditions <sup>p</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, …<sup>m</sup> � <sup>2</sup>, m <sup>≥</sup> <sup>3</sup>, , Eq. (5) suffers from singularity;

p tðÞ¼ <sup>I</sup><sup>θ</sup>1φð Þþ <sup>t</sup> <sup>C</sup>1<sup>t</sup>

p tðÞ¼ <sup>I</sup><sup>θ</sup>1φð Þþ <sup>t</sup> <sup>C</sup>1<sup>t</sup>

t θ1�1 λ1

<sup>0</sup> Gð Þ t; φð Þs ds:

t θ1�1 λ1

<sup>0</sup> Gð Þ t;s φð Þs ds:

p tðÞ¼ <sup>I</sup><sup>θ</sup>1φð Þþ <sup>t</sup>

, 0 ≤ η ≤ s ≤ t ≤ 1,

, 0 ≤ η ≤ s≤ t ≤ 1,

<sup>θ</sup>1�<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> Cmt

<sup>θ</sup>1�<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> Cmt

θ1�1

θ1�1

<sup>I</sup><sup>θ</sup>1φ ηð Þ� <sup>I</sup><sup>θ</sup>1φð Þ<sup>1</sup> � �

<sup>I</sup><sup>θ</sup>1φ ηð Þ� <sup>I</sup><sup>θ</sup>1φð Þ<sup>1</sup> � �

<sup>0</sup> G1ð Þ t;s H1ð Þ s; p sð Þ; q sð Þ ds,

<sup>0</sup> G1ð Þ t;s H1ð Þ s; p sð Þ; q sð Þ ds,

<sup>0</sup> G2ð Þ t;s H2ð Þ s; p sð Þ; q sð Þ ds,

<sup>0</sup> G2ð Þ t;s H2ð Þ s; p sð Þ; q sð Þ ds,

where G1ð Þ t;s , G2ð Þ t;s are Green's functions, which can be similarly computed like in Theorem

where G1ð Þ t;s , G2ð Þ t;s are Green's functions, which can be similarly computed like in Theorem

<sup>λ</sup><sup>1</sup> ¼ G1ð Þ 1;s , for all s∈ I,

<sup>λ</sup><sup>2</sup> ¼ G2ð Þ 1;s , for all s∈ I;

<sup>2</sup> Gð Þ 1;s for every θ s∈ ð Þ 0; 1 ;

<sup>2</sup> Gð Þ 1;s for every θ s∈ ð Þ 0; 1 ;

;γ<sup>2</sup> <sup>¼</sup> θθ2�<sup>1</sup> � �:

<sup>λ</sup><sup>1</sup> ¼ G1ð Þ 1;s , for all s∈ I,

3.1. Further, they are continuous on I � I and satisfy the following properties:

<sup>λ</sup><sup>2</sup> ¼ G2ð Þ 1;s , for all s∈ I;

<sup>2</sup> Gð Þ 1;s for every θ s∈ ð Þ 0; 1 ;

<sup>2</sup> Gð Þ 1;s for every θ s∈ ð Þ 0; 1 ;

; <sup>γ</sup><sup>2</sup> <sup>¼</sup> θθ2�<sup>1</sup> � �:

½ � <sup>t</sup>ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> ð Þ <sup>t</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

½ � <sup>t</sup>ð Þ <sup>1</sup>�<sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

½ � <sup>t</sup>ð Þ <sup>1</sup>�<sup>s</sup> <sup>θ</sup>1�<sup>1</sup> <sup>þ</sup> ð Þ <sup>t</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

½ � <sup>t</sup>ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

p tðÞ¼ <sup>I</sup><sup>θ</sup>1φð Þþ <sup>t</sup> <sup>C</sup>1<sup>t</sup>

therefore, we have C<sup>2</sup> ¼ C<sup>3</sup> ¼ … ¼ Cn ¼ 0: Hence, Eq. (5) becomes

p tðÞ¼ <sup>I</sup><sup>θ</sup>1φð Þþ <sup>t</sup> <sup>C</sup>1<sup>t</sup>

therefore, we have C<sup>2</sup> ¼ C<sup>3</sup> ¼ … ¼ Cn ¼ 0: Hence, Eq. (5) becomes

Applying boundary condition p tð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>η</sup> and <sup>d</sup> <sup>¼</sup> <sup>1</sup> � ηθ

p tðÞ¼ <sup>Ð</sup> <sup>1</sup>

where Gð Þ t;s is Green's function given in Eq. (4).

In view of Theorem 3.1 and using <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> � ηθ1�<sup>1</sup>, <sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>ξ</sup><sup>θ</sup>2�<sup>1</sup>

system of integral equations to the proposed system (1) is given as

8 < :

8 < :

p tðÞ¼ <sup>Ð</sup> <sup>1</sup>

In view of Theorem 3.1 and using <sup>λ</sup><sup>1</sup> <sup>¼</sup><sup>1</sup> � ηθ1�<sup>1</sup>, <sup>λ</sup><sup>2</sup> <sup>¼</sup><sup>1</sup> � <sup>ξ</sup><sup>θ</sup>2�<sup>1</sup>

system of integral equations to the proposed system (1) is given as

p tðÞ¼ <sup>Ð</sup> <sup>1</sup>

q tðÞ¼ <sup>Ð</sup> <sup>1</sup>

q tðÞ¼ <sup>Ð</sup> <sup>1</sup>

3.1. Further, they are continuous on I � I and satisfy the following properties:

where Gð Þ t;s is Green's function given in Eq. (4).

i. max<sup>t</sup> <sup>∈</sup><sup>I</sup>∣G1ð Þ <sup>t</sup>;<sup>s</sup> <sup>∣</sup> <sup>≤</sup> ð Þ <sup>λ</sup>1þ<sup>1</sup> ð Þ <sup>1</sup>�<sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

min<sup>t</sup>∈½ � <sup>θ</sup>;1�<sup>θ</sup> <sup>G</sup>2ð Þ <sup>t</sup>;<sup>s</sup> <sup>≥</sup> <sup>γ</sup>2ð Þ<sup>s</sup>

ii. min<sup>t</sup>∈½ � <sup>θ</sup>;1�<sup>θ</sup> <sup>G</sup>1ð Þ <sup>t</sup>;<sup>s</sup> <sup>≥</sup> <sup>γ</sup>1ð Þ<sup>s</sup>

max<sup>t</sup> <sup>∈</sup><sup>I</sup>∣G1ð Þ <sup>t</sup>;<sup>s</sup> <sup>∣</sup> <sup>≤</sup> ð Þ <sup>λ</sup>2þ<sup>1</sup> ð Þ <sup>1</sup>�<sup>s</sup> <sup>θ</sup>2�<sup>1</sup>

ii. min<sup>t</sup>∈½ � <sup>θ</sup>;1�<sup>θ</sup> <sup>G</sup>1ð Þ <sup>t</sup>;<sup>s</sup> <sup>≥</sup> <sup>γ</sup>1ð Þ<sup>s</sup>

i. max<sup>t</sup> <sup>∈</sup><sup>I</sup>∣G1ð Þ <sup>t</sup>;<sup>s</sup> <sup>∣</sup> <sup>≤</sup> ð Þ <sup>λ</sup>1þ1ð Þ <sup>1</sup>�<sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

min<sup>t</sup>∈½ � <sup>θ</sup>;1�<sup>θ</sup> <sup>G</sup>2ð Þ <sup>t</sup>;<sup>s</sup> <sup>≥</sup> <sup>γ</sup>2ð Þ<sup>s</sup>

max<sup>t</sup> <sup>∈</sup><sup>I</sup>∣G1ð Þ <sup>t</sup>;<sup>s</sup> <sup>∣</sup> <sup>≤</sup> ð Þ <sup>λ</sup>2þ1ð Þ <sup>1</sup>�<sup>s</sup> <sup>θ</sup>2�<sup>1</sup>

Further, taking that <sup>γ</sup> <sup>¼</sup> inf <sup>γ</sup><sup>1</sup> <sup>¼</sup> θθ1�<sup>1</sup>

Further, taking that <sup>γ</sup> <sup>¼</sup>inf <sup>γ</sup><sup>1</sup> <sup>¼</sup> θθ1�<sup>1</sup>

p tðÞ¼ <sup>I</sup><sup>θ</sup>1φð Þþ <sup>t</sup>

p tðÞ¼ <sup>Ð</sup> <sup>1</sup>

Applying boundary condition p tð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>η</sup> and <sup>d</sup>¼<sup>1</sup> � ηθ

, 0 ≤ s ≤ t ≤ η ≤ 1,

http://dx.doi.org/10.5772/intechopen.75523

, 0 ≤ s≤ t ≤ η ≤ 1,

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

Existence Theory of Differential Equations of Arbitrary Order

<sup>θ</sup>1�<sup>m</sup>: (5)

: (6)

<sup>θ</sup>1�<sup>m</sup>: (5)

: (6)

<sup>1</sup> in Eq. (6), one has

, the corresponding coupled

, the corresponding coupled

<sup>1</sup> in Eq. (6), one has

(4)

(4)

39

39

(7)

(7)

(8)

(8)

1 Γð Þ θ<sup>1</sup>

Gð Þ¼ t;s

Gð Þ¼ t;s


Remark 2.6. As � is an equivalence relation, therefore defines a set C<sup>f</sup> ¼ f g p∈ E : p � f for f ∈ C. Obviously, one can derive that C<sup>f</sup> ⊂ C for f ≻0: Remark 2.6. As � is an equivalence relation, therefore defines a set C<sup>f</sup> ¼ f g p∈ E : p� f for f ∈ C. Obviously, one can derive that C<sup>f</sup> ⊂ C for f ≻0:

Definition 2.7. The operator S : C ! C is said to be λ concave for every θ, λ∈ ð Þ 0; 1 , p∈ C, if and only if <sup>S</sup>ð Þ <sup>λ</sup><sup>p</sup> <sup>⪰</sup>θ<sup>λ</sup>Sp: Definition 2.7. The operator S : C ! C is said to be λ concave for every θ, λ∈ ð Þ 0; 1 , p∈ C, if and only if <sup>S</sup>ð Þ <sup>λ</sup><sup>p</sup> <sup>⪰</sup>θ<sup>λ</sup>Sp:

Definition 2.8. The operator S : C ! C is said to be to be increasing if p, q∈ C, p⪯q gives that Sp⪯Sq: Definition 2.8. The operatorS : C ! C is said to be to be increasing if p, q∈ C, p⪯q gives that Sp⪯Sq:

Lemma 2.9. [17, 28] Assume that S : C ! C is increasing λ�concave operator for a normal cone C produced by Banach space E, such that there exists p≻ 0 with Sf ∈ C<sup>f</sup> . Then, S has a unique fixed point p∈ C<sup>f</sup> Lemma 2.9. [17, 28] Assume that S : C ! C is increasing λ�concave operator for a normal cone C produced by Banach space E, such that there exists p≻ 0 with Sf ∈ C<sup>f</sup> . Then, S has a unique fixed point p∈ C<sup>f</sup>

Theorem 2.10. [30] Let E be a Banach space with C⊆B, which is closed and convex. Let E be a relatively open subset of C with 0∈E and S : E ! C be a continuous and compact operator. Then. Theorem 2.10. [30] Let E be a Banach space with C⊆B, which is closed and convex. Let E be a relatively open subset of C with 0∈Eand S : E ! C be a continuous and compact operator. Then.


Lemma 2.11. [30] For a Banach space E together with a cone C, there exist two relatively open subsets A1 and A<sup>2</sup> of E such that 0 ∈ A<sup>1</sup> ⊂ A<sup>1</sup> ⊂ A2. Moreover, for a completely continuous operator S : C ∩ A2\A<sup>1</sup> � � ! <sup>C</sup>, one of the given conditions holds: Lemma 2.11. [30] For a Banach space E together with a cone C, there exist two relatively open subsets A1 and A<sup>2</sup> of E such that 0 ∈ A<sup>1</sup> ⊂ A<sup>1</sup> ⊂ A2. Moreover, for a completely continuous operator S : C ∩ <sup>2</sup>\A<sup>1</sup> � � ! <sup>C</sup>, one of the given conditions holds:


Then, S has at least one fixed point in C ∩ A2\A<sup>1</sup> � �: Then, S has at least one fixed point in C ∩ A2\A<sup>1</sup> � �:

#### 3. Main results 3. Main results

Theorem 3.1. Let <sup>φ</sup> <sup>∈</sup>Cð Þ ½ � <sup>0</sup>; <sup>1</sup> ; <sup>R</sup> , <sup>η</sup><sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> and <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> � ηθ1�<sup>1</sup> <sup>&</sup>lt; 1, and then the unique solution to BVP of linear FDE Theorem 3.1. Let <sup>φ</sup> <sup>∈</sup>Cð Þ ½ �0; <sup>1</sup> ; <sup>R</sup> , <sup>η</sup>∈ð Þ <sup>0</sup>; <sup>1</sup> and <sup>λ</sup><sup>1</sup> <sup>¼</sup><sup>1</sup> � ηθ1�<sup>1</sup> <sup>&</sup>lt; 1, and then the unique solution to BVP of linear FDE

$$\begin{cases} \mathcal{B}^{\theta\_1} p(t) = q(t), t \in \mathcal{I}, \ \theta\_1 \in (m-1, m]\_{\succ} \\ p^{(j)}(t)\_{t=0} = 0, p(t)|\_{t=1} = p(t)|\_{t=\eta'} \ j = 0, 1, 2, \cdots \\ \end{cases} \tag{2}$$

is given by is given by

$$p(t) = \int\_0^1 \mathbf{G}(t, s)\varphi(s)ds,\tag{3}$$

(2)

where Gð Þ t;s is the Green's function defined by where Gð Þ t;s is the Green's function defined by

Definition 2.5. [17, 28] A closed and convex set C of E is said to be a normal cone if it obeys the given

Definition 2.5. [17, 28] A closed and convex set Cof E is said to be a normal cone if it obeys the given

Remark 2.6. As � is an equivalence relation, therefore defines a set C<sup>f</sup> ¼ f g p∈ E : p� f for f ∈ C.

Definition 2.7. The operator S : C ! C is said to be λ concave for every θ, λ∈ð Þ 0; 1 , p∈ C, if and

Definition 2.8. The operator S : C ! C is said to be to be increasing if p, q∈ C, p⪯q gives that

Lemma 2.9. [17, 28] Assume that S : C ! C is increasing λ�concave operator for a normal cone C produced by Banach space E, such that there exists p≻ 0 with Sf ∈ C<sup>f</sup> . Then, S has a unique fixed point

Theorem 2.10. [30] Let E be a Banach space with C⊆B, which is closed and convex. Let E be a relatively open subset of C with 0∈E and S : E ! C be a continuous and compact operator. Then.

Lemma 2.11. [30] For a Banach space E together with a cone C, there exist two relatively open subsets A1 and A<sup>2</sup> of E such that 0 ∈ A<sup>1</sup> ⊂ A<sup>1</sup> ⊂ A2. Moreover, for a completely continuous operator

Remark 2.6. As � is an equivalence relation, therefore defines a set C<sup>f</sup> ¼ f g p∈ E : p � f for f ∈ C.

2. p � q, for all p, q∈ E yields that there exist constants a, b> 0 such that ap⪯q⪯bq:

Definition 2.7. The operator S : C ! C is said to be λ concave for every θ, λ∈ ð Þ 0; 1 , p∈ C, if and

Definition 2.8. The operator S : C ! C is said to be to be increasing if p, q∈ C, p⪯q gives that

Lemma 2.9. [17, 28] Assume that S : C ! C is increasing λ�concave operator for a normal cone C produced by Banach space E, such that there exists p≻ 0 with Sf ∈ C<sup>f</sup> . Then, S has a unique fixed point

Theorem 2.10. [30] Let E be a Banach space with C⊆B, which is closed and convex. Let E be a relatively open subset of C with 0∈E and S : E ! C be a continuous and compact operator. Then.

Lemma 2.11. [30] For a Banach space E together with a cone C, there exist two relatively open subsets A1 and A<sup>2</sup> of E such that 0 ∈ A<sup>1</sup> ⊂ A<sup>1</sup> ⊂ A2. Moreover, for a completely continuous operator

� �:

� �:

Theorem 3.1. Let <sup>φ</sup> <sup>∈</sup>Cð Þ ½ � <sup>0</sup>; <sup>1</sup> ; <sup>R</sup> , <sup>η</sup><sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> and <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> � ηθ1�<sup>1</sup> <sup>&</sup>lt; 1, and then the unique solution

Theorem 3.1. Let <sup>φ</sup> <sup>∈</sup>Cð Þ ½ � <sup>0</sup>; <sup>1</sup> ; <sup>R</sup> , <sup>η</sup><sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> and <sup>λ</sup><sup>1</sup> <sup>¼</sup><sup>1</sup> � ηθ1�<sup>1</sup> <sup>&</sup>lt; 1, and then the unique solution

<sup>p</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>,p tð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>η</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, <sup>⋯</sup><sup>m</sup> � <sup>2</sup>, m <sup>≥</sup> <sup>3</sup>,

<sup>p</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>,p tð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>η</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, <sup>⋯</sup><sup>m</sup> � <sup>2</sup>, m <sup>≥</sup> <sup>3</sup>,

ð1 0

Gð Þ t;s φð Þs ds, (3)

Gð t;s φð Þs ds, (3)

(2)

(2)

<sup>D</sup><sup>θ</sup><sup>1</sup> p tðÞ¼ <sup>φ</sup>ð Þ<sup>t</sup> , t∈I, <sup>θ</sup><sup>1</sup> <sup>∈</sup> ð � <sup>m</sup> � <sup>1</sup>; <sup>m</sup> ,

<sup>D</sup><sup>θ</sup><sup>1</sup> p tðÞ¼ <sup>φ</sup>ð Þ<sup>t</sup> , t∈I, <sup>θ</sup><sup>1</sup> <sup>∈</sup> ð � <sup>m</sup> � <sup>1</sup>; <sup>m</sup> ,

p tðÞ¼

ð1 0

p tðÞ¼

2. p � q, for all p, q∈ E yields that there exist constants a, b > 0 such that ap⪯q⪯bq:

1. For 0⪯p⪯q∈E, there exists β > 0, such that pk k<sup>E</sup> ≤ βk kq <sup>E</sup>;

1. For 0⪯p⪯q∈E, there exists β > 0, such that pk k<sup>E</sup> ≤ βk kq <sup>E</sup>;

Obviously, one can derive that C<sup>f</sup> ⊂ C for f ≻0:

Obviously, one can derive that C<sup>f</sup> ⊂ C for f ≻0:

38 Differential Equations - Theory and Current Research

38 Differential Equations - Theory and Current Research

1. The operator S has a fixed point in E,

2. There exist w ∈ ∂ℰ and λ∈ð Þ 0; 1 with w ¼ λSw:

1. The operator S has a fixed point in E,

Then, S has at least one fixed point in C ∩ A2\A<sup>1</sup>

Then, S has at least one fixed point in C ∩ <sup>2</sup>\A<sup>1</sup>

� � ! <sup>C</sup>, one of the given conditions holds:

2. There exist w ∈ ∂ℰ and λ∈ð Þ 0; 1 with w ¼ λSw:

1. ∥Sp∥ ≤ ∥p∥ for all p ∈ C ∩ ∂A1; ∥Sp∥ ≥ ∥p∥, for all p∈ C ∩ ∂A2; 2. ∥Sp∥ ≥ ∥p∥ for all p ∈ C ∩ ∂A1; ∥Sp∥ ≤ ∥p∥, for all p∈ C ∩ ∂A<sup>2</sup>

� � ! <sup>C</sup>, one of the given conditions holds:

1. ∥Sp∥ ≤ ∥p∥ for all p ∈ C ∩ ∂A1; ∥Sp∥ ≥ ∥p∥, for all p∈C ∩ ∂A2; 2. ∥Sp∥ ≥ ∥p∥ for all p ∈ C ∩ ∂A1; ∥Sp∥ ≤ ∥p∥, for all p∈C ∩ ∂A<sup>2</sup>

properties:

properties:

only if <sup>S</sup>ð Þ <sup>λ</sup><sup>p</sup> <sup>⪰</sup>θ<sup>λ</sup>Sp:

Sp⪯Sq:

p∈ C<sup>f</sup>

only if <sup>S</sup>ð Þ <sup>λ</sup><sup>p</sup> <sup>⪰</sup>θ<sup>λ</sup>Sp:

Sp⪯Sq:

p∈ C<sup>f</sup>

S : C ∩ A2\A<sup>1</sup>

S : C ∩ A2\A<sup>1</sup>

3. Main results

3. Main results

to BVP of linear FDE

to BVP of linear FDE

is given by

is given by

(

(

$$\mathbf{G}(t,s) = \frac{1}{\Gamma(\theta\_1)} \begin{cases} \frac{1}{\lambda\_1} \left[ -\left[ t(1-s) \right]^{\theta\_1 - 1} + \left[ t(\eta - s) \right]^{\theta\_1 - 1} \right] + (t - s)^{\theta\_1 - 1}, & 0 \le s \le t \le \eta \le 1, \\\\ \frac{1}{\lambda\_1} \left[ -\left[ t(1-s) \right]^{\theta\_1 - 1} + \left[ t(\eta - s) \right]^{\theta\_1 - 1} \right], & 0 \le t \le s \le \eta \le 1, \\\\ -\frac{1}{\lambda\_1} \left[ t(1-s) \right]^{\theta\_1 - 1} + (t - s)^{\theta\_1 - 1}, & 0 \le \eta \le s \le t \le 1, \\\\ -\frac{1}{\lambda\_1} \left[ t(1-s) \right]^{\theta\_1 - 1}, & 0 \le \eta \le t \le s \le 1. \end{cases} \tag{4}$$

Proof. In view of Lemma 2.3, we may write Eq. (2) as Proof. In view of Lemma 2.3, we may write Eq. (2) as

$$p(t) = \mathcal{Z}^{\theta\_1} \varphi(t) + \mathcal{C}\_1 t^{\theta\_1 - 1} + \mathcal{C}\_2 t^{\theta\_1 - 2} + \dots + \mathcal{C}\_m t^{\theta\_1 - m}.\tag{5}$$

In view of conditions <sup>p</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, …<sup>m</sup> � <sup>2</sup>, m <sup>≥</sup> <sup>3</sup>, , Eq. (5) suffers from singularity; therefore, we have C<sup>2</sup> ¼ C<sup>3</sup> ¼ … ¼ Cn ¼ 0: Hence, Eq. (5) becomes In view of conditions <sup>p</sup>ð Þ<sup>j</sup> ð Þ<sup>t</sup> <sup>t</sup>¼<sup>0</sup> <sup>¼</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, …<sup>m</sup> � <sup>2</sup>, m <sup>≥</sup> <sup>3</sup>, , Eq. (5) suffers from singularity; therefore, we have C<sup>2</sup> ¼ C<sup>3</sup> ¼ … ¼ Cn ¼ 0: Hence, Eq. (5) becomes

$$p(t) = \mathcal{T}^{\theta\_1} \varphi(t) + \mathcal{C}\_1 t^{\theta\_1 - 1}. \tag{6}$$

Applying boundary condition p tð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>η</sup> and <sup>d</sup> <sup>¼</sup> <sup>1</sup> � ηθ <sup>1</sup> in Eq. (6), one has Applying boundary condition p tð Þj j <sup>t</sup>¼<sup>1</sup> <sup>¼</sup> p tð Þ <sup>t</sup>¼<sup>η</sup> and <sup>d</sup> <sup>¼</sup><sup>1</sup> � ηθ <sup>1</sup> in Eq. (6), one has

$$\begin{split} p(t) &= \mathcal{Z}^{\theta\_1} \boldsymbol{\varrho}(t) + \frac{t^{\theta\_1 - 1}}{\lambda\_1} \left[ \mathcal{Z}^{\theta\_1} \boldsymbol{\varrho}(\boldsymbol{\eta}) - \mathcal{Z}^{\theta\_1} \boldsymbol{\varrho}(1) \right] \\ p(t) &= \int\_0^1 \mathbf{G}(t, s) \boldsymbol{\varrho}(s) ds. \end{split} \tag{7}$$

where Gð Þ t;s is Green's function given in Eq. (4). where Gð Þ t;s is Green's function given in Eq. (4).

> 8 < :

In view of Theorem 3.1 and using <sup>λ</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> � ηθ1�<sup>1</sup>, <sup>λ</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup> � <sup>ξ</sup><sup>θ</sup>2�<sup>1</sup> , the corresponding coupled system of integral equations to the proposed system (1) is given as In view of Theorem 3.1 and using <sup>λ</sup><sup>1</sup> <sup>¼</sup><sup>1</sup> � ηθ1�<sup>1</sup>, <sup>λ</sup><sup>2</sup> <sup>¼</sup><sup>1</sup> � <sup>ξ</sup><sup>θ</sup>2�<sup>1</sup> , the corresponding coupled system of integral equations to the proposed system (1) is given as

$$\begin{cases} p(t) = \int\_0^1 \mathbf{G}\_1(t, s) \mathcal{H}\_1(s, p(s), q(s)) ds, \\\\ q(t) = \int\_0^1 \mathbf{G}\_2(t, s) \mathcal{H}\_2(s, p(s), q(s)) ds, \end{cases} \tag{8}$$

where G1ð Þ t;s , G2ð Þ t;s are Green's functions, which can be similarly computed like in Theorem 3.1. Further, they are continuous on I � I and satisfy the following properties: where G1ð Þ t;s , G2ð Þ t;s are Green's functions, which can be similarly computed like in Theorem 3.1. Further, they are continuous on I � I and satisfy the following properties:


$$\min\_{t \in [\theta, 1-\theta]} \mathbf{G}\_2(t, s) \ge \frac{\gamma\_2(s)}{2} \mathbf{G}(1, s) \text{ for every } \theta \text{ } s \in (0, 1);$$

Further, taking that <sup>γ</sup> <sup>¼</sup> inf <sup>γ</sup><sup>1</sup> <sup>¼</sup> θθ1�<sup>1</sup> ; <sup>γ</sup><sup>2</sup> <sup>¼</sup> θθ2�<sup>1</sup> � �: Further, taking that <sup>γ</sup> <sup>¼</sup> inf <sup>γ</sup><sup>1</sup> <sup>¼</sup> θθ1�<sup>1</sup> ; <sup>γ</sup><sup>2</sup> <sup>¼</sup> θθ2�<sup>1</sup> � �: Let us define a Banach space by E ¼ f g p tð Þjp ∈Cð ÞI endowed with a norm k kp <sup>E</sup> ¼ max<sup>t</sup><sup>∈</sup> <sup>I</sup>∣p tð Þ∣. Further, in the norm for the product space, we define it as k k ð Þ <sup>p</sup>; <sup>q</sup> <sup>E</sup>�<sup>E</sup> <sup>¼</sup> k k<sup>p</sup> <sup>E</sup> <sup>þ</sup> k k<sup>q</sup> <sup>E</sup>. Clearly, <sup>E</sup> � <sup>E</sup>; k k� <sup>E</sup>�<sup>E</sup> � � is a Banach space. Onward, we define the cone <sup>C</sup> <sup>⊂</sup> <sup>E</sup> � <sup>E</sup> by Let us define a Banach space by E ¼ f g p tð Þjp ∈Cð ÞI endowed with a norm k kp<sup>E</sup> ¼ max<sup>t</sup><sup>∈</sup> <sup>I</sup>∣p tð Þ∣. Further, in the norm for the product space, we define it as k k ð Þ <sup>p</sup>; <sup>q</sup> <sup>E</sup>�<sup>E</sup> <sup>¼</sup> k k<sup>p</sup> <sup>E</sup> <sup>þ</sup> k k<sup>q</sup> <sup>E</sup>. Clearly, <sup>E</sup> � <sup>E</sup>; k k� <sup>E</sup>�<sup>E</sup> � � is a Banach space. Onward, we define the cone <sup>C</sup><sup>⊂</sup> � <sup>E</sup> by

$$\mathbf{C} = \left\{ (p, q) \in \mathbf{E} \times \mathbf{E} : \min\_{t \in \mathbf{I}} \left[ p(t) + q(t) \right] \succeq \gamma \|(p, q)\|\_{\mathbf{E} \times \mathbf{E}} \right\}.$$

Consider an operator S : E � E ! E � E defined by Consider an operator S : � E ! E � E defined by

$$\begin{split} \mathcal{S}(p,q)(t) &= \left( \bigcup\_{0}^{1} \mathbf{G}\_{1}(t,s)\mathcal{H}\_{1}(s,p(s),q(s))ds, \; \int\_{0}^{1} \mathbf{G}\_{2}(t,s)\mathcal{H}\_{2}(s,p(s),q(s))ds \right) \\ &= (\mathcal{S}\_{1}p(t), \mathcal{S}\_{2}q(t)). \end{split} \tag{9}$$

Let us consider

Let us consider

max t∈ I

max t∈ I

Then, we consider t<sup>1</sup> < t<sup>2</sup> ∈ I, such that

Then, we consider t<sup>1</sup> < t<sup>2</sup> ∈ I, such that

∣S1ð Þ p; q ð Þ� t<sup>2</sup> S1ð Þp; q ð Þ t<sup>1</sup> ∣ ¼

By the same fashion, we obtain for S<sup>2</sup> as

∣S2ð Þ p; q ð Þ� t<sup>2</sup> S2ð Þ p; q ð Þ t<sup>1</sup> ∣ ≤

By the same fashion, we obtain for S<sup>2</sup> as

S ¼ ð Þ S1; S<sup>2</sup> : C ! C is completely continuous.

S ¼ ð Þ S1; S<sup>2</sup> : C ! C is completely continuous.

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>φ</sup>1ð Þ<sup>s</sup> ds <sup>&</sup>lt; <sup>∞</sup>, <sup>Λ</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup>

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>φ</sup>1ð Þ<sup>s</sup> ds <sup>&</sup>lt;∞, <sup>Λ</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup>

<sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>φ</sup>2ð Þ<sup>s</sup> ds <sup>&</sup>lt;∞, <sup>Λ</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup>

<sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>φ</sup>2ð Þ<sup>s</sup> ds <sup>&</sup>lt; <sup>∞</sup>, <sup>Λ</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup>

∣S2ð Þ p; q ð Þ� t<sup>2</sup> S2ð Þ p; q ð Þ t<sup>1</sup> ∣ ≤

along with the following conditions:

1 0

1 0

along with the following conditions:

φj ,ψ<sup>j</sup>

φj ,ψ<sup>j</sup>

(9)

i. <sup>Δ</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup>

ii. <sup>Δ</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup>

1 0

i. <sup>Δ</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup>

ii. <sup>Δ</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup>

1 0

∣S1ð Þ p; q ð Þ� t<sup>2</sup> S1ð Þ p; q ð Þ t<sup>1</sup> ∣ ¼

∣H1ð Þ t; p tð Þ; q tð Þ ∣ ≤M1, max

∣H1ð Þ t; p tð Þ; q tð Þ ∣ ≤M1, max

t θ1�1 <sup>2</sup> � t

4

ðt2 0

M<sup>1</sup> <sup>λ</sup>1Γð Þ <sup>θ</sup><sup>1</sup> <sup>þ</sup> <sup>1</sup> <sup>t</sup>

M<sup>2</sup> <sup>λ</sup>2Γð Þ <sup>θ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>t</sup>

4

M<sup>1</sup> <sup>λ</sup>1Γð Þ <sup>θ</sup><sup>1</sup> <sup>þ</sup> <sup>1</sup> <sup>t</sup>

M<sup>2</sup> <sup>λ</sup>2Γð Þ <sup>θ</sup><sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>t</sup>

, σjð Þ j ¼ 1; 2 : ð Þ! 0; 1 Rþ∪f g0 for t∈ ð Þ 0; 1 , p, q ≥ 0 such that

, σjð Þ j ¼ 1; 2 : ð Þ! 0; 1 Rþ∪f g0 for t∈ ð Þ 0; 1 , p, q ≥ 0 such that

ð1 0

ð1 0

� � � �

≤ M<sup>1</sup> Γð Þ θ<sup>1</sup>

þ M<sup>1</sup> Γð Þ θ<sup>1</sup>

≤

� � � �

≤ M<sup>1</sup> Γð Þ θ<sup>1</sup>

þ M<sup>1</sup> Γð Þ θ<sup>1</sup>

≤

t∈ I

t∈ I

ð Þ Gðt2;sÞ � G1ð Þ t1;s H1ðs; p sð Þ; q sð ÞÞds

ðη 0

θ1�1 1 � �

ð Þ Gðt2;sÞ � G1ð Þ t1;s H1ðs; p sð Þ; q sð ÞÞds

ds � ðt1 0

ds � �

ds � ðt1 0

ds � �

θ1�1 1 � � <sup>η</sup><sup>θ</sup><sup>1</sup> � <sup>λ</sup><sup>1</sup>

θ1�1 1 � � <sup>η</sup><sup>θ</sup><sup>1</sup> � <sup>λ</sup><sup>1</sup>

θ2�1 1 � � <sup>ξ</sup><sup>θ</sup><sup>2</sup> � <sup>λ</sup><sup>2</sup>

θ2�1 1 � � <sup>ξ</sup><sup>θ</sup><sup>2</sup> � <sup>λ</sup><sup>2</sup>

θ2�1 <sup>2</sup> � t

The right hand sides of Eqs. (12) and (13) are approaching to zero at t<sup>1</sup> ! t2: Thus, the operator S is equi-continuous. Therefore, thanks to the Arzelá-Ascoli theorem, we receive that

Theorem 3.3. Due to continuity of H<sup>1</sup> and H<sup>2</sup> on I � Rþ∪f g0 � Rþ∪f g0 ! Rþ, there exist

∣H1ð Þt; p tð Þ; q tð Þ ∣ ≤ φ1ð Þþ t ψ1ð Þ ∣p tð Þ∣ þ σ<sup>1</sup> q tð Þ∣; ∣H2ð Þt; p tð Þ; q tð Þ ∣ ≤ φ2ð Þþ t ψ2ð Þ ∣p tð Þ∣ þ σ<sup>2</sup> q tð Þ∣,

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>1ð Þþ <sup>s</sup> <sup>σ</sup>1ð Þ<sup>s</sup> � �ds <sup>&</sup>lt; <sup>1</sup>;

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>1ð Þþ <sup>s</sup> <sup>σ</sup>1ð Þ<sup>s</sup> � �ds <sup>&</sup>lt; <sup>1</sup>;

<sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>2ð Þþ <sup>s</sup> <sup>σ</sup>2ð Þ<sup>s</sup> � �ds <sup>&</sup>lt; <sup>1</sup>

<sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>2ð Þþ <sup>s</sup> <sup>σ</sup>2ð Þ<sup>s</sup> � �ds <sup>&</sup>lt; <sup>1</sup>

� � � � :

� � � � :

1 � 2Λ<sup>1</sup>

; <sup>2</sup>Δ<sup>2</sup> 1 � 2Λ<sup>2</sup>

; <sup>2</sup>Δ<sup>2</sup> 1 � 2Λ<sup>2</sup>

1 � 2Λ<sup>1</sup>

ð Þ <sup>η</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

ðη 0

ds � � <sup>2</sup>

ð Þ <sup>t</sup><sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

h i � � :

� � <sup>þ</sup> <sup>λ</sup><sup>1</sup> <sup>t</sup>

h i � � :

� � <sup>þ</sup> <sup>λ</sup><sup>2</sup> <sup>t</sup>

h i � � : (13)

� � <sup>þ</sup> <sup>λ</sup><sup>2</sup> <sup>t</sup>

h i � � : (13)

θ1�1 1 � �

λ1

ð Þ <sup>t</sup><sup>2</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

θ1�1 <sup>2</sup> � t

λ1

t θ1�1 <sup>2</sup> � t

ðt2 0

ð Þ <sup>t</sup><sup>2</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

θ1�1 <sup>2</sup> � t

> θ2�1 <sup>2</sup> � t

The right hand sides of Eqs. (12) and (13) are approaching to zero at t<sup>1</sup> ! t2: Thus, the operator S is equi-continuous. Therefore, thanks to the Arzelá-Ascoli theorem, we receive that

Theorem 3.3. Due to continuity of H<sup>1</sup> and H<sup>2</sup> on I � Rþ∪f g0 � Rþ∪f g0 ! Rþ, there exist

∣H1ð Þ t; p tð Þ; q tð Þ ∣ ≤ φ1ð Þþ t ψ1ð Þt ∣p tð Þ∣ þ σ1ð Þt ∣q tð Þ∣; ∣H2ð Þ t; p tð Þ; q tð Þ ∣ ≤ φ2ð Þþ t ψ2ð Þt ∣p tð Þ∣ þ σ2ð Þt ∣q tð Þ∣,

> 1 0

1 0

1 0

1 0

<sup>E</sup> <sup>¼</sup> ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> : <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; min <sup>2</sup>Δ<sup>1</sup>

<sup>E</sup> <sup>¼</sup> ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>C</sup> : <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; min <sup>2</sup>Δ<sup>1</sup>

are satisfied. Then, the system(1) has at least one solution pð Þ ; q which lies in

are satisfied. Then, the system (1) has at least one solution pð Þ ; q which lies in

∣H2ð Þ t; p tð Þ; q tð Þ ∣ ≤M2:

� � � � � � � �

ds � ð1 0

∣H2ð Þ t; p tð Þ; q tð Þ ∣ ≤M2:

Existence Theory of Differential Equations of Arbitrary Order

http://dx.doi.org/10.5772/intechopen.75523

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

ds � ð1 0

ð Þ <sup>t</sup><sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

� � <sup>þ</sup> <sup>λ</sup><sup>1</sup> <sup>t</sup>

ð Þ <sup>η</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

ds � � <sup>2</sup>

ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

θ1 <sup>2</sup> � t θ1 1

ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

θ1 <sup>2</sup> � t θ1 1

> θ2 <sup>2</sup> � t θ2 1

θ2 <sup>2</sup> � t θ2 1

3 5

(12)

(12)

3 5 41

41

It is to be noted that the fixed points of the operator S correspond with the solution of the system (1) under consideration. It is to be noted that the fixed points of the operator S correspond with the solution of the system (1) under consideration.

Theorem 3.2. Under the continuity of H1, H<sup>2</sup> : I � Rþ∪f g0 � Rþ∪f g0 ! Rþf g0 , the operator S satisfies that Sð Þ C ⊂ C and S : C ! C is completely continuous. Theorem 3.2. Under the continuity of H1, H<sup>2</sup> : I � Rþ∪f g0 � Rþ∪f g0 ! Rþf g0 , the operator S satisfies that Sð Þ C ⊂ C and S : C ! C is completely continuous.

Proof. To derive Sð Þ C ⊂ C, let ð Þ p; q ∈ C, and then we have Proof. To deriveSð Þ C ⊂ C, let ð Þ p; q ∈ C, and then we have

$$\mathcal{S}\_1(p(t),q(t)) = \int\_0^1 \mathbf{G}\_1(t,s)\mathcal{H}\_1(s,p(s),q(s))ds \ge \nu\_1 \int\_0^1 \mathbf{G}\_1(1,s)\mathcal{H}\_1(s,p(s)), q(s)ds.\tag{10}$$

Also, we get Also, we get

$$\mathcal{S}\_1(p(t),q(t)) = \int\_0^1 \mathbf{G}\_1(t,s)\mathcal{H}\_1(s,p(s),q(s))ds \le \int\_0^1 \mathbf{G}\_1(1,s)\mathcal{H}\_1(s,p(s)),q(s)ds.\tag{11}$$

Thus, from Eqs. (10) and (11), we have Thus, from Eqs. (10) and (11), we have

$$\mathcal{S}\_1(p(t), q(t)) \ge \gamma \| \mathcal{S}\_1(p, q) \|\_{\mathbf{E}^\nu} \quad \text{for every} \ t \in \mathbf{I}.$$

Similarly, we can obtain Similarly, we can obtain

$$\mathcal{S}\_2(p(t), q(t)) \succeq \gamma \| \mathcal{S}\_2(p, q) \|\_{\mathbb{E}\prime} \quad \text{for every} \ t \in \mathcal{I}.$$

$$\text{Thus } \mathcal{S}\_1(p(t), q(t)) + \mathcal{S}\_2(p(t), q(t)) \succeq \gamma \|(p, q)\|\_{\mathbb{E}\times\mathbb{E}^\nu} \text{ for all } t \in \mathcal{I},$$

$$\min\_{t \in \mathcal{I}} \left[ \mathcal{S}\_1(p(t), q(t)) + \mathcal{S}\_2(p(t), q(t)) \right] \succeq \gamma \|(p, q)\|\_{\mathbb{E}\times\mathbb{E}^\nu}.$$

Hence, we have Sð Þ p; q ∈ C ) Sð Þ C ⊂ C: Hence, we have Sð Þ p; q ∈ C ) Sð Þ C ⊂ C: Let us consider Let us consider

Let us define a Banach space by E ¼ f g p tð Þjp ∈Cð ÞI endowed with a norm k kp <sup>E</sup> ¼ max<sup>t</sup><sup>∈</sup> <sup>I</sup>∣p tð Þ∣. Further, in the norm for the product space, we define it as k k ð Þ <sup>p</sup>; <sup>q</sup> <sup>E</sup>�<sup>E</sup> <sup>¼</sup> k k<sup>p</sup> <sup>E</sup> <sup>þ</sup> k k<sup>q</sup> <sup>E</sup>. Clearly,

Let us define a Banach space byE ¼ f g p tð Þjp ∈Cð ÞI endowed with a norm k kp<sup>E</sup> ¼ max<sup>t</sup><sup>∈</sup> <sup>I</sup>∣p tð Þ∣. Further, in the norm for the product space, we define it as k k ð Þ <sup>p</sup>; <sup>q</sup> <sup>E</sup>�<sup>E</sup> <sup>¼</sup> k kp<sup>E</sup> <sup>þ</sup> k k<sup>q</sup> <sup>E</sup>. Clearly,

� �

It is to be noted that the fixed points of the operator S correspond with the solution of the

It is to be noted that the fixed points of the operator S correspond with the solution of the

Theorem 3.2. Under the continuity of H1, H<sup>2</sup> : I � Rþ∪f g0 � Rþ∪f g0 ! Rþf g0 , the operator S

Theorem 3.2. Under the continuity of H1, H<sup>2</sup> : I � Rþ∪f g0 � Rþ∪f g0 ! Rþf g0 , the operator S

G1ð Þ t;s H1ð Þ s; p sð Þ; q sð Þ ds ≥ γ<sup>1</sup>

G1ð Þ t;s H1ð Þ s; p sð Þ; q sð Þ ds ≥ γ<sup>1</sup>

G1ð Þ t;s H1ð Þ s; p sð Þ; q sð Þ ds ≤

G1ð Þ t;s H1ð Þ s; p sð Þ; q sð Þ ds ≤

S1ð Þ p tð Þ; q tð Þ ≥ γ∥S1ð Þ p; q ∥E, for every t ∈I:

S1ð Þ p tð Þ; q tð Þ ≥ γ∥S1ð Þ p; q∥E, for every t ∈I:

S2ð Þ p tð Þ; q tð Þ ≥ γ∥S2ð Þ p; q∥E, for every t ∈I:

Thus S1ð Þþ p tð Þ; q tð Þ S2ð Þ p tð Þ; q tð Þ ≥ γ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>, for all t ∈I,

<sup>t</sup><sup>∈</sup> <sup>I</sup> ½ � <sup>S</sup>1ð Þþ p tð Þ; q tð Þ <sup>S</sup>2ð Þ p tð Þ; q tð Þ <sup>≥</sup> <sup>γ</sup>∥ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup>:

S2ð Þ p tð Þ; q tð Þ ≥ γ∥S2ð Þ p; q ∥E, for every t ∈I:

Thus S1ð Þþ p tð Þ; q tð Þ S2ð Þ p tð Þ; q tð Þ ≥ γ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>, for all t ∈I,

<sup>t</sup><sup>∈</sup> <sup>I</sup> ½ � <sup>S</sup>1ð Þþ p tð Þ; q tð Þ <sup>S</sup>2ð Þ p tð Þ; q tð Þ <sup>≥</sup> <sup>γ</sup>∥ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup>:

<sup>t</sup><sup>∈</sup> <sup>I</sup> ½ � p tð Þþ q tð Þ <sup>≥</sup> <sup>γ</sup>k k ð Þ <sup>p</sup>; <sup>q</sup> <sup>E</sup>�<sup>E</sup>

� �

<sup>t</sup><sup>∈</sup> <sup>I</sup> ½ � p tð Þþ q tð Þ <sup>≥</sup> <sup>γ</sup>k k ð Þ <sup>p</sup>; <sup>q</sup> <sup>E</sup>�<sup>E</sup>

ð 1 ð 1

0

0

ð 1 ð 1

0

ð 1

0

0

ð 1

0

:

:

1 A: 1 A:

G1ð Þ 1;s H1ð Þ s; p sð Þ ,q sð Þds: (10)

G1ð Þ 1;s H1ð Þ s; p sð Þ ,q sð Þds: (10)

G1ð Þ 1;s H1ð Þ s; p sð Þ ,q sð Þds: (11)

G1ð Þ 1;s H1ð Þ s; p sð Þ ,q sð Þds: (11)

(9)

(9)

G2ðt;sÞH2ðs; p sð Þ; q sð ÞÞds

G2ðt;sÞH2ðs; p sð Þ; q sð ÞÞds

� � is a Banach space. Onward, we define the cone <sup>C</sup> <sup>⊂</sup> <sup>E</sup> � <sup>E</sup> by

C ¼ ð Þ p; q∈�E : min

� � is a Banach space. Onward, we define the cone <sup>C</sup> <sup>⊂</sup> <sup>E</sup>� <sup>E</sup> by

G1ð Þ t;s H1ðs; p sð Þ; q sð ÞÞds;

G1ð Þ t;s H1ðs; p sð Þ; q sð ÞÞds;

C ¼ ð Þ p; q ∈E � E : min

Consider an operator S : E � E ! E � E defined by

Consider an operator S : � E ! � E defined by

0 @

0 @

Sð Þp; q ðÞ¼ t

ð 1

0

¼ ð Þ S1p tð Þ; S2q tð Þ :

¼ ð Þ S1p tð Þ; S2q tð Þ :

satisfies that Sð Þ C ⊂ C and S : C ! C is completely continuous.

Proof. To deriveSð Þ C ⊂ C, let ð Þ p; q ∈ C, and then we have

0

ð 1

satisfies that Sð Þ C ⊂ C and S : C ! C is completely continuous.

ð 1

0

ð 1

0

Proof. To derive Sð Þ C ⊂ C, let ð Þ p; q ∈ C, and then we have

ð 1

0

ð 1

0

min

Hence, we have Sð Þ p; q ∈C ) Sð Þ C ⊂ C:

min

Hence, we have Sð Þ p; q ∈ C ) Sð Þ C ⊂ C:

Sð Þ p; q ðÞ¼ t

40 Differential Equations - Theory and Current Research

40 Differential Equations - Theory and Current Research

system (1) under consideration.

system (1) under consideration.

S1ð Þ¼ p tð Þ; q tð Þ

S1ð Þ¼ p tð Þ; q tð Þ

S1ð Þ¼ p tð Þ; q tð Þ

S1ð Þ¼ p tð Þ; q tð Þ

Thus, from Eqs. (10) and (11), we have

Thus, from Eqs. (10) and (11), we have

Similarly, we can obtain

Similarly, we can obtain

Also, we get

Also, we get

<sup>E</sup> � <sup>E</sup>; k k� <sup>E</sup>�<sup>E</sup>

� <sup>E</sup>; k k� <sup>E</sup>�<sup>E</sup>

$$\max\_{t \in \mathbf{I}} |\mathcal{H}\_1(t, p(t), q(t))| \le \mathcal{M}\_1, \quad \max\_{t \in \mathbf{I}} |\mathcal{H}\_2(t, p(t), q(t))| \le \mathcal{M}\_2.$$

Then, we consider t<sup>1</sup> < t<sup>2</sup> ∈ I, such that Then, we considert<sup>1</sup> < t<sup>2</sup> ∈ I, such that

t∈ I

$$\begin{split} \left| \mathcal{S}\_{1}(p,q)(t\_{2}) - \mathcal{S}\_{1}(p,q)(t\_{1}) \right| &= \left| \int\_{0}^{1} (\mathbf{G}(t\_{2},s) - \mathbf{G}\_{1}(t\_{1},s)) \mathcal{H}\_{1}(s,p(s),q(s)) ds \right| \\ &\leq \frac{\mathcal{M}\_{1}}{\Gamma(\theta\_{1})} \left[ \frac{\left(t\_{2}^{\theta\_{1}-1} - t\_{1}^{\theta\_{1}-1}\right)}{\lambda\_{1}} \left( \int\_{0}^{\eta} (\eta-s)^{\theta\_{1}-1} ds - \int\_{0}^{1} (1-s)^{\theta\_{1}-1} ds \right) \right] \\ &\quad + \frac{\mathcal{M}\_{1}}{\Gamma(\theta\_{1})} \left[ \int\_{0}^{t\_{2}} (t\_{2}-s)^{\theta\_{1}-1} ds - \int\_{0}^{t\_{1}} (t\_{1}-s)^{\theta\_{1}-1} ds \right] \\ &\leq \frac{\mathcal{M}\_{1}}{\lambda\_{1}\Gamma(\theta\_{1}+1)} \left[ \left(t\_{2}^{\theta\_{1}-1} - t\_{1}^{\theta\_{1}-1}\right) \left(\eta^{\theta\_{1}} - \lambda\_{1}\right) + \lambda\_{1} \left(t\_{2}^{\theta\_{1}} - t\_{1}^{\theta\_{1}}\right) \right]. \end{split} \tag{12}$$

By the same fashion, we obtain for S<sup>2</sup> as By the same fashion, we obtain for S<sup>2</sup> as

$$|\mathcal{S}\_2(p,q)(t\_2) - \mathcal{S}\_2(p,q)(t\_1)| \le \frac{\mathcal{M}\_2}{\lambda\_2 \Gamma(\theta\_2 + 1)} \left[ \left(t\_2^{\theta\_2 - 1} - t\_1^{\theta\_2 - 1}\right) \left(\xi^{\theta\_2} - \lambda\_2\right) + \lambda\_2 \left(t\_2^{\theta\_2} - t\_1^{\theta\_2}\right) \right]. \tag{13}$$

The right hand sides of Eqs. (12) and (13) are approaching to zero at t<sup>1</sup> ! t2: Thus, the operator S is equi-continuous. Therefore, thanks to the Arzelá-Ascoli theorem, we receive that S ¼ ð Þ S1; S<sup>2</sup> : C ! C is completely continuous. The right hand sides of Eqs. (12) and (13) are approaching to zero at t<sup>1</sup> ! t2: Thus, the operator S is equi-continuous. Therefore, thanks to the Arzelá-Ascoli theorem, we receive that S ¼ ð Þ S1; S<sup>2</sup> : C ! C is completely continuous.

Theorem 3.3. Due to continuity of H<sup>1</sup> and H<sup>2</sup> on I � Rþ∪f g0 � Rþ∪f g0 ! Rþ, there exist ,ψ<sup>j</sup> , σjð Þ j ¼ 1; 2 : ð Þ! 0; 1 Rþ∪f g0 for t∈ ð Þ 0; 1 , p, q ≥ 0 such that Theorem 3.3. Due to continuity of H<sup>1</sup> and H<sup>2</sup> onI � Rþ∪f g0 � Rþ∪f g0 ! Rþ, there exist φj ,ψ<sup>j</sup> , σjð Þ j ¼ 1; 2 : ð Þ! 0; 1 Rþ∪f g0 for t∈ ð Þ 0; 1 , p, q ≥ 0 such that

$$\begin{aligned} |\mathcal{H}\_1(t, p(t), q(t))| &\leq \quad \varphi\_1(t) + \psi\_1(t)|p(t)| + \sigma\_1(t)|q(t)|\varphi(t)|\\ |\mathcal{H}\_2(t, p(t), q(t))| &\leq \quad \varphi\_2(t) + \psi\_2(t)|p(t)| + \sigma\_2(t)|q(t)|\varphi(t)| \end{aligned}$$

along with the following conditions: along with the following conditions:

φj

$$\mathbf{i}\mathbf{i} \qquad \Lambda\_1 = \underset{\mathbf{0}}{\int} \mathbf{G}\_1(1,\mathbf{s})\boldsymbol{\varrho}\_1(\mathbf{s})d\mathbf{s} < \underset{\mathbf{0}}{\Leftrightarrow} \Lambda\_1 = \underset{\mathbf{0}}{\int} \mathbf{G}\_1(1,\mathbf{s})\left[\boldsymbol{\psi}\_1(\mathbf{s}) + \boldsymbol{\sigma}\_1(\mathbf{s})\right]d\mathbf{s} < 1;$$

$$\ddot{\mathbf{u}}.\quad \Delta\_2 = \int\_0^1 \mathbf{G}\_2(1, s)\boldsymbol{\rho}\_2(s)ds<\Leftrightarrow\quad \Lambda\_2 = \int\_0^1 \mathbf{G}\_2(1, s)\left[\boldsymbol{\psi}\_2(s) + \boldsymbol{\sigma}\_2(s)\right]ds<1.$$

are satisfied. Then, the system (1) has at least one solution pð Þ ; q which lies in are satisfied. Then, the system(1) has at least one solution pð Þ ; q which lies in

$$\mathcal{E} = \left\{ (p, q) \in \mathbf{C} : \|(p, q)\|\_{\mathbf{E} \times \mathbf{E}} < \min \left( \frac{2\Lambda\_1}{1 - 2\Lambda\_1}, \frac{2\Lambda\_2}{1 - 2\Lambda\_2} \right) \right\}.$$

Proof. Let <sup>E</sup> <sup>¼</sup> f g ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> : <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>r</sup> with min <sup>2</sup>Δ<sup>1</sup> <sup>1</sup>�2Λ<sup>1</sup> ; <sup>2</sup>Δ<sup>2</sup> 1�2Λ<sup>2</sup> � � <sup>&</sup>lt; <sup>r</sup>: Proof. Let <sup>E</sup> <sup>¼</sup> f g ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>C</sup> : <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>r</sup> with min <sup>2</sup>Δ<sup>1</sup> <sup>1</sup>�2Λ<sup>1</sup> ; <sup>2</sup>Δ<sup>2</sup> 1�2Λ<sup>2</sup> � � <sup>&</sup>lt; <sup>r</sup>:

Define the operator S : E ! C as in Eq. (9). Define the operator S : E ! C as in Eq. (9).

Let ð Þ p; q ∈E that is ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < r: Then, we have Let ð Þ p; q∈E that is ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < r: Then, we have

$$\begin{split} |\mathcal{S}\_{1}(p,q)(t)| &= \max\_{t \in \mathcal{I}} \left| \int\_{0}^{1} \mathbf{G}\_{1}(t,s)\mathcal{H}\_{1}(s,p(s),q(s))ds \right| \\ &\leq \left( \int\_{0}^{1} \mathbf{G}\_{1}(1,s)\boldsymbol{\rho}\_{1}(s)ds + \int\_{0}^{1} \mathbf{G}\_{1}(1,s)\boldsymbol{\psi}\_{1}(s)|p(s)|ds + \int\_{0}^{1} \mathbf{G}\_{1}(1,s)\boldsymbol{\sigma}\_{1}(s)|q(s)|ds \right) \\ &\leq \int\_{0}^{1} \mathbf{G}\_{1}(1,s)\boldsymbol{\rho}\_{1}(s)ds + r \left[ \int\_{0}^{1} \mathbf{G}\_{1}(1,s)\left[\boldsymbol{\psi}\_{1}(s) + \boldsymbol{\sigma}\_{1}(s)\right]ds \right] = \boldsymbol{\Delta}\_{1} + r\boldsymbol{\Lambda}\_{1} \leq \frac{r}{2}. \end{split} \tag{14}$$

Thus, from Eq. (14), we have Thus, from Eq. (14), we have

$$\|\|\mathcal{S}\_1(p,q)\|\|\_{\mathbb{E}} \le \frac{r}{2}.\tag{15}$$

Assume that the given hypothesis holds:

Assume that the given hypothesis holds:

(H2) For all t∈ I, we have

(H2) For all t∈ I, we have

(H3) For all t∈ I such that

(H3) For all t∈ I such that

and

and

(14)

have

have

f tðÞ¼ t

<sup>θ</sup>1�<sup>1</sup>; t <sup>θ</sup>2�<sup>1</sup> � �.

<sup>θ</sup>1�<sup>1</sup>; t <sup>θ</sup>2�<sup>1</sup> � �.

f tðÞ¼ t

Analogously, we also get

Analogously, we also get

defined by

defined by

Suppose that

Suppose that

≥ τ<sup>λ</sup> ð 1

0

≥ τ<sup>λ</sup> ð 1

0

(H1) The nonlinear functions H<sup>1</sup> and H<sup>2</sup> are continuous on I � Rþ∪f g0 � Rþ∪f g0 ! Rþ∪f g0

(H1) The nonlinear functions H<sup>1</sup> and H<sup>2</sup> are continuous on I � Rþ∪f g0 � Rþ∪f g0 ! Rþ∪f g0

H1ð Þ¼ t; p; q 6¼ 0, H2ð Þ t;p; q 6¼ 0, at ð Þ¼ p; q ð Þ 0; 0

H1ð Þ t;p; q 6¼ 1, H2ð Þ t; 1; 1 6¼ 1; atð Þ¼ p; q ð Þ 1; 1 ;

(H4) For p, q ≥ 0, there exist real numbers 0 < λ, μ < 1, such that for each t∈I, τ∈ ð Þ 0; 1 , we

<sup>H</sup>1ð Þ <sup>t</sup>; <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>≥</sup> τλH1ð Þ <sup>t</sup>;p; <sup>q</sup> , <sup>H</sup>2ð Þ <sup>t</sup>; <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>≥</sup> <sup>τ</sup><sup>μ</sup>H2ð Þ <sup>t</sup>; <sup>p</sup>; <sup>q</sup> :

Theorem 3.4. Under the assumptions Hð Þ� <sup>1</sup> ð Þ H<sup>4</sup> , the BVP (1) has a unique solution in C<sup>f</sup> where

H1ð Þ¼ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at ð Þ¼ p; q ð Þ 0; 0

H1ð Þ t; p; q 6¼ 1, H2ð Þ t; 1; 1 6¼ 1; at ð Þ¼ p; q ð Þ 1; 1 ;

(H4) For p, q ≥ 0, there exist real numbers 0 < λ, μ < 1, such that for each t∈I, τ∈ ð Þ 0; 1 , we

<sup>H</sup>1ð Þ <sup>t</sup>; <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>≥</sup> τλH1ð Þ <sup>t</sup>; <sup>p</sup>; <sup>q</sup> , <sup>H</sup>2ð Þ <sup>t</sup>; <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>≥</sup> <sup>τ</sup><sup>μ</sup>H2ð Þ <sup>t</sup>; <sup>p</sup>; <sup>q</sup> :

Theorem 3.4. Under the assumptions Hð Þ� <sup>1</sup> ð Þ H<sup>4</sup> , the BVP (1) has a unique solution in C<sup>f</sup> where

ð 1 ð 1

0

<sup>G</sup>1ð Þ <sup>t</sup>;<sup>s</sup> <sup>H</sup>1ð Þ <sup>s</sup>; p sð Þ; q sð Þ ds <sup>¼</sup> <sup>τ</sup><sup>λ</sup>S1ð Þ <sup>p</sup>; <sup>q</sup> ð Þ<sup>t</sup> <sup>≥</sup> τκS1ð Þ <sup>p</sup>; <sup>q</sup> ð Þ<sup>t</sup> ,

0

<sup>G</sup>1ð Þ <sup>t</sup>;<sup>s</sup> <sup>H</sup>1ð Þ <sup>s</sup>; p sð Þ; q sð Þ ds <sup>¼</sup> <sup>τ</sup><sup>λ</sup>S1ð Þ <sup>p</sup>; <sup>q</sup> ð Þ<sup>t</sup> <sup>≥</sup> τκS1ð Þ <sup>p</sup>; <sup>q</sup> ð Þ<sup>t</sup> ,

<sup>S</sup>2ð Þ <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> ð Þ<sup>t</sup> <sup>≥</sup> <sup>τ</sup><sup>κ</sup>S2ð Þ <sup>p</sup>; <sup>q</sup> ð Þ<sup>t</sup> :

<sup>S</sup>2ð Þ <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> ð Þ<sup>t</sup> <sup>≥</sup> <sup>τ</sup><sup>κ</sup>S2ð Þ <sup>p</sup>; <sup>q</sup> ð Þ<sup>t</sup> :

In view of partial order <sup>⪰</sup>onE� <sup>E</sup> induced by the cone <sup>C</sup>, we get <sup>S</sup>ð Þ <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>⪰</sup>τκSðp tð Þ; <sup>q</sup> ð ÞÞ t , τ∈ð Þ 0; 1 , pð Þ ; q ∈ C: Which yields that S is τ� concave and nondecreasing operator with respect to the partial order by using hypothesis ð Þ H<sup>4</sup> . Hence, taking f ∈ C for each t ∈I

In view of partial order <sup>⪰</sup> on <sup>E</sup> � <sup>E</sup> induced by the cone <sup>C</sup>, we get <sup>S</sup>ð Þ <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>⪰</sup>τκSðp tð Þ; <sup>q</sup> ð ÞÞ t , τ∈ð Þ 0; 1 , pð Þ ; q ∈ C: Which yields that S is τ� concave and nondecreasing operator with respect to the partial order by using hypothesis ð Þ H<sup>4</sup> . Hence, taking f ∈ C for each t ∈I

<sup>θ</sup>2�<sup>1</sup> � � <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>t</sup> ; <sup>f</sup> <sup>2</sup>ð Þ<sup>t</sup> � �:

<sup>θ</sup>2�<sup>1</sup> � � <sup>¼</sup> <sup>f</sup> <sup>1</sup>ð Þ<sup>t</sup> ; <sup>f</sup> <sup>2</sup>ð Þ<sup>t</sup> � �:

� �, <sup>H</sup>2ð Þ <sup>t</sup>; <sup>p</sup>; <sup>q</sup> <sup>≤</sup> <sup>H</sup><sup>1</sup> <sup>t</sup>; <sup>p</sup>1; <sup>q</sup><sup>1</sup>

� �, <sup>H</sup>2ð Þ <sup>t</sup>; <sup>p</sup>; <sup>q</sup> <sup>≤</sup> <sup>H</sup><sup>1</sup> <sup>t</sup>; <sup>p</sup>1; <sup>q</sup><sup>1</sup>

Existence Theory of Differential Equations of Arbitrary Order

http://dx.doi.org/10.5772/intechopen.75523

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

43

43

G1ð Þ t;s H1ð Þ s; τp sð Þ; τq sð Þ ds

G1ð Þ t;s H1ð Þ s; τp sð Þ; τq sð Þ ds

� �;

� �;

0 ≤ p ≤ p1, 0 ≤ q ≤ q<sup>1</sup> ) H1ð Þ t; p; q ≤ H<sup>1</sup> t; p1; q<sup>1</sup>

0 ≤ p ≤ p1, 0 ≤ q ≤ q<sup>1</sup> ) H1ð Þ t; p; q ≤ H<sup>1</sup> t; p1; q<sup>1</sup>

Proof. Let max <sup>λ</sup>; <sup>μ</sup> � � <sup>¼</sup> <sup>κ</sup> and ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup>. For each <sup>t</sup> <sup>∈</sup>I, using ð Þ <sup>H</sup><sup>4</sup> , we have

f tðÞ¼ t

θ1�1 ; t

θ1�1 ; t

f tðÞ¼ t

S1ð Þ τp; τq ðÞ¼ t

S1ð Þτp; τq ðÞ¼ t

Proof. Let max <sup>λ</sup>; <sup>μ</sup> � � <sup>¼</sup> <sup>κ</sup> and ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup>. For each <sup>t</sup> <sup>∈</sup>I, using ð Þ <sup>H</sup><sup>4</sup> , we have

Similarly, one can derive that Similarly, one can derive that

$$\|\|\mathcal{S}\_2(p,q)\|\|\_{\mathbb{E}} \le \frac{r}{2}.\tag{16}$$

Thus, from Eqs. (15) and (16), we get Thus, from Eqs. (15) and (16), we get

$$\|\|\mathcal{S}(p,q)\|\|\_{\mathbf{E}\times\mathbf{E}}\leq r.\tag{17}$$

3 5

Therefore, Sð Þ p; q ⊆E: Hence, by Theorem 3.2 the operator S : E ! E is completely continuous. Consider the eigenvalue problem: Therefore, Sð Þ p; q ⊆E: Hence, by Theorem 3.2 the operator S : E ! E is completely continuous. Consider the eigenvalue problem:

$$
\rho(p,q) = \rho \mathcal{S}(p,q), \text{ with } \rho \in (0,1). \tag{18}
$$

Under the assumption that ð Þ p; q is a solution of Eq. (18) for r ∈ð Þ 0; 1 , we have Under the assumption that ð Þ p; q is a solution of Eq. (18) for r ∈ð Þ 0; 1 , we have

$$\begin{split} |p(t)| &\leq \rho \max\_{t \in \mathcal{I}} \left| \operatorname{\bf G}\_{1}(t,s) | \mathcal{H}\_{1}(s,p(s),q(s))ds \right| \\ &\leq \rho \left[ \int\_{0}^{1} \mathsf{G}\_{1}(1,s)\rho\_{1}(s)ds + \int\_{0}^{1} \mathsf{G}\_{1}(1,s)(\psi\_{1}(s)|p(s)| + \sigma\_{1}(s)|q(s)|)ds \right] \\ &\leq \rho(\Delta\_{1} + r\Lambda\_{1}) \\ \text{which implies that } \|p\|\_{\mathbb{E}} < \frac{r}{2}. \end{split}$$

Similarly, we can obtain that ∥q∥<sup>E</sup> < <sup>r</sup> <sup>2</sup> , so ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < r, which implies that ð Þ p; q does not belong to ∂E for all r ∈ ð Þ 0; 1 : Therefore, due to Theorem 2.10, S has a fixed point in E Similarly, we can obtain that ∥q∥<sup>E</sup> < <sup>r</sup> <sup>2</sup> , so ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < r, which implies that ð Þ p; q does not belong to ∂E for all r ∈ ð Þ 0; 1 : Therefore, due to Theorem 2.10, S has a fixed point in E

Assume that the given hypothesis holds: Assume that the given hypothesis holds:

(H1) The nonlinear functions H<sup>1</sup> and H<sup>2</sup> are continuous on I � Rþ∪f g0 � Rþ∪f g0 ! Rþ∪f g0 (H1) The nonlinear functions H<sup>1</sup> and H<sup>2</sup> are continuous on I � Rþ∪f g0 � Rþ∪f g0 ! Rþ∪f g0

(H2) For all t∈ I, we have (H2) For all t∈ I, we have

$$\mathcal{H}\_1(t, p, q) := \neq 0, \ \mathcal{H}\_2(t, p, q) \neq 0, \ \text{at} \ (p, q) = (0, 0).$$

and

and

Proof. Let <sup>E</sup> <sup>¼</sup> f g ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> : <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>r</sup> with min <sup>2</sup>Δ<sup>1</sup>

Proof. Let <sup>E</sup> <sup>¼</sup> f g ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>C</sup> : <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>r</sup> with min <sup>2</sup>Δ<sup>1</sup>

G1ð Þ 1;s φ1ð Þs ds þ

G1ð Þ 1;s φ1ð Þs ds þ

G1ð Þ 1;s φ1ð Þs ds þ r

Let ð Þ p; q ∈E that is ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < r: Then, we have

ð1 0

� � � �

G1ð Þ 1;s φ1ð Þs ds þ r

G1ð Þ t;s H1ðs; p sð Þ; q sð ÞÞds

ð1 0

G1ð Þ t;s H1ðs; p sð Þ; q sð ÞÞds

ð1 0

> ð1 0

ð1 0

∥S1ð Þ p; q ∥<sup>E</sup> ≤

∥S2ð Þ p; q ∥<sup>E</sup> ≤

Therefore, Sð Þ p; q ⊆E: Hence, by Theorem 3.2 the operator S : E ! E is completely continuous.

Therefore, Sð Þ p; q ⊆E: Hence, by Theorem 3.2 the operator S : E ! E is completely continuous.

Under the assumption that ð Þ p; q is a solution of Eq. (18) for r ∈ð Þ 0; 1 , we have

G1ð Þ t;s j j H1ðs; p sð Þ; q sð ÞÞds

G1ð Þ t;s j j H1ðs; p sð Þ; q sð ÞÞds

ð 1

0

r 2 :

belong to ∂E for all r ∈ ð Þ 0; 1 : Therefore, due to Theorem 2.10, S has a fixed point in E

Under the assumption that ð Þ p; q is a solution of Eq. (18) for r ∈ð Þ 0; 1 , we have

ð 1

0

r 2 :

belong to ∂E for all r ∈ ð Þ 0; 1 : Therefore, due to Theorem 2.10, S has a fixed point in E

Define the operator S : E ! C as in Eq. (9).

42 Differential Equations - Theory and Current Research

42 Differential Equations - Theory and Current Research

t ∈I

≤ ð1 0

≤ ð1 0

Thus, from Eq. (14), we have

Similarly, one can derive that

∣S1ð Þ p; q ð Þt ∣ ¼ max

≤ ð1 0

∣S1ð Þ p; q ð Þt ∣ ¼ max

≤ ð1 0

Thus, from Eq. (14), we have

Similarly, one can derive that

Thus, from Eqs. (15) and (16), we get

Thus, from Eqs. (15) and (16), we get

Consider the eigenvalue problem:

Consider the eigenvalue problem:

∣p tð Þ∣ ≤ r max t∈ I ð 1

> 2 4

> > 0

Similarly, we can obtain that ∥q∥<sup>E</sup> < <sup>r</sup>

≤ r ð 1

≤ rð Þ Δ<sup>1</sup> þ rΛ<sup>1</sup>

0

≤ r ð 1

Similarly, we can obtain that ∥q∥<sup>E</sup> < <sup>r</sup>

0

∣p tð Þ∣ ≤ r max t∈ I ð 1

> 2 4

which implies that ∥p∥<sup>E</sup> <

≤ rð Þ Δ<sup>1</sup> þ rΛ<sup>1</sup>

G1ð Þ 1;s φ1ð Þs ds þ

which implies that ∥p∥<sup>E</sup> <

G1ð Þ1;s φ1ð Þs ds þ

0

Let ð Þ p; q ∈E that is ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < r: Then, we have

Define the operator S : E ! C as in Eq. (9).

ð1 0

t ∈I

� � � �

<sup>1</sup>�2Λ<sup>1</sup> ; <sup>2</sup>Δ<sup>2</sup> 1�2Λ<sup>2</sup> � �

<sup>1</sup>�2Λ<sup>1</sup> ; <sup>2</sup>Δ<sup>2</sup> 1�2Λ<sup>2</sup> � �

� � � �

G1ð1;sÞψ1ð Þj s p sð Þjds þ

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>1ð Þþ <sup>s</sup> <sup>σ</sup>1ð Þ<sup>s</sup> � �ds � �

> r 2

∥S1ð Þ p; q ∥<sup>E</sup> ≤

∥S2ð Þ p; q ∥<sup>E</sup> ≤

r 2

� �

� � � �

G1ð1;sÞψ1ð Þj s p sð Þjds þ

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>1ð Þþ <sup>s</sup> <sup>σ</sup>1ð Þ<sup>s</sup> � �ds � �

> r 2

> r 2

� �

< r:

< r:

ð1 0 ð1 0

G1ð1;sÞσ1ð Þj s q sð Þjds

¼ Δ<sup>1</sup> þ rΛ<sup>1</sup> ≤

: (15)

: (16)

r 2 : r 2 :

G1ð1;sÞσ1ð Þj s q sð Þjds

(14)

(14)

¼ Δ<sup>1</sup> þ rΛ<sup>1</sup> ≤

: (15)

: (16)

3 5 3 5

∥Sð Þ p; q ∥<sup>E</sup>�<sup>E</sup> ≤ r: (17)

ð Þ¼ p; q rSð Þ p; q , with r ∈ð Þ 0; 1 : (18)

∥Sð Þ p; q ∥<sup>E</sup>�<sup>E</sup> ≤ r: (17)

ð Þ¼ p; q rSð Þ p; q , with r ∈ð Þ 0; 1 : (18)

G1ð1;sÞðψ1ð Þj s p sð Þj þ σ1ð Þj s q sð ÞjÞds

G1ð1;sÞðψ1ð Þj s p sð Þj þ σ1ð Þj s q sð ÞjÞds

<sup>2</sup> , so ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < r, which implies that ð Þ p; q does not

<sup>2</sup> , so ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < r, which implies that ð Þ p; q does not

$$\mathcal{H}\_1(t, p, q) \neq 1, \ \mathcal{H}\_2(t, 1, 1) \neq 1; \ \text{at} \ (p, q) = (1, 1);$$

(H3) For all t∈ I such that (H3) For all t∈ I such that

$$0 \le p \le p\_{1\prime} \quad 0 \le q \le q\_1 \Rightarrow \mathcal{H}\_1(t, p, q) \le \mathcal{H}\_1(t, p\_1, q\_1), \ \ \mathcal{H}\_2(t, p, q) \le \mathcal{H}\_1(t, p\_1, q\_1);$$

(H4) For p, q ≥ 0, there exist real numbers 0 < λ, μ < 1, such that for each t∈I, τ∈ ð Þ 0; 1 , we have (H4) For p, q ≥ 0, there exist real numbers 0 < λ, μ < 1, such that for each t∈I, τ∈ ð Þ 0; 1 , we have

$$
\mathcal{H}\_1(t, \tau p, \tau q) \ge \tau^{\lambda} \mathcal{H}\_1(t, p, q), \quad \mathcal{H}\_2(t, \tau p, \tau q) \ge \tau^{\mu} \mathcal{H}\_2(t, p, q).
$$

Theorem 3.4. Under the assumptions Hð Þ� <sup>1</sup> ð Þ H<sup>4</sup> , the BVP (1) has a unique solution in C<sup>f</sup> where f tðÞ¼ t <sup>θ</sup>1�<sup>1</sup>; t <sup>θ</sup>2�<sup>1</sup> � �. Theorem 3.4. Under the assumptions Hð Þ� <sup>1</sup> ð Þ H<sup>4</sup> , the BVP (1) has a unique solution in C<sup>f</sup> where f tðÞ¼ t <sup>θ</sup>1�<sup>1</sup>; t <sup>θ</sup>2�<sup>1</sup> � �.

Proof. Let max <sup>λ</sup>; <sup>μ</sup> � � <sup>¼</sup> <sup>κ</sup> and ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup>. For each <sup>t</sup> <sup>∈</sup>I, using ð Þ <sup>H</sup><sup>4</sup> , we have Proof. Let max <sup>λ</sup>; <sup>μ</sup> � � <sup>¼</sup> <sup>κ</sup> and ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup>. For each <sup>t</sup>∈I, using ð Þ <sup>H</sup><sup>4</sup> , we have

$$\mathcal{S}\_1(\tau p, \tau q)(t) = \int\_0^1 \mathbf{G}\_1(t, s) \mathcal{H}\_1(s, \tau p(s), \tau q(s)) ds$$

$$\tau \ge \tau^{\lambda} \int\_0^1 \mathbf{G}\_1(t, s) \mathcal{H}\_1(s, p(s), q(s)) ds = \tau^{\lambda} \mathcal{S}\_1(p, q)(t) \ge \tau^{\kappa} \mathcal{S}\_1(p, q)(t),$$

Analogously, we also get Analogously, we also get

≥ τ<sup>λ</sup>

$$
\mathcal{S}\_2(\pi p, \pi q)(t) \ge \pi^\kappa \mathcal{S}\_2(p, q)(t) \dots
$$

In view of partial order <sup>⪰</sup> on <sup>E</sup> � <sup>E</sup> induced by the cone <sup>C</sup>, we get <sup>S</sup>ð Þ <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>⪰</sup>τκSðp tð Þ; <sup>q</sup> ð ÞÞ t , τ∈ð Þ 0; 1 , pð Þ ; q ∈ C: Which yields that S is τ� concave and nondecreasing operator with respect to the partial order by using hypothesis ð Þ H<sup>4</sup> . Hence, taking f ∈ C for each t ∈I defined by In view of partial order <sup>⪰</sup>on � <sup>E</sup> induced by the cone <sup>C</sup>, we get <sup>S</sup>ð Þ <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>⪰</sup>τκSðp tð Þ; <sup>q</sup> ð ÞÞ t , τ∈ð Þ 0; 1 , pð Þ ; q ∈ C: Which yields that S is τ� concave and nondecreasing operator with respect to the partial order by using hypothesis ð Þ H<sup>4</sup> . Hence, taking f ∈ C for each t ∈I defined by

$$f(t) = \left(t^{\theta\_1 - 1}, t^{\theta\_2 - 1}\right) = \left(f\_1(t), f\_2(t)\right).$$

Suppose that Suppose that

#### 44 Differential Equations - Theory and Current Research 44 Differential Equations - Theory and Current Research

$$\mathbf{w}\_1 = \max\left\{ \frac{1}{\Gamma(\theta\_1)} \prod\_{0}^{1} \mathcal{H}\_1(s, 1, 1) ds, \frac{1}{\Gamma(\theta\_2)} \int\_0^1 \mathcal{H}\_2(s, 1, 1) ds \right\}$$

and

and

$$\mathbf{w}\_2 = \max\left\{ \frac{1}{\Gamma(\theta\_1)} \int\_0^1 \mathbf{L}(s) \mathcal{H}\_1(s, 0, 0) ds, \frac{1}{\Gamma(\theta\_2)} \int\_0^1 \mathbf{K}(s) \mathcal{H}\_2(s, 0, 0) ds \right\}.$$

Also, from Green's functions, we can obtain that Also, from Green's functions, we can obtain that

$$\mathbf{L}(\mathbf{s}) = (1 - s)^{\theta\_1 - 1} \left( \frac{1 + \lambda\_1}{\lambda\_1} \right), \quad \mathbf{K}(\mathbf{s}) = (1 - s)^{\theta\_2 - 1} \left( \frac{1 + \lambda\_2}{\lambda\_2} \right). \tag{19}$$

9 = ;

> 9 = ;:

S2f tð Þ ≥ νf <sup>2</sup>ð Þt :

S2f tð Þ ≥ νf <sup>2</sup>ð Þt :

νf ⪯Sf ⪯μf ,

νf ⪯S ⪯μf ,

which implies that Sf ∈ C<sup>f</sup> . So, thanks to Lemma 2.9, we see that the operator S is concave; hence, it has at most one fixed point ð Þ p; q ∈ C<sup>f</sup> which is the corresponding solution of BVPs (1).

(C1) Hjð Þj ¼ 1; 2 : I � Rþ∪f g0 � Rþ∪f g0 ! Rþ∪f g0 is uniformly bounded and continuous on I

which implies that Sf ∈ C<sup>f</sup> . So, thanks to Lemma 2.9, we see that the operator S is concave; hence, it has at most one fixed point ð Þ p; q ∈ C<sup>f</sup> which is the corresponding solution of BVPs (1).

(C1) Hjð Þ j ¼ 1; 2 : I � Rþ∪f g0 � Rþ∪f g0 ! Rþ∪f g0 is uniformly bounded and continuous on I

ð 1 ð 1

0

max t ∈I

G2ð Þ 1;s ds < ∞;

G2ð Þ 1;s ds < ∞;

H2ð Þ t; p; q <sup>p</sup> <sup>þ</sup> <sup>q</sup> ,

H2ð Þ t; p; q <sup>p</sup> <sup>þ</sup> <sup>q</sup> ,

H2ð Þ t;p; q

H2ð Þ t; p; q

<sup>p</sup> <sup>þ</sup> <sup>q</sup> , where <sup>ϱ</sup><sup>∈</sup> f g <sup>0</sup>; <sup>∞</sup> :

<sup>p</sup><sup>þ</sup> <sup>q</sup> , where <sup>ϱ</sup><sup>∈</sup> f g <sup>0</sup>; <sup>∞</sup> :

0

max t ∈I

pþq!ϱ inf t∈I

G2ð Þ t;s ds:

Theorem 3.5. Assume that the conditions Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> together with given assumptions are satisfied:

G2ð Þ t;s ds:

0

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>&</sup>gt; 1 and

0 @

<sup>A</sup> <sup>&</sup>gt; <sup>1</sup>, <sup>H</sup>2,<sup>∞</sup> <sup>γ</sup><sup>2</sup>

2 1�ð θ

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>&</sup>gt; 1 and

0 @

θ

2 1�ð θ

θ

G2ð Þ 1;s ds

1 A > 1:

1 A > 1:

G2ð Þ 1;s ds

G1ð Þ 1;s ds < ∞, 0 <

<sup>p</sup> <sup>þ</sup> <sup>q</sup> , <sup>H</sup>2, <sup>ϱ</sup> <sup>¼</sup> lim

G1ð Þ t;s ds, δ<sup>2</sup> ¼ max

t∈I ð 1

> 1 Ð 1�θ θ

1

G2ð Þ 1;s ds

0

t∈I ð 1

Theorem 3.5. Assume that the conditions Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> together with given assumptions are satisfied:

1 Ð 1�θ θ

1

<sup>A</sup> <sup>&</sup>gt; <sup>1</sup>, <sup>H</sup>2,<sup>∞</sup> <sup>γ</sup><sup>2</sup>

<sup>2</sup> ¼ lim pþg!ϱ

<sup>p</sup> <sup>þ</sup> <sup>q</sup> , <sup>H</sup>2, <sup>ϱ</sup> <sup>¼</sup> lim

G1ð Þ 1;s ds < ∞, 0 <

pþq!ϱ inf t∈I

<sup>2</sup> ¼ lim pþg!ϱ

Sf tð Þ⪰νf : (21)

Sf tð Þ⪰νf : (21)

Existence Theory of Differential Equations of Arbitrary Order

http://dx.doi.org/10.5772/intechopen.75523

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

45

45

Thus, we have

Thus, we have

From Eqs. (20) and (21), we produce

From Eqs. (20) and (21), we produce

Now, we define the following:

Now, we define the following:

(C3) Let these limits hold:

Hϱ <sup>1</sup> ¼ lim pþq!ϱ

(H5) H1, <sup>0</sup> γ<sup>2</sup>

1 Ð 1�θ θ

(H5) H1, <sup>0</sup> γ<sup>2</sup>

H1,<sup>ϱ</sup> ¼ lim pþq!ϱ inf t∈ I

H1,<sup>ϱ</sup> ¼ lim pþq!ϱ inf t∈ I

Hϱ <sup>1</sup> ¼ lim pþq!ϱ

(C3) Let these limits hold:

δ<sup>1</sup> ¼ max t∈I ð 1

> 1 Ð 1�θ θ

(C2) Green's functions G1ð Þ 1;s , G2ð Þ 1;s satisfy

max t∈ I

0

δ<sup>1</sup> ¼ max t∈I ð 1

H2, <sup>0</sup> γ<sup>2</sup> 2 1�ð θ

(H6) There exists constant α > 0 such that

0 @

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>&</sup>gt; <sup>1</sup>, <sup>H</sup>1,<sup>∞</sup> <sup>γ</sup><sup>2</sup>

H2, <sup>0</sup> γ<sup>2</sup> 2 1�ð θ

0 @

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>&</sup>gt; <sup>1</sup>, <sup>H</sup>1,<sup>∞</sup> <sup>γ</sup><sup>2</sup>

0

θ

Moreover, H1,<sup>0</sup> ¼ H2, <sup>0</sup> ¼ H1,<sup>∞</sup> ¼ H2,<sup>∞</sup> ¼ ∞ also hold:

Moreover, H1,<sup>0</sup> ¼ H2, <sup>0</sup> ¼ H1,<sup>∞</sup> ¼ H2,<sup>∞</sup> ¼ ∞ also hold:

(H6) There exists constant α > 0 such that

0 < ð 1

(C2) Green's functions G1ð Þ 1;s , G2ð Þ 1;s satisfy

0

0 < ð 1

0

H1ð Þ t;p; q <sup>p</sup> <sup>þ</sup> <sup>q</sup> , <sup>H</sup><sup>ϱ</sup>

H1ð Þ t; p; q

H1ð Þ t; p; q <sup>p</sup> <sup>þ</sup> <sup>q</sup> , <sup>H</sup><sup>ϱ</sup>

max t∈ I

H1ð Þ t; p; q

G1ð Þ t;s ds, δ<sup>2</sup> ¼ max

G2ð Þ 1;s ds

θ

with respect to t.

with respect to t.

Due to nondecreasing property of H1, H<sup>2</sup> in view of ð Þ H<sup>3</sup> , we get μ > 0, ν > 0. Therefore, applying (19) together with ð Þ H<sup>4</sup> , one has Due to nondecreasing property of H1, H<sup>2</sup> in view of ð Þ H<sup>3</sup> , we get μ > 0, ν > 0. Therefore, applying (19) together with ð Þ H<sup>4</sup> , one has

$$\mathcal{S}\_1 h(t) = \int\_0^1 \mathbf{G}\_1(t, s) \mathcal{H}\_1 \{s, f\_1(s), f\_2(s)\} ds$$

$$= \int\_0^1 \mathbf{G}\_1(t, s) \mathcal{H}\_1 \{s, s^{\theta\_1 - 1}, s^{\theta\_2 - 1}\} ds \le \int\_0^1 \mathbf{G}\_1(t, s) \mathcal{H}\_1 (s, 1, 1) ds$$

$$\le \left(\frac{1}{\Gamma(\theta\_1)} \prod\_{0}^1 (1 - s)^{\theta\_1 - 1} \mathcal{H}\_1 (s, 1, 1) ds\right) t^{\theta\_1 - 1} \le \mu f\_1(t).$$

Similarly, we can get Similarly, we can get

$$
\mathcal{S}\_2 f(t) \le \mu f\_2(t).
$$

Then, we obtain Then, we obtain

$$
\mathcal{S}f \preceq \mu f.\tag{20}
$$

Like the aforesaid process, applying Eq. (19) together with ð Þ H<sup>4</sup> , for each t∈ I, one has Like the aforesaid process, applying Eq. (19) together with ð Þ H<sup>4</sup> , for each t∈ I, one has

$$\begin{aligned} \mathcal{S}\_1 f(t) &= \int\_0^1 \mathbf{G}\_1(t, s) \mathcal{H}\_1 \{s, s^{\theta\_1 - 1}, s^{\theta\_2 - 1}\} ds \geq \int\_0^1 \mathbf{G}\_1(t, s) \mathcal{H}\_1(s, 0, 0) ds \\ &\geq \left(\frac{1}{\Gamma \theta\_1} \int\_0^1 \mathbf{L}(s) \mathcal{H}\_1(s, 0, 0) ds\right) t^{\theta\_1 - 1} \geq \nu h\_1(t), \end{aligned}$$

With same fashion, we can obtain With same fashion, we can obtain

$$
\mathcal{S}\_2 f(t) \ge \nu f\_2(t).
$$

Thus, we have Thus, we have

w<sup>1</sup> ¼ max

8 < :

Also, from Green's functions, we can obtain that

w<sup>2</sup> ¼ max

w<sup>2</sup> ¼ max

44 Differential Equations - Theory and Current Research

44 Differential Equations - Theory and Current Research

applying (19) together with ð Þ H<sup>4</sup> , one has

applying (19) together with ð Þ H<sup>4</sup> , one has

¼ ð 1

S1f tðÞ¼

With same fashion, we can obtain

With same fashion, we can obtain

ð 1

ð 1

0

S1f tðÞ¼

0

Similarly, we can get

Then, we obtain

Similarly, we can get

Then, we obtain

0

¼ ð 1

≤

0 @

0

and

and

1 Γð Þ θ<sup>1</sup>

8 < :

8 < :

w<sup>1</sup> ¼ max

1 Γð Þ θ<sup>1</sup>

8 < :

<sup>L</sup>ð Þ¼ <sup>s</sup> ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>1</sup>

Also, from Green's functions, we can obtain that

G1ð Þ t;s H<sup>1</sup> s;s

1 Γð Þ θ<sup>1</sup>

0 @

≤

ð 1

1 Γð Þ θ<sup>1</sup>

G1ð Þ t;s H<sup>1</sup> s;s

0

G1ð Þ t;s H<sup>1</sup> s;s

≥

0 @

G1ð Þ t;s H<sup>1</sup> s;s

ð 1

1 Γð Þ θ<sup>1</sup>

0

<sup>L</sup>ð Þ¼ <sup>s</sup> ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>1</sup>

ð 1

1 Γð Þ θ<sup>1</sup> ð 1

0

<sup>H</sup>1ð Þ <sup>s</sup>; <sup>1</sup>; <sup>1</sup> ds; <sup>1</sup>

<sup>L</sup>ð Þ<sup>s</sup> <sup>H</sup>1ð Þ <sup>s</sup>; <sup>0</sup>; <sup>0</sup> ds; <sup>1</sup>

<sup>L</sup>ð Þ<sup>s</sup> <sup>H</sup>1ð Þ <sup>s</sup>; <sup>0</sup>; <sup>0</sup> ds; <sup>1</sup>

Due to nondecreasing property of H1, H<sup>2</sup> in view of ð Þ H<sup>3</sup> , we get μ > 0, ν > 0. Therefore,

Due to nondecreasing property of H1, H<sup>2</sup> in view of ð Þ H<sup>3</sup> , we get μ > 0, ν > 0. Therefore,

ð 1

0

λ1 � �

> ð 1

S1h tðÞ¼

θ1�1 ;s <sup>θ</sup>2�<sup>1</sup> � �ds ≤

ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

0

S2f tð Þ ≤ μf <sup>2</sup>ð Þt :

Like the aforesaid process, applying Eq. (19) together with ð Þ H<sup>4</sup> , for each t∈ I, one has

θ1�1 ;s <sup>θ</sup>2�<sup>1</sup> � �ds ≥

S2f tð Þ≤ μf <sup>2</sup>ð Þt :

Like the aforesaid process, applying Eq. (19) together with ð Þ H<sup>4</sup> , for each t∈ I, one has

θ1�1 ;s <sup>θ</sup>2�<sup>1</sup> � �ds ≥

> 1 Γθ<sup>1</sup> ð 1

0 @

≥

0

1 Γθ<sup>1</sup> ð 1

0

ð 1

0

H1ð Þ s; 1; 1 ds

Γð Þ θ<sup>2</sup>

<sup>H</sup>1ð Þ <sup>s</sup>; <sup>1</sup>; <sup>1</sup> ds; <sup>1</sup>

Γð Þ θ<sup>2</sup>

ð 1

H2ð Þ s; 1; 1 ds

ð 1

0

ð 1

0

, <sup>K</sup>ð Þ¼ <sup>s</sup> ð Þ <sup>1</sup>� <sup>s</sup> <sup>θ</sup>2�<sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

Kð Þs H2ð Þ s; 0; 0 ds

Kð Þs H2ð Þ s; 0; 0 ds

λ2 � �

λ2 � �

9 = ;

H2ð Þ s; 1; 1 ds

9 = ;: 9 = ;:

9 = ;

: (19)

: (19)

0

Γð Þ θ<sup>2</sup>

ð 1

0

Γð Þ θ<sup>2</sup>

, <sup>K</sup>ð Þ¼ <sup>s</sup> ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>2�<sup>1</sup> <sup>1</sup> <sup>þ</sup> <sup>λ</sup><sup>2</sup>

<sup>G</sup>1ð Þ <sup>t</sup>;<sup>s</sup> <sup>H</sup><sup>1</sup> <sup>s</sup>; <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> ; <sup>f</sup> <sup>2</sup>ð Þ<sup>s</sup> � �ds

1 At

ð 1

0

H1ð Þ s; 1; 1 ds

ð 1

0

Lð Þs H1ð Þ s; 0; 0 ds

Lð Þs H1ð Þ s; 0; 0 ds

G1ð Þ t;s H1ð Þ s; 1; 1 ds

1 At

<sup>G</sup>1ð Þ <sup>t</sup>;<sup>s</sup> <sup>H</sup><sup>1</sup> <sup>s</sup>; <sup>f</sup> <sup>1</sup>ð Þ<sup>s</sup> ; <sup>f</sup> <sup>2</sup>ð Þ<sup>s</sup> � �ds

<sup>θ</sup>1�<sup>1</sup> <sup>≤</sup> <sup>μ</sup><sup>f</sup> <sup>1</sup>ð Þ<sup>t</sup> :

<sup>θ</sup>1�<sup>1</sup> <sup>≤</sup> <sup>μ</sup><sup>f</sup> <sup>1</sup>ð Þ<sup>t</sup> :

G1ð Þ t;s H1ð Þ s; 1; 1 ds

Sf ⪯μf : (20)

S⪯μf : (20)

G1ð Þ t;s H1ð Þ s; 0; 0 ds

1 At

<sup>θ</sup>1�<sup>1</sup> <sup>≥</sup> <sup>ν</sup>h1ð Þ<sup>t</sup> ,

<sup>θ</sup>1�<sup>1</sup> <sup>≥</sup> <sup>ν</sup>h1ð Þ<sup>t</sup> ,

G1ð Þ t;s H1ð Þ s; 0; 0 ds

1 At

ð 1

0

0

ð 1

0

λ1 � �

S1h tðÞ¼

θ1�1 ;s <sup>θ</sup>2�<sup>1</sup> � �ds ≤

ð Þ <sup>1</sup> � <sup>s</sup> <sup>θ</sup>1�<sup>1</sup>

ð 1

0

$$\mathcal{S}f(t) \succeq \mathsf{v}f. \tag{21}$$

From Eqs. (20) and (21), we produce From Eqs. (20) and (21), we produce

$$
\nu f \preceq \mathcal{S}f \preceq \mu f,
$$

which implies that Sf ∈ C<sup>f</sup> . So, thanks to Lemma 2.9, we see that the operator S is concave; hence, it has at most one fixed point ð Þ p; q ∈ C<sup>f</sup> which is the corresponding solution of BVPs (1). which implies that Sf ∈ C<sup>f</sup> . So, thanks to Lemma 2.9, we see that the operator S is concave; hence, it has at most one fixed point ð Þ p; q ∈ C<sup>f</sup> which is the corresponding solution of BVPs (1).

Now, we define the following: Now, we define the following:

(C1) Hjð Þ j ¼ 1; 2 : I � Rþ∪f g0 � Rþ∪f g0 ! Rþ∪f g0 is uniformly bounded and continuous on I with respect to t. (C1) Hjð Þ j ¼ 1; 2 : I � Rþ∪f g0 � Rþ∪f g0 ! Rþ∪f g0 is uniformly bounded and continuous on I with respect to t.

(C2) Green's functions G1ð Þ 1;s , G2ð Þ 1;s satisfy (C2) Green's functions G1ð Þ 1;s , G2ð Þ 1;s satisfy

0 <

$$0 < \int\_0^1 \mathbf{G}\_1(1, s) ds < \underset{\ast}{\Leftrightarrow} 0 < \int\_0^1 \mathbf{G}\_2(1, s) ds < \ast \ast$$

(C3) Let these limits hold: (C3) Let these limits hold:

Hϱ

$$\begin{aligned} \mathcal{H}\_1^0 &= \lim\_{p+q \to q} \max\_{t \in \mathbf{I}} \frac{\mathcal{H}\_1(t, p, q)}{p+q}, \ \mathcal{H}\_2^0 = \lim\_{p+q \to q} \max\_{t \in \mathbf{I}} \frac{\mathcal{H}\_2(t, p, q)}{p+q}, \\\mathcal{H}\_{1, q} &= \lim\_{p+q \to q} \inf\_{t \in \mathbf{I}} \frac{\mathcal{H}\_1(t, p, q)}{p+q}, \ \mathcal{H}\_{2, q} = \lim\_{p+q \to q} \inf\_{t \in \mathbf{I}} \frac{\mathcal{H}\_2(t, p, q)}{p+q}, \ \text{where } \varrho \in \{0, \infty\}. \end{aligned}$$

$$\delta\_1 = \max\_{t \in \mathbf{I}} \left\{ \mathbf{G}\_1(t, s) ds, \ \delta\_2 = \max\_{t \in \mathbf{I}} \left\{ \mathbf{G}\_2(t, s) ds, \ \ \ \right\} \right\}$$

Theorem 3.5. Assume that the conditions Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> together with given assumptions are satisfied: Theorem 3.5. Assume that the conditions Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> together with given assumptions are satisfied:

$$\begin{aligned} \left(\mathsf{H}\_{5}\right)\mathcal{H}\_{1,0}\left(\mathcal{V}\_{1}^{2}\int\_{\theta}^{1-\theta}\mathsf{G}\_{1}(1,s)ds\right) &> 1, \mathsf{H}\_{1,\bowtie}\left(\mathcal{V}\_{1}^{2}\int\_{\theta}^{1-\theta}\mathsf{G}\_{1}(1,s)ds\right) > 1 \text{ and} \\ &\quad \mathscr{H}\_{2,0}\left(\mathcal{V}\_{2}^{2}\int\_{\theta}^{1-\theta}\mathsf{G}\_{2}(1,s)ds\right) > 1, \mathscr{H}\_{2,\bowtie}\left(\mathcal{V}\_{2}^{2}\int\_{\theta}^{1-\theta}\mathsf{G}\_{2}(1,s)ds\right) > 1. \end{aligned}$$

Moreover, H1,<sup>0</sup> ¼ H2, <sup>0</sup> ¼ H1,<sup>∞</sup> ¼ H2,<sup>∞</sup> ¼ ∞ also hold: Moreover, H1,<sup>0</sup> ¼ H2, <sup>0</sup> ¼ H1,<sup>∞</sup> ¼ H2,<sup>∞</sup> ¼ ∞ also hold:

(H6) There exists constant α > 0 such that (H6) There exists constant α > 0 such that

$$\max\_{t \in \mathbf{I}\_\nu(p,q) \in \partial \mathbf{C}\_\pi} \mathcal{H}\_1(t, p, q) < \frac{\alpha}{2\delta\_1}$$

and

and

$$\max\_{t \in \mathbf{I}\_{\mathsf{L}}(p,q) \in \partial \mathbf{C}\_{\mathsf{a}}} \mathcal{H}\_2(t,p,q) < \frac{\alpha}{2\delta\_2}.$$

Then, the system (1) of BVPs has at least two positive solutions pð Þ ; q ,ð Þ p; q which obeying Then, the system(1) of BVPs has at least two positive solutions pð Þ ; q ,ð Þ p; q which obeying

$$0 < \|(p,q)\|\_{\mathbf{E}\times\mathbf{E}} < \alpha < \|(\overline{p},\overline{q})\|\_{\mathbf{E}\times\mathbf{E}}.\tag{22}$$

α 2δ<sup>1</sup>

α 2δ<sup>2</sup> : <sup>∥</sup>Sð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup>, for ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> <sup>∩</sup> <sup>∂</sup>Ωα: (25)

<sup>∥</sup>Sð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup>,for ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>C</sup> <sup>∩</sup> <sup>∂</sup>Ωα: (25)

� �: Hence, we conclude that the

� �: Hence, we conclude that the

http://dx.doi.org/10.5772/intechopen.75523

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

47

47

Existence Theory of Differential Equations of Arbitrary Order

Now, applying Lemma 2.11 to Eqs. (23) and (25) yields that S has a fixed point

Now, applying Lemma 2.11 to Eqs. (23) and (25) yields that S has a fixed point

system of BVPs (1) has at least two positive solutions ð Þ p; q ,ð Þ p; q such that ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> 6¼ α and

system of BVPs (1) has at least two positive solutions ð Þ p; q ,ð Þ p; q such that ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> 6¼ α and

Theorem 3.6. Consider that Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> together with the following hypothesis are satisfied:

Theorem 3.6. Consider that Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> together with the following hypothesis are satisfied:

<sup>1</sup>H1ð Þ t; p; q >

γ2

γ2

<sup>2</sup>H2ð Þ t; p; q >

α 2

<sup>1</sup>H1ð Þ t;p; q >

<sup>2</sup>H2ð Þ t;p; q >

α 2

0 < ∥ð Þ p; q ∥ <sup>E</sup>�<sup>E</sup> < α < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>:

0 < ∥ð Þ p; q ∥ <sup>E</sup>�<sup>E</sup> < α < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>:

Analogously, we deduce from Theorem 3.5 and 3.6 the following results for multiplicity of

Theorem 3.7. Under the conditions Cð Þ� <sup>1</sup> ð Þ <sup>C</sup><sup>3</sup> , there exist <sup>2</sup>k positive numbers <sup>a</sup>j, <sup>b</sup>aj, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…<sup>k</sup> with <sup>a</sup><sup>1</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>1</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>1</sup> <sup>&</sup>lt; <sup>a</sup><sup>2</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>2</sup> <sup>&</sup>lt; <sup>b</sup>a2…a<sup>k</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>k</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>k</sup> and <sup>a</sup><sup>1</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>1</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>1</sup> <sup>&</sup>lt; <sup>a</sup><sup>2</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>2</sup> <sup>&</sup>lt;

Analogously, we deduce from Theorem 3.5 and 3.6 the following results for multiplicity of

Theorem 3.7. Under the conditions Cð Þ� <sup>1</sup> ð Þ <sup>C</sup><sup>3</sup> , there exist <sup>2</sup>k positive numbers <sup>a</sup>j, <sup>b</sup>aj, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…<sup>k</sup> with <sup>a</sup><sup>1</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>1</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>1</sup> <sup>&</sup>lt; <sup>a</sup><sup>2</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>2</sup> <sup>&</sup>lt; <sup>b</sup>a2…a<sup>k</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>k</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>k</sup> and <sup>a</sup><sup>1</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>1</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>1</sup> <sup>&</sup>lt; <sup>a</sup><sup>2</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>2</sup> <sup>&</sup>lt;

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>≥</sup> <sup>a</sup>j, for tð Þ ;p; <sup>q</sup>∈<sup>I</sup> � <sup>γ</sup>1aj;a<sup>j</sup>

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>≥</sup> <sup>a</sup>j, for tð Þ ; <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>I</sup> � <sup>γ</sup>1aj; <sup>a</sup><sup>j</sup>

<sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>≥</sup> <sup>a</sup>j, for tð Þ ; <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>I</sup> � <sup>γ</sup>1aj; <sup>a</sup><sup>j</sup>

<sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>≥</sup> <sup>a</sup>j, for tð Þ ; <sup>p</sup>; <sup>q</sup>∈<sup>I</sup> � <sup>γ</sup>1aj; <sup>a</sup><sup>j</sup>

<sup>H</sup>1ð Þ <sup>t</sup>; p tð Þ; q tð Þ <sup>δ</sup><sup>1</sup> <sup>≤</sup> <sup>b</sup>ai, for tð Þ ; <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>I</sup> � <sup>γ</sup>1ba<sup>j</sup>

<sup>H</sup>1ð Þ <sup>t</sup>; p tð Þ; q tð Þδ<sup>1</sup> <sup>≤</sup> <sup>b</sup>ai, for tð Þ ;p; <sup>q</sup>∈<sup>I</sup> � <sup>γ</sup>1ba<sup>j</sup>

1�ð θ

0 @

0 @

α 2

0 @

0 @ θ

1�ð θ

α 2

θ

G1ð Þ 1;s ds

1�ð θ

θ

1�ð θ

θ

G2ð Þ 1;s ds

1 A

G1ð Þ 1;s ds

G2ð Þ 1;s ds

1 A

� � � <sup>γ</sup>2aj; <sup>a</sup><sup>j</sup>

� � � <sup>γ</sup>2aj; <sup>a</sup><sup>j</sup>

� �, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…k;

� � � <sup>γ</sup>2aj; <sup>a</sup><sup>j</sup>

� � � <sup>γ</sup>2aj; <sup>a</sup><sup>j</sup>

; baj h i � <sup>γ</sup>2aj; <sup>a</sup><sup>i</sup>

; baj h i � <sup>γ</sup>2aj; <sup>a</sup><sup>i</sup>

� �, and

� �, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…k;

� �, and

� �, and

� �, and

�1 ,

1 A

1 A

�1 ,

�1

�1

� � and a fixed point in ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> <sup>∩</sup> <sup>Ω</sup>λ\Ω<sup>α</sup>

� � and a fixed point in ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> <sup>∩</sup> <sup>Ω</sup>λ\Ω<sup>α</sup>

ð Þ p; q ∈ C ∩ Ωα\C<sup>ε</sup>

ð Þ p; q ∈ C ∩ Ωα\C<sup>ε</sup>

such that

such that

∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> 6¼ α. Thus, relation (22) holds.

∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> 6¼ α. Thus, relation (22) holds.

(H8) There exist r > 0 such that

(H8) There exist r> 0 such that

(H7) δ1H1,<sup>0</sup> < 1, δ1H1,<sup>∞</sup> < 1; δ2H1, <sup>0</sup> < 1, and δ2H2,<sup>∞</sup> < 1;

(H7) δ1H1,<sup>0</sup> < 1, δ1H1,<sup>∞</sup> < 1; δ2H1, <sup>0</sup> < 1, and δ2H2,<sup>∞</sup> < 1;

max <sup>t</sup><sup>∈</sup> <sup>I</sup>,ð Þ <sup>p</sup>;<sup>q</sup> <sup>∈</sup> <sup>∂</sup>C<sup>α</sup>

max <sup>t</sup><sup>∈</sup> <sup>I</sup>,ð Þ <sup>p</sup>;<sup>q</sup> <sup>∈</sup> <sup>∂</sup>C<sup>α</sup>

Proof. Proof is like the proof of Theorem 3.4.

solutions to the system (1) of BVPs.

Proof. Proof is like the proof of Theorem 3.4.

Ð 1 0

Ð 1 0

Ð 1 0

Ð 1 0

solutions to the system (1) of BVPs.

<sup>b</sup>a2…a<sup>k</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>k</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>k</sup> such that.

<sup>b</sup>a2…a<sup>k</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>k</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>k</sup> such that.

(H9) H1ð Þ t; p tð Þ; q tð Þ γ<sup>1</sup>

(H10) H2ð Þ t; p tð Þ; q tð Þ γ<sup>2</sup>

(H9) H1ð Þ t; p tð Þ; q tð Þ γ<sup>1</sup>

(H10) H2ð Þ t; p tð Þ; q tð Þ γ<sup>2</sup>

γ2

max <sup>t</sup><sup>∈</sup> <sup>I</sup>,ð Þ <sup>p</sup>;<sup>q</sup> <sup>∈</sup> <sup>∂</sup>C<sup>α</sup>

max <sup>t</sup><sup>∈</sup> <sup>I</sup>,ð Þ <sup>p</sup>;<sup>q</sup> <sup>∈</sup> <sup>∂</sup>C<sup>α</sup>

γ2

Then, the proposed coupled system of BVPs (1) has at least two positive solutions.

Then, the proposed coupled system of BVPs (1) has at least two positive solutions.

Proof. Assume that ð Þ H<sup>5</sup> holds, and consider e, α, λ such that 0 < e < α < λ. Further we define a set by Proof. Assume that ð Þ H<sup>5</sup> holds, and consider e, α, λ such that 0<e< α < λ. Further we define a set by

$$\Omega\_r = \{ (\mu, \upsilon) \in \mathbf{E} \times \mathbf{E} : \|(\mu, \upsilon)\|\_{\mathbf{E} \times \mathbf{E}} < r \}, \text{ where } r \in \{ \epsilon, \alpha, \lambda \}.$$

Now, if Now, if

$$\mathcal{H}\_{1,0}\left(\gamma\_1^2 \int\_{\theta}^{1-\theta} \mathbf{G}\_1(1,s)ds\right) > 1 \quad \text{and} \quad \mathcal{H}\_{2,0}\left(\gamma\_2^2 \int\_{\theta}^{1-\theta} \mathbf{G}\_2(1,s)ds\right) > 1.$$

Then, obviously, we can obtain that Then, obviously, we can obtain that

$$\|\mathcal{S}(p,q)\|\_{\mathbf{E}\times\mathbf{E}} \ge \|(p,q)\|\_{\mathbf{E}\times\mathbf{E}'} \text{ for } (p,q)\in\mathbf{C}\cap\partial\Omega\_{\varepsilon}.\tag{23}$$

$$\text{Now, if } \mathcal{H}\_{\mathbf{l},\bowtie} \left( \gamma\_1^2 \int\_{\theta}^{1-\theta} \mathbf{G}\_1(1,\mathbf{s}) ds \right) > 1 \text{ and } \mathcal{H}\_{\mathbf{2},\bowtie} \left( \gamma\_2^2 \int\_{\theta}^{1-\theta} \mathbf{G}\_2(1,\mathbf{s}) ds \right) > 1.$$

Then, like the proof of Eq. (23), we have Then, like the proof of Eq. (23), we have

$$\|\|S(p,q)\|\_{\mathbf{E}\times\mathbf{E}} \ge \|(p,q)\|\_{\mathbf{E}\times\mathbf{E}\nu} \quad \text{for} \ (p,q) \in \mathbf{C} \cap \partial\Omega\_{\mathbb{L}}.\tag{24}$$

� � �

2 :

Also, from ð Þ H<sup>5</sup> and ð Þ p; q ∈ C ∩ ∂Ωα, we get Also, from ð Þ H<sup>5</sup> and ð Þ p; q∈ C ∩ ∂Ωα, we get

$$\begin{aligned} |\mathcal{S}\_1(p,q)(t)| &= \left| \int\_0^1 \mathbf{G}\_1(t,s)\mathcal{H}(s,\mu(s),v(s))ds \right| \\ &\le \int\_0^1 \mathbf{G}\_1(1,s)|\mathcal{H}\_1(s,p(s),q(s))|ds. \end{aligned}$$

From which we have From which we have

$$\|\|\mathcal{S}\_1(p,q)\|\|\_{\mathbf{E}\times\mathbf{E}} < \frac{\alpha}{2\varrho\_1} \int\_0^1 \mathbf{G}\_1(1,s)ds = \frac{\alpha}{2}.$$

Similarly, we have <sup>∥</sup>S1ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>α</sup> <sup>2</sup> as ð Þ p; q ∈ C ∩ ∂Ωα. Hence, we have Similarly, we have <sup>∥</sup>S1ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>α</sup> <sup>2</sup> as ð Þ p; q ∈C ∩ ∂Ωα. Hence, we have

$$\|\mathcal{S}(p,q)\|\_{\mathbf{E}\times\mathbf{E}} < \|(p,q)\|\_{\mathbf{E}\times\mathbf{E}'} \text{ for } (p,q) \in \mathbf{C}\cap\partial\Omega\_a. \tag{25}$$

Now, applying Lemma 2.11 to Eqs. (23) and (25) yields that S has a fixed point ð Þ p; q ∈ C ∩ Ωα\C<sup>ε</sup> � � and a fixed point in ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> <sup>∩</sup> <sup>Ω</sup>λ\Ω<sup>α</sup> � �: Hence, we conclude that the system of BVPs (1) has at least two positive solutions ð Þ p; q ,ð Þ p; q such that ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> 6¼ α and ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> 6¼ α. Thus, relation (22) holds. Now, applying Lemma 2.11 to Eqs. (23) and (25) yields that S has a fixed point ð Þ p; q ∈ C ∩ Ωα\C<sup>ε</sup> � � and a fixed point inð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> <sup>∩</sup> <sup>Ω</sup>λ\Ω<sup>α</sup> � �: Hence, we conclude that the system of BVPs (1) has at least two positive solutions ð Þ p; q ,ð Þ p; q such that ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> 6¼ α and ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> 6¼ α. Thus, relation (22) holds.

Theorem 3.6. Consider that Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> together with the following hypothesis are satisfied: Theorem 3.6. Consider that Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> together with the following hypothesis are satisfied:


$$\begin{aligned} \max\_{t \in \mathcal{I}\_{\nu}(p,q) \in \partial \mathbb{C}\_{a}} \mathcal{V}\_{1}^{2} \mathcal{H}\_{1}(t,p,q) &> \frac{\alpha}{2} \left( \int\_{\theta}^{1-\theta} \mathbf{G}\_{1}(1,s) ds \right)^{-1}, \\\max\_{t \in \mathcal{I}\_{\nu}(p,q) \in \partial \mathbb{C}\_{a}} \mathcal{V}\_{2}^{2} \mathcal{H}\_{2}(t,p,q) &> \frac{\alpha}{2} \left( \int\_{\theta}^{1-\theta} \mathbf{G}\_{2}(1,s) ds \right)^{-1} \end{aligned}$$

such that such that

max <sup>t</sup>∈<sup>I</sup>,ð Þ <sup>p</sup>;<sup>q</sup> <sup>∈</sup>∂C<sup>α</sup>

max <sup>t</sup>∈<sup>I</sup>,ð Þ <sup>p</sup>;<sup>q</sup> <sup>∈</sup>∂C<sup>α</sup>

max <sup>t</sup><sup>∈</sup> <sup>I</sup>,ð Þ <sup>p</sup>;<sup>q</sup> <sup>∈</sup> <sup>∂</sup>C<sup>α</sup>

Then, the system (1) of BVPs has at least two positive solutions pð Þ ; q ,ð Þ p; q which obeying

max <sup>t</sup><sup>∈</sup> <sup>I</sup>,ð Þ <sup>p</sup>;<sup>q</sup> <sup>∈</sup> <sup>∂</sup>C<sup>α</sup>

and

and

define a set by

Now, if

define a set by

Now, if H1,<sup>∞</sup> γ<sup>2</sup>

From which we have

From which we have

H1, <sup>0</sup> γ<sup>2</sup> 1 1�ð θ

46 Differential Equations - Theory and Current Research

46 Differential Equations - Theory and Current Research

Then, obviously, we can obtain that

1 Ð 1�θ θ

Now, if H1,<sup>∞</sup> γ<sup>2</sup>

Then, like the proof of Eq. (23), we have

1 Ð 1�θ θ

Also, from ð Þ H<sup>5</sup> and ð Þ p; q ∈ C ∩ ∂Ωα, we get

Also, from ð ÞH<sup>5</sup> andð Þ p; q ∈ C ∩ ∂Ωα, we get

Similarly, we have <sup>∥</sup>S1ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>α</sup>

Similarly, we have <sup>∥</sup>S1ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>&</sup>lt; <sup>α</sup>

0 @

Then, obviously, we can obtain that

θ

H1, <sup>0</sup> γ<sup>2</sup> 1 1�ð θ

0 @

G1ð Þ 1;s ds !

Then, like the proof of Eq. (23), we have

G1ð Þ 1;s ds !

G1ð Þ 1;s ds

θ

<sup>∣</sup>S1ð Þ <sup>p</sup>; <sup>q</sup> ð Þ<sup>t</sup> <sup>∣</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

≤ Ð <sup>1</sup>

∥S1ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> <

∥S1ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> <

≤ Ð <sup>1</sup>

1

1

G1ð Þ 1;s ds

> 1 and H2,<sup>∞</sup> γ<sup>2</sup>

> 1 and H2,<sup>∞</sup> γ<sup>2</sup>

� � �

<sup>∣</sup>S1ð Þ <sup>p</sup>; <sup>q</sup> ð Þ<sup>t</sup> <sup>∣</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> G1ð Þ 1;s ∣H1ð Þ s; p sð Þ; q sð Þ ∣ds:

<sup>0</sup> G1ð Þ 1;s ∣H1ð Þ s; p sð Þ; q sð Þ ∣ds:

α 2ϱ<sup>1</sup> ð1 0

� � �

α 2ϱ<sup>1</sup> ð1 0

Now, if

H1ð Þ t; p; q <

H1ð Þ t;p; q <

H2ð Þ t;p; q <

H2ð Þ t; p; q <

Proof. Assume that ð Þ H<sup>5</sup> holds, and consider e, α, λ such that 0 < e < α < λ. Further we

Proof. Assume that ð Þ H<sup>5</sup> holds, and consider e, α, λ such that 0e< α < λ. Further we

Ω<sup>r</sup> ¼ f g ð Þ u; v ∈� E : ∥ð Þ u; v ∥<sup>E</sup>�<sup>E</sup> < r , where r∈f g e; α; λ :

<sup>A</sup> <sup>&</sup>gt; 1 and <sup>H</sup>2, <sup>0</sup> <sup>γ</sup><sup>2</sup>

Then, the system(1) of BVPs has at least two positive solutions pð Þ ; q ,ð Þ p; q which obeying

Ω<sup>r</sup> ¼ f g ð Þ u; v ∈E � E : ∥ð Þ u; v ∥<sup>E</sup>�<sup>E</sup> < r , where r∈f g e; α; λ :

<sup>A</sup> <sup>&</sup>gt; 1 and <sup>H</sup>2, <sup>0</sup> <sup>γ</sup><sup>2</sup>

2 Ð 1�θ θ

2 Ð 1�θ θ

α 2δ<sup>1</sup>

α 2δ<sup>1</sup>

α 2δ<sup>2</sup> :

α 2δ<sup>2</sup> :

2 1�ð θ

0 @

0 @

θ

2 1�ð θ

θ

<sup>∥</sup>Sð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≥</sup> <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup>,forð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>C</sup> <sup>∩</sup> <sup>∂</sup>Ωε: (23)

G2ð Þ 1;s ds !

<sup>∥</sup>Sð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≥</sup> <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup>,forð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> <sup>∩</sup> <sup>∂</sup>Ωλ: (24)

<sup>∥</sup>Sð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≥</sup> <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup>, for ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> <sup>∩</sup> <sup>∂</sup>Ωε: (23)

G2ð Þ 1;s ds !

<sup>∥</sup>Sð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≥</sup> <sup>∥</sup>ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup>, for ð Þ <sup>p</sup>; <sup>q</sup> <sup>∈</sup> <sup>C</sup> <sup>∩</sup> <sup>∂</sup>Ωλ: (24)

<sup>0</sup> G1ð Þ t;s Hðs; u sð Þ; v sð ÞÞds

<sup>0</sup> G1ð Þ t;s Hðs; u sð Þ; v sð ÞÞds

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds <sup>¼</sup> <sup>α</sup>

<sup>2</sup> as ð Þ p; q∈ C ∩ ∂Ωα. Hence, we have

<sup>2</sup> as ð Þ p; q ∈ C ∩ ∂Ωα. Hence, we have

G2ð Þ 1;s ds

> 1:

> 1:

� � � � � �

2 :

2 :

<sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds <sup>¼</sup> <sup>α</sup>

1 A > 1:

1 A > 1:

G2ð Þ 1;s ds

0 < ∥ð Þ p; q ∥ <sup>E</sup>�<sup>E</sup> < α < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>: (22)

0 < ∥ð Þ p; q ∥ <sup>E</sup>�<sup>E</sup> < α < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>: (22)

$$0 < \|(p,q)\|\_{\mathbf{E}\times\mathbf{E}} < \alpha < \|(\overline{p},\overline{q})\|\_{\mathbf{E}\times\mathbf{E}}.$$

Then, the proposed coupled system of BVPs (1) has at least two positive solutions. Then, the proposed coupled system of BVPs (1) has at least two positive solutions.

Proof. Proof is like the proof of Theorem 3.4. Proof. Proof is like the proof of Theorem 3.4.

Analogously, we deduce from Theorem 3.5 and 3.6 the following results for multiplicity of solutions to the system (1) of BVPs. Analogously, we deduce from Theorem 3.5 and 3.6 the following results for multiplicity of solutions to the system (1) of BVPs.

Theorem 3.7. Under the conditions Cð Þ� <sup>1</sup> ð Þ <sup>C</sup><sup>3</sup> , there exist <sup>2</sup>k positive numbers <sup>a</sup>j, <sup>b</sup>aj, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…<sup>k</sup> with <sup>a</sup><sup>1</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>1</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>1</sup> <sup>&</sup>lt; <sup>a</sup><sup>2</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>2</sup> <sup>&</sup>lt; <sup>b</sup>a2…a<sup>k</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>k</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>k</sup> and <sup>a</sup><sup>1</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>1</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>1</sup> <sup>&</sup>lt; <sup>a</sup><sup>2</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>2</sup> <sup>&</sup>lt; <sup>b</sup>a2…a<sup>k</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>k</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>k</sup> such that. Theorem 3.7. Under the conditions Cð Þ� <sup>1</sup> ð Þ <sup>C</sup><sup>3</sup> , there exist <sup>2</sup>k positive numbers <sup>a</sup>j, <sup>b</sup>aj, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…<sup>k</sup> with <sup>a</sup><sup>1</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>1</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>1</sup> <sup>&</sup>lt; <sup>a</sup><sup>2</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>2</sup> <sup>&</sup>lt; <sup>b</sup>a2…a<sup>k</sup> <sup>&</sup>lt; <sup>γ</sup>1ba<sup>k</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>k</sup> and <sup>a</sup><sup>1</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>1</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>1</sup> <sup>&</sup>lt; <sup>a</sup><sup>2</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>2</sup> <sup>&</sup>lt; <sup>b</sup>a2…a<sup>k</sup> <sup>&</sup>lt; <sup>γ</sup>2ba<sup>k</sup> <sup>&</sup>lt; <sup>b</sup>a<sup>k</sup> such that.

(H9) H1ð Þ t; p tð Þ; q tð Þ γ<sup>1</sup> Ð 1 0 <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>≥</sup> <sup>a</sup>j, for tð Þ ; <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>I</sup> � <sup>γ</sup>1aj; <sup>a</sup><sup>j</sup> � � � <sup>γ</sup>2aj; <sup>a</sup><sup>j</sup> � �, and <sup>H</sup>1ð Þ <sup>t</sup>; p tð Þ; q tð Þ <sup>δ</sup><sup>1</sup> <sup>≤</sup> <sup>b</sup>ai, for tð Þ ; <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>I</sup> � <sup>γ</sup>1ba<sup>j</sup> ; baj h i � <sup>γ</sup>2aj; <sup>a</sup><sup>i</sup> � �, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…k; (H10) H2ð Þ t; p tð Þ; q tð Þ γ<sup>2</sup> Ð 1 0 <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>≥</sup> <sup>a</sup>j, for tð Þ ; <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>I</sup> � <sup>γ</sup>1aj; <sup>a</sup><sup>j</sup> � � � <sup>γ</sup>2aj; <sup>a</sup><sup>j</sup> � �, and (H9) H1ð Þ t; p tð Þ; q tð Þ γ<sup>1</sup> Ð 1 0 <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>≥</sup> <sup>a</sup>j, for tð Þ ; <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>I</sup> � <sup>γ</sup>1aj; <sup>a</sup><sup>j</sup> � � � <sup>γ</sup>2aj; <sup>a</sup><sup>j</sup> � �, and <sup>H</sup>1ð Þ <sup>t</sup>; p tð Þ; q tð Þ <sup>δ</sup><sup>1</sup> <sup>≤</sup> <sup>b</sup>ai, for tð Þ ;p; <sup>q</sup> <sup>∈</sup><sup>I</sup> � <sup>γ</sup>1ba<sup>j</sup> ; baj h i � <sup>γ</sup>2aj; <sup>a</sup><sup>i</sup> � �, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…k; (H10) H2ð Þ t; p tð Þ; q tð Þ γ<sup>2</sup> Ð 1 0 <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> ds ! <sup>≥</sup> <sup>a</sup>j, for tð Þ ;p; <sup>q</sup>∈<sup>I</sup> � <sup>γ</sup>1aj;a<sup>j</sup> � � � <sup>γ</sup>2aj; <sup>a</sup><sup>j</sup> � �, and

$$\mathcal{H}\_1(t, p(t), q(t))\delta\_2 \le \widehat{\mathbf{a}}\_{j\prime} for \ (t, p, q) \in \mathbf{I} \times \left[\boldsymbol{\gamma}\_1 \mathbf{a}\_{\boldsymbol{\gamma}}, \mathbf{a}\_{\boldsymbol{\gamma}}\right] \times \left[\boldsymbol{\gamma}\_2 \widehat{\mathbf{a}}\_{j\prime}, \widehat{\mathbf{a}}\_{\boldsymbol{\gamma}}\right], \ j = 1, 2...k.$$

there exist a solution ð Þ p; q ∈E � E of system (26) which satisfy

the spectral radius rð Þ M is defined by

the spectral radius rð Þ M is defined by

8 >< >: 8 >< >:

two operators such that

two operators such that

and if the matrix

and if the matrix

Hyers-Ulam stable.

Hyers-Ulam stable.

that

that

Further, the matrix will converge to zero if rð Þ M < 1:.

Further, the matrix will converge to zero if rð Þ M < 1:.

�

�

there exist a solution ð Þ p; q∈� E of system (26) which satisfy

for all pð Þ ; <sup>q</sup> , p<sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ<sup>∈</sup> <sup>E</sup> � <sup>E</sup>,

for all pð Þ ; <sup>q</sup> , p<sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ<sup>∈</sup> <sup>E</sup> � <sup>E</sup>,

For the stability results, the following should be hold:

For the stability results, the following should be hold:

M ¼

Proof. In view of Theorem 4.3, we have

Proof. In view of Theorem 4.3, we have

Ð 1

Ð 1

M ¼

is converging to zero. Then, the solutions of (1) are Hyers-Ulam stable.

<sup>∥</sup>p<sup>∗</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>C</sup>1r<sup>1</sup> <sup>þ</sup> <sup>C</sup>2r2, <sup>∥</sup>q<sup>∗</sup> � <sup>q</sup>∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>C</sup>3r<sup>1</sup> <sup>þ</sup> <sup>C</sup>4r2:

<sup>∥</sup>p<sup>∗</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>C</sup>1r<sup>1</sup> <sup>þ</sup> <sup>C</sup>2r2, <sup>∥</sup>q<sup>∗</sup> � <sup>q</sup>∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>C</sup>3r<sup>1</sup> <sup>þ</sup> <sup>C</sup>4r2:

Definition 4.2. If <sup>λ</sup>i, for i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, n be the (real or complex) eigenvalues of a matrix <sup>M</sup><sup>∈</sup> <sup>C</sup><sup>n</sup>�n, then

rð Þ¼ M max jλ<sup>i</sup> f g j; for i ¼ 1; 2;⋯; n :

Theorem 4.3. ([31, Theorem 4]) Consider a Banach space � E withS1,S<sup>2</sup> : E� E ! E � E be the

<sup>∥</sup>S1ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup><sup>1</sup> <sup>p</sup><sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>C</sup>1∥<sup>p</sup> � <sup>p</sup>∗∥<sup>E</sup>�<sup>E</sup> <sup>þ</sup> <sup>C</sup>2∥<sup>q</sup> � <sup>q</sup>∗∥<sup>E</sup>�<sup>E</sup>, <sup>∥</sup>S2ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup><sup>2</sup> <sup>p</sup><sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>C</sup>3∥<sup>p</sup> � <sup>p</sup>∗∥<sup>E</sup>�<sup>E</sup> <sup>þ</sup> <sup>C</sup>4∥<sup>q</sup> � <sup>q</sup>∗∥<sup>E</sup>�<sup>E</sup>,

Definition 4.2. If <sup>λ</sup>i, for i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, n be the (real or complex) eigenvalues of a matrix <sup>M</sup><sup>∈</sup> <sup>C</sup><sup>n</sup>�n, then

rð Þ¼ M max jλ<sup>i</sup> f g j; for i ¼ 1; 2; ⋯; n :

Theorem 4.3. ([31, Theorem 4]) Consider a Banach space E � E with S1, S<sup>2</sup> : E � E ! E � E be the

<sup>∥</sup>S1ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup><sup>1</sup> <sup>p</sup><sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>C</sup>1∥<sup>p</sup> � <sup>p</sup>∗∥<sup>E</sup>�<sup>E</sup> <sup>þ</sup> <sup>C</sup>2∥<sup>q</sup> � <sup>q</sup>∗∥<sup>E</sup>�<sup>E</sup>, <sup>∥</sup>S2ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup><sup>2</sup> <sup>p</sup><sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>C</sup>3∥<sup>p</sup> � <sup>p</sup>∗∥<sup>E</sup>�<sup>E</sup> <sup>þ</sup> <sup>C</sup>4∥<sup>q</sup> � <sup>q</sup>∗∥<sup>E</sup>�<sup>E</sup>,

> <sup>M</sup> <sup>¼</sup> <sup>C</sup><sup>1</sup> <sup>C</sup><sup>2</sup> C<sup>3</sup> C<sup>4</sup> � �

<sup>M</sup> <sup>¼</sup> <sup>C</sup><sup>1</sup> <sup>C</sup><sup>2</sup> C<sup>3</sup> C<sup>4</sup> � �

converges to zero ([31, Theorem 1]), then the fixed points corresponding to operatorial system (26) are

(H13) Under the continuity of Hi, i ¼ 1, 2, there exist ai, bi ∈cð Þ 0; 1, i ¼ 1, 2 and ð Þ p; q ,ð Þ p; q such

∣Hið Þ� t; p; q Hið Þ t;p;q ∣ ≤aið Þt∣p � p∣ þ bið Þt ∣q � q∣, i ¼ 1, 2:

In this section, we study Hyers-Ulam stability for the solutions of our proposed system.

converges to zero ([31, Theorem 1]), then the fixed points corresponding to operatorial system (26) are

(H13) Under the continuity of Hi, i ¼ 1, 2, there exist ai, bi ∈cð Þ 0; 1 , i ¼ 1, 2 and ð Þ p; q ,ð Þ p; q such

∣Hið Þ� t; p; q Hið Þ t; p; q ∣ ≤ aið Þt ∣p � p∣ þ bið Þt ∣q � q∣, i ¼ 1, 2:

In this section, we study Hyers-Ulam stability for the solutions of our proposed system.

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>b</sup>2ð Þ<sup>s</sup> ds " #

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>b</sup>2ð Þ<sup>s</sup> ds " #

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> G1ð Þ 1;s b1ð Þs ds

<sup>0</sup> G1ð Þ 1;s b1ð Þs ds

Thanks to Definition 4.1 and Theorem 4.3, the respective results are received.

Theorem 4.4. Suppose that the assumptions Hð Þ <sup>13</sup> along with condition that matrix

Ð 1

Ð 1

is converging to zero. Then, the solutions of (1) are Hyers-Ulam stable.

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

Thanks to Definition 4.1 and Theorem 4.3, the respective results are received.

Theorem 4.4. Suppose that the assumptions Hð Þ <sup>13</sup> along with condition that matrix

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

(29)

(29)

49

49

Existence Theory of Differential Equations of Arbitrary Order

http://dx.doi.org/10.5772/intechopen.75523

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

(30)

(30)

Then, system (1) of BVPs has at least k solutions pj ; qj � �, satisfying Then, system(1) of BVPs has at least k solutions pj ; qj � �, satisfying

$$\|\mathbf{a}\_{j} \le \|\left(p\_{j}, q\_{j}\right)\|\_{\mathbf{E}\times\mathbf{E}} \le \widehat{\mathbf{a}}\_{j} \quad j = 1, 2...k.$$

Further, if assumptions Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> hold such that there exist 2k positive numbers bj, bbj, j ¼ 1, 2…k, with Further, if assumptions Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> hold such that there exist 2k positive numbers bj, bbj, j ¼ 1, 2…k, with

$$\mathbf{b}\_1 < \widehat{\mathbf{b}}\_1 < \mathbf{b}\_2 < \widehat{\mathbf{b}}\_2 \dots < \mathbf{b}\_k < \widehat{\mathbf{b}}\_{k'}$$

together with following hypothesis hold: together with following hypothesis hold:

(H11) H1ð Þ t; p; q and H2ð Þ t; p; q are nondecreasing on 0; bb<sup>k</sup> h i for all t <sup>∈</sup>I; (H11) H1ð Þ t; p; q and H2ð Þ t; p; q are nondecreasing on 0; bb<sup>k</sup> h i for all t <sup>∈</sup>I;

$$\mathcal{H}(H\_{11})\mathcal{H}\_{1}(t,p(t),q(t))\left(\mathcal{V}\_{1}\int\_{0}^{1-\theta}\mathbf{G}\_{1}(1,\mathsf{s})d\mathsf{s}\right)\geq\mathbf{b}\_{j},\mathcal{H}\_{1}(t,p(t),q(t))\delta\_{1}\leq\widehat{\mathbf{b}}\_{j},\ j=1,2\ldots k,\tag{26}$$

$$\mathcal{H}\_{2}(t,u(t),v(t))\left(\mathcal{V}\_{2}\int\_{0}^{1-\theta}\mathbf{G}\_{2}(1,\mathsf{s})d\mathsf{s}\right)\geq\mathbf{b}\_{j},\mathcal{H}\_{2}(t,p(t),q(t))\delta\_{2}\leq\widehat{\mathbf{b}}\_{j},\ j=1,2\ldots k.$$

Then, system (1) of BVPs has at least k solutions pj ; qj � �, satisfying Then, system(1) of BVPs has at least k solutions pj ; qj � �, satisfying

$$\mathbf{b}\_{j} \le \| \left( p\_{j}, q\_{j} \right) \|\_{\mathbf{E} \times \mathbf{E}} \le \widehat{\mathbf{b}}\_{j}, \ j = 1, 2...k.$$

#### 4. Hyers-Ulam stability 4. Hyers-Ulam stability

Definition 4.1. ([31, Definition 2]) Consider a Banach space E � E such that S1, S<sup>2</sup> : E � E ! E � E be the two operators. Then, the operator system provided by Definition 4.1. ([31, Definition 2]) Consider a Banach space � E such that S1, S<sup>2</sup> : E� E ! � E be the two operators. Then, the operator system provided by

$$\begin{cases} p(t) = \mathcal{S}\_1(p, q)(t), \\ q(t) = \mathcal{S}\_2(p, q)(t) \end{cases} \tag{27}$$

(27)

(28)

is called Hyers-Ulam stability if we can find C <sup>i</sup>ð Þ i ¼ 1; 2; 3; 4 > 0, such that for each rið Þ i ¼ 1; 2 > 0 and for each solution p<sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ <sup>∈</sup><sup>E</sup> � <sup>E</sup> of the inequalities given by is called Hyers-Ulam stability if we can find C <sup>i</sup>ð Þ i ¼ 1;2; 3; 4 > 0, such that for each rið Þ i ¼ 1; 2 > 0 and for each solution p<sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ <sup>∈</sup>E� <sup>E</sup> of the inequalities given by

$$\begin{cases} \|p^\* - \mathcal{H}\_1(p^\*, q^\*)\|\_{\mathbf{E}\times\mathbf{E}} \le \rho\_{1\prime} \\ \|q^\* - \mathcal{H}\_2(p^\*, q^\*)\|\_{\mathbf{E}\times\mathbf{E}} \le \rho\_{2\prime} \end{cases} \tag{28}$$

there exist a solution ð Þ p; q ∈E � E of system (26) which satisfy there exist a solution ð Þ p; q ∈E � E of system (26) which satisfy

�

$$\begin{cases} \|p^\*-\overline{p}\|\_{\mathbb{E}\times\mathbb{E}} \le \mathcal{C}\_1\rho\_1 + \mathcal{C}\_2\rho\_2, \\\ \|q^\*-\overline{q}\|\_{\mathbb{E}\times\mathbb{E}} \le \mathcal{C}\_3\rho\_1 + \mathcal{C}\_4\rho\_2. \end{cases} \tag{29}$$

Definition 4.2. If <sup>λ</sup>i, for i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, n be the (real or complex) eigenvalues of a matrix <sup>M</sup><sup>∈</sup> <sup>C</sup><sup>n</sup>�n, then the spectral radius rð Þ M is defined by Definition 4.2. If <sup>λ</sup>i, for i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, n be the (real or complex) eigenvalues of a matrix <sup>M</sup><sup>∈</sup> <sup>C</sup><sup>n</sup>�n, then the spectral radius rð Þ M is defined by

$$\rho(\mathbf{M}) = \max\{|\lambda\_i|, \text{ for } i = 1, 2, \dots, n\}.$$

Further, the matrix will converge to zero if rð Þ M < 1:. Further, the matrix will converge to zero if rð Þ M < 1:.

Theorem 4.3. ([31, Theorem 4]) Consider a Banach space E � E with S1, S<sup>2</sup> : E � E ! E � E be the two operators such that Theorem 4.3. ([31, Theorem 4]) Consider a Banach space � E withS1,S<sup>2</sup> : � E ! � E be the two operators such that

$$\begin{cases} \|\mathcal{S}\_1(p,q) - \mathcal{S}\_1(p^\*,q^\*)\|\_{\mathbf{E}\times\mathbf{E}} \le \mathcal{C}\_1 \|p - p^\*\|\_{\mathbf{E}\times\mathbf{E}} + \mathcal{C}\_2 \|q - q^\*\|\_{\mathbf{E}\times\mathbf{E}}\\ \|\mathcal{S}\_2(p,q) - \mathcal{S}\_2(p^\*,q^\*)\|\_{\mathbf{E}\times\mathbf{E}} \le \mathcal{C}\_3 \|p - p^\*\|\_{\mathbf{E}\times\mathbf{E}} + \mathcal{C}\_4 \|q - q^\*\|\_{\mathbf{E}\times\mathbf{E}}\\ \text{for all } (p,q), (p^\*,q^\*) \in \mathbf{E} \times\mathbf{E}, \end{cases} \tag{30}$$

and if the matrix and if the matrix

that

8 >< >:

<sup>H</sup>1ð Þ <sup>t</sup>; p tð Þ; q tð Þ <sup>δ</sup><sup>2</sup> <sup>≤</sup> <sup>b</sup>aj, for tð Þ ; <sup>p</sup>; <sup>q</sup> <sup>∈</sup><sup>I</sup> � <sup>γ</sup>1aj; <sup>a</sup><sup>j</sup>

<sup>H</sup>1ð Þ <sup>t</sup>; p tð Þ; q tð Þδ<sup>2</sup> <sup>≤</sup> <sup>b</sup>aj, for tð Þ ;p; <sup>q</sup>∈<sup>I</sup> � <sup>γ</sup>1aj; <sup>a</sup><sup>j</sup>

a<sup>j</sup> ≤ ∥ pj

; qj � �

; qj � �

a<sup>j</sup> ≤ ∥pj

Then, system (1) of BVPs has at least k solutions pj

Then, system(1) of BVPs has at least k solutions pj

48 Differential Equations - Theory and Current Research

48 Differential Equations - Theory and Current Research

together with following hypothesis hold:

together with following hypothesis hold:

ðH11ÞH1ð Þ t; p tð Þ; q tð Þ γ<sup>1</sup>

ðH11ÞH1ð Þ t; p tð Þ; q tð Þ γ<sup>1</sup>

H2ð Þ t; u tð Þ; v tð Þ γ<sup>2</sup>

4. Hyers-Ulam stability

4. Hyers-Ulam stability

(H11) H1ð Þ t; p; q and H2ð Þ t; p; q are nondecreasing on 0; bb<sup>k</sup>

0 @

0 @

Then, system (1) of BVPs has at least k solutions pj

Then, system(1) of BVPs has at least k solutions pj

H2ð Þ t; u tð Þ; v tð Þ γ<sup>2</sup>

be the two operators. Then, the operator system provided by

be the two operators. Then, the operator system provided by

and for each solution p<sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ <sup>∈</sup><sup>E</sup> � <sup>E</sup> of the inequalities given by

�

and for each solution p<sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ <sup>∈</sup>� <sup>E</sup> of the inequalities given by

1�ð θ

0 @

(H11) H1ð Þ t;p; q and H2ð Þ t;p; q are nondecreasing on0; bb<sup>k</sup>

θ

1�ð θ

θ

b<sup>j</sup> ≤ ∥ pj

1�ð θ

0 @

θ

G1ð Þ 1;s ds

G2ð Þ 1;s ds

1�ð θ

θ

; qj � �

; qj � �

b<sup>j</sup> ≤ ∥pj

�

�

�

with

with

� � � <sup>γ</sup>2ba<sup>j</sup>

� � � <sup>γ</sup>2ba<sup>j</sup>

, satisfying

, satisfying

<sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>b</sup>aj, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…k:

; qj � �

Further, if assumptions Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> hold such that there exist 2k positive numbers bj, bbj, j ¼ 1, 2…k,

b<sup>1</sup> < bb<sup>1</sup> < b<sup>2</sup> < bb2… < b<sup>k</sup> < bbk,

1

; qj � �

1

Definition 4.1. ([31, Definition 2]) Consider a Banach space E � E such that S1, S<sup>2</sup> : E � E ! E � E

Definition 4.1. ([31, Definition 2]) Consider a Banach space E� E such that S1,S<sup>2</sup> : � E ! E � E

p tðÞ¼ S1ð Þ p; q ð Þt , q tðÞ¼ S2ð Þ p; q ð Þt

is called Hyers-Ulam stability if we can find C <sup>i</sup>ð Þ i ¼ 1; 2; 3; 4 > 0, such that for each rið Þ i ¼ 1; 2 > 0

<sup>∥</sup>p<sup>∗</sup> � <sup>H</sup><sup>1</sup> <sup>p</sup><sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>r</sup>1, <sup>∥</sup>q<sup>∗</sup> � <sup>H</sup><sup>2</sup> <sup>p</sup><sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>r</sup>2,

p tðÞ¼ S1ð Þ p; q ð Þt , q tðÞ¼ S2ð Þ p; q ð Þt

is called Hyers-Ulam stability if we can find C <sup>i</sup>ð Þ i ¼ 1; 2; 3; 4 > 0, such that for each rið Þ i ¼ 1; 2 > 0

<sup>∥</sup>p<sup>∗</sup> � <sup>H</sup><sup>1</sup> <sup>p</sup><sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>r</sup>1, <sup>∥</sup>q<sup>∗</sup> � <sup>H</sup><sup>2</sup> <sup>p</sup><sup>∗</sup>; <sup>q</sup><sup>∗</sup> ð Þ∥<sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>r</sup>2,

<sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>b</sup>bj, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…k:

; qj � �

, satisfying

<sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>b</sup>bj, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…k:

1

G2ð Þ 1;s ds

G1ð Þ 1;s ds

<sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>b</sup>aj, j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>…k:

Further, if assumptions Cð Þ� <sup>1</sup> ð Þ C<sup>3</sup> hold such that there exist 2k positive numbers bj, bbj, j ¼ 1, 2…k,

b<sup>1</sup> < bb<sup>1</sup> < b<sup>2</sup> < bb2… < b<sup>k</sup> < bbk,

1

; qj � �

h i

h i

for all t ∈I;

for all t ∈I;

A ≥ bj, H2ð Þ t; p tð Þ; q tð Þ δ<sup>2</sup> ≤ bbj, j ¼ 1, 2…k:

, satisfying

A ≥ bj, H2ð Þ t; p tð Þ; q tð Þ δ<sup>2</sup> ≤ bbj, j ¼ 1, 2…k:

A ≥ bj, H1ð Þ t; p tð Þ; q tð Þ δ<sup>1</sup> ≤ bbj, j ¼ 1, 2…k, (26)

A ≥ bj, H1ð Þ t; p tð Þ; q tð Þδ<sup>1</sup> ≤ bbj, j ¼ 1, 2…k, (26)

(27)

(27)

(28)

(28)

; baj h i

; baj h i

, j ¼ 1, 2…k:

, j ¼ 1, 2…k:

$$\mathbf{M} = \begin{bmatrix} \mathcal{C}\_1 & \mathcal{C}\_2 \\ \mathcal{C}\_3 & \mathcal{C}\_4 \end{bmatrix}$$

converges to zero ([31, Theorem 1]), then the fixed points corresponding to operatorial system (26) are Hyers-Ulam stable. converges to zero ([31, Theorem 1]), then the fixed points corresponding to operatorial system (26) are Hyers-Ulam stable.

For the stability results, the following should be hold: For the stability results, the following should be hold:

(H13) Under the continuity of Hi, i ¼ 1, 2, there exist ai, bi ∈cð Þ 0; 1 , i ¼ 1, 2 and ð Þ p; q ,ð Þ p; q such (H13) Under the continuity of Hi, i ¼ 1, 2, there exist ai, bi ∈cð Þ 0; 1, i ¼ 1, 2 and ð Þ p; q ,ð Þ p; q such that

$$|\mathcal{H}\_i(t, p, q) - \mathcal{H}\_i(t, \overline{p}, \overline{q})| \le a\_i(t)|p - \overline{p}| + b\_i(t)|q - \overline{q}|, \ i = 1, 2.1$$

In this section, we study Hyers-Ulam stability for the solutions of our proposed system. Thanks to Definition 4.1 and Theorem 4.3, the respective results are received. In this section, we study Hyers-Ulam stability for the solutions of our proposed system. Thanks to Definition 4.1 and Theorem 4.3, the respective results are received.

Theorem 4.4. Suppose that the assumptions Hð Þ <sup>13</sup> along with condition that matrix Theorem 4.4. Suppose that the assumptions Hð Þ <sup>13</sup> along with condition that matrix

$$\mathbf{M} = \begin{bmatrix} \int\_0^1 \mathbf{G}\_1(1, \mathbf{s}) a\_1(\mathbf{s}) d\mathbf{s} & \int\_0^1 \mathbf{G}\_1(1, \mathbf{s}) b\_1(\mathbf{s}) d\mathbf{s} \\\int\_0^1 \mathbf{G}\_2(1, \mathbf{s}) a\_2(\mathbf{s}) d\mathbf{s} & \int\_0^1 \mathbf{G}\_2(1, \mathbf{s}) b\_2(\mathbf{s}) d\mathbf{s} \end{bmatrix}$$

is converging to zero. Then, the solutions of (1) are Hyers-Ulam stable. is converging to zero. Then, the solutions of (1) are Hyers-Ulam stable.

Proof. In view of Theorem 4.3, we have Proof. In view of Theorem 4.3, we have

$$\begin{split} & \| \| \mathcal{S}\_{1}(p,q) - \mathcal{S}\_{1}(\overline{p},\overline{q}) \|\_{\mathbb{E} \times \mathbb{E}} \leq \int\_{0}^{1} \mathsf{G}\_{1}(1,s) a\_{1}(s) \| p - \overline{p} \|\_{\mathbb{E} \times \mathbb{E}} ds + \int\_{0}^{1} \mathsf{G}\_{1}(1,s) b\_{1}(s) \| q - \overline{q} \|\_{\mathbb{E} \times \mathbb{E}} ds \\ & \| \| \mathcal{S}\_{2}(p,q) - \mathcal{S}\_{2}(\overline{p},\overline{q}) \|\_{\mathbb{E} \times \mathbb{E}} \leq \int\_{0}^{1} \mathsf{G}\_{2}(1,s) a\_{2}(s) \| p - \overline{p} \|\_{\mathbb{E} \times \mathbb{E}} ds + \int\_{0}^{1} \mathsf{G}\_{2}(1,s) b\_{2}(s) \| q - \overline{q} \|\_{\mathbb{E} \times \mathbb{E}} ds. \end{split}$$

From which we get From which we get

$$\begin{split} \|\mathscr{S}\_{1}(p,q) - \mathscr{S}\_{1}(\overline{p},\overline{q})\|\_{\mathbb{E}\times\mathbb{E}} &\leq \Big[\int\_{0}^{1} \mathbf{G}\_{1}(1,s)a\_{1}(s)ds\Big] \|p-\overline{p}\|\_{\mathbb{E}\times\mathbb{E}} + \Big[\int\_{0}^{1} \mathbf{G}\_{1}(1,s)b\_{1}(s)ds\Big] \|q-\overline{q}\|\_{\mathbb{E}\times\mathbb{E}}\\ \|\mathscr{S}\_{2}(p,q) - \mathscr{S}\_{2}(\overline{p},\overline{q})\|\_{\mathbb{E}\times\mathbb{E}} &\leq \Big[\int\_{0}^{1} \mathbf{G}\_{2}(1,s)a\_{2}(s)ds\Big] \|p-\overline{p}\|\_{\mathbb{E}\times\mathbb{E}} + \Big[\int\_{0}^{1} \mathbf{G}\_{2}(1,s)b\_{2}(s)ds\Big] \|q-\overline{q}\|\_{\mathbb{E}\times\mathbb{E}}. \end{split} \tag{31}$$

Hence, we get Hence, we get

$$\|\mathcal{S}(p,q) - \mathcal{S}(\overline{p}, \overline{q})\|\_{\mathbf{E}\times\mathbf{E}} \leq \mathbf{M} \|(p,q) - (\overline{p}, \overline{q})\|\_{\mathbf{E}\times\mathbf{E}'} \tag{32}$$

It is obvious that H1ð Þ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at pð Þ¼ ; q ð Þ 0; 0 , and H1ð Þ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at pð Þ¼ ; q ð Þ 1; 1 . Also, an easy computation yields that H1, H<sup>2</sup> are nondecreasing for

Thus, all the assumption of Theorem 3.4 is fulfilled, so the coupled system (33) has a unique positive

<sup>20</sup> cos∣p tð Þ<sup>∣</sup> <sup>þ</sup>

<sup>60</sup> sin∣p tð Þ<sup>∣</sup> <sup>þ</sup>

<sup>p</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>q</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, at t <sup>¼</sup> <sup>0</sup>,

� �, qð Þ¼ <sup>1</sup> <sup>q</sup>

� �, qð Þ¼ <sup>1</sup> <sup>q</sup>

1 2

40 þ t

> t 40 þ t

t 40 þ t

t 2 50 þ t 2

t 2 50 þ t 2

<sup>G</sup>jð Þ¼ <sup>1</sup>;<sup>s</sup> <sup>6</sup>:<sup>65710</sup> ð Þ <sup>1</sup> � <sup>s</sup> <sup>5</sup>

<sup>G</sup>jð Þ¼ <sup>1</sup>;<sup>s</sup> <sup>6</sup>:<sup>65710</sup> ð Þ <sup>1</sup> � <sup>s</sup> <sup>5</sup>

<sup>50</sup> , <sup>ψ</sup>1ðÞ¼<sup>t</sup> <sup>t</sup>

<sup>50</sup> , <sup>ψ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup>

<sup>2</sup> ,λ<sup>1</sup> ¼ λ<sup>2</sup> ¼ 0:17677: Thus, by computation, we have

∣H2ð Þ t;p; q ∣ ≤

<sup>H</sup>1ð Þ <sup>t</sup>; <sup>p</sup>; <sup>q</sup> , <sup>H</sup>2ð Þ <sup>t</sup>; <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>≥</sup> <sup>τ</sup><sup>3</sup>

Thus, all the assumption of Theorem 3.4 is fulfilled, so the coupled system (33) has a unique positive

<sup>H</sup>1ð Þ <sup>t</sup>;p; <sup>q</sup> , <sup>H</sup>2ð Þ <sup>t</sup>; <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>≥</sup> <sup>τ</sup><sup>3</sup>

It is obvious that H1ð Þ t;p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at pð Þ¼ ; q ð Þ 0; 0 , and H1ð Þ t; p; q 6¼ 0, H2ð Þ t;p; q 6¼ 0, at pð Þ¼ ; q ð Þ 1; 1 . Also, an easy computation yields thatH1, H<sup>2</sup> are nondecreasing for

> t 2

t 2

t

t

1 2 � �:

1 2 � �:

<sup>20</sup> cos∣p tð Þ<sup>∣</sup> <sup>þ</sup>

<sup>60</sup> sin∣p tð Þ<sup>∣</sup> <sup>þ</sup>

<sup>p</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>q</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, at t <sup>¼</sup> <sup>0</sup>,

<sup>20</sup> cos∣p tð Þ<sup>∣</sup> <sup>þ</sup>

<sup>60</sup> sin∣p tð Þ<sup>∣</sup> <sup>þ</sup>

<sup>20</sup> , <sup>ψ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup>

2

Γ <sup>7</sup> 2

> ð1 0

Γ <sup>7</sup> 2

Further, we see that max 0f :007626; 0:00185g ¼ 0:007626. So, all the conditions of Theorem 3.3 are

Further, we see that max 0f :007626; 0:00185g ¼ 0:007626. So, all the conditions of Theorem 3.3 are

E ¼ f g ð Þ p; q ∈ C : ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < 0:007626 :

E ¼ f g ð Þ p; q ∈ C : ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < 0:007626 :

<sup>20</sup> sin∣q tð Þ∣, t<sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> ,

<sup>20</sup> sinq tð Þ∣, t<sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> ,

<sup>60</sup> cosq tð Þ∣, t<sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> ,

<sup>60</sup> cos∣q tð Þ∣, t<sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> ,

t 2 <sup>20</sup> sin∣q tð Þ<sup>∣</sup>

<sup>20</sup> cos∣p tð Þ<sup>∣</sup> <sup>þ</sup>

<sup>60</sup> sin∣p tð Þ<sup>∣</sup> <sup>þ</sup>

<sup>20</sup> , <sup>ψ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup>

t

2

� � , for j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>:

ð1 0

2

<sup>60</sup> cos∣q tð Þ∣:

2

� � , for j <sup>¼</sup> <sup>1</sup>, <sup>2</sup>:

t

t 2 <sup>20</sup> sin∣q tð Þ<sup>∣</sup>

<sup>60</sup> , <sup>σ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup>

<sup>60</sup> cos∣q tð Þ∣:

<sup>60</sup> , <sup>σ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup>

2

G2ð Þ 1;s φ2ð Þs ds ¼ 0:000924 < ∞:

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>2ð Þþ <sup>s</sup> <sup>σ</sup>2ð Þ<sup>s</sup> � �ds <sup>¼</sup> <sup>0</sup>:<sup>00289</sup> <sup>&</sup>lt; <sup>1</sup>:

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>2ð Þþ <sup>s</sup> <sup>σ</sup>2ð Þ<sup>s</sup> � �ds <sup>¼</sup> <sup>0</sup>:<sup>00289</sup> <sup>&</sup>lt; <sup>1</sup>:

G2ð Þ 1;s φ2ð Þs ds ¼ 0:000924 < ∞:

<sup>20</sup> , <sup>σ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup>

<sup>20</sup> , <sup>σ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup>

2

60. Also,

60. Also,

H2ð Þ t; p; q :

H2ð Þ t; p; q :

Existence Theory of Differential Equations of Arbitrary Order

http://dx.doi.org/10.5772/intechopen.75523

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

(35)

(35)

51

51

each t∈ð Þ 0; 1 : Moreover, for τ, t∈ð Þ 0; 1 , and p, q ≥ 0, we see that max 3f g ; 2 ¼ 3,

each t∈ð Þ 0; 1 : Moreover, for τ, t∈ð Þ 0; 1 , and p, q≥ 0, we see that max 3f g ;2 ¼ 3,

<sup>H</sup>1ð Þ <sup>t</sup>; <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>≥</sup> <sup>τ</sup><sup>3</sup>

Example 5.3. Consider the following system of BVPs:

8

>>>>>>>>><

D7

8

>>>>>>>>><

>>>>>>>>>:

<sup>40</sup> , <sup>φ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup>

D7 <sup>2</sup>q tðÞ¼ <sup>t</sup> 2 50 þ t 2

pð Þ¼ 1 p

>>>>>>>>>:

3 <sup>4</sup>; t 4 3 � �:

<sup>H</sup>1ð Þ <sup>t</sup>; <sup>τ</sup>p; <sup>τ</sup><sup>q</sup> <sup>≥</sup> <sup>τ</sup><sup>3</sup>

3 <sup>4</sup>; t 4 3 � �:

<sup>2</sup>p tðÞ¼ <sup>t</sup>

D7

D7 <sup>2</sup>q tðÞ¼ <sup>t</sup> 2 50 þ t 2

Example 5.3. Consider the following system of BVPs:

40 þ t

<sup>2</sup>p tðÞ¼ <sup>t</sup>

1 2

pð Þ¼ 1 p

∣H1ð Þ t; p; q ∣ ≤

∣H1ð Þ t;p; q ∣ ≤

∣H2ð Þ t; p; q ∣ ≤

<sup>2</sup> , λ<sup>1</sup> ¼ λ<sup>2</sup> ¼ 0:17677: Thus, by computation, we have

2

G1ð Þ 1;s φ1ð Þs ds ¼ 0:003577 < ∞, Δ<sup>2</sup> ¼

G1ð Þ 1;s φ1ð Þs ds ¼ 0:003577 <∞, Δ<sup>2</sup> ¼

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>1ð Þþ <sup>s</sup> <sup>σ</sup>1ð Þ<sup>s</sup> � �ds <sup>¼</sup> <sup>0</sup>:<sup>03092853</sup> <sup>&</sup>lt; <sup>1</sup>, <sup>Λ</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

satisfied. So, the BVP (34) has at least one solution and the solution lies in

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>1ð Þþ <sup>s</sup> <sup>σ</sup>1ð Þ<sup>s</sup> � �ds <sup>¼</sup> <sup>0</sup>:<sup>03092853</sup> <sup>&</sup>lt; <sup>1</sup>, <sup>Λ</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

satisfied. So, the BVP (34) has at least one solution and the solution lies in

2

<sup>40</sup> , <sup>φ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup>

solution in B<sup>f</sup> where f tðÞ¼ t

solution in B<sup>f</sup> where f tðÞ¼ t

From system (33), we see that

From system (33), we see that

and

and

(31)

(33)

(34)

where <sup>φ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup>

<sup>η</sup> <sup>¼</sup><sup>ξ</sup> <sup>¼</sup> <sup>1</sup>

Upon computation, we get

Δ<sup>1</sup> ¼ ð1 0

Upon computation, we get

where <sup>φ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup>

Similarly, we can also compute.

Similarly, we can also compute.

Δ<sup>1</sup> ¼ ð1 0

<sup>Λ</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

<sup>Λ</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup>

<sup>η</sup> <sup>¼</sup> <sup>ξ</sup> <sup>¼</sup> <sup>1</sup>

where M ¼ Ð 1 <sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup> <sup>0</sup> G1ð Þ 1;s b1ð Þs ds Ð 1 <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup> <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>b</sup>2ð Þ<sup>s</sup> ds " #. Hence, we received the required results. where M ¼ Ð 1 <sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup> <sup>0</sup> G1ð Þ 1;s b1ð Þs ds Ð 1 <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup> <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>b</sup>2ð Þ<sup>s</sup> ds " #. Hence, we received the required results.

#### 5. Illustrative examples 5. Illustrative examples

Example 5.1. Consider the given system of BVPs Example 5.1. Consider the given system of BVPs

$$\begin{cases} \mathcal{B}^{\frac{2}{3}}p(t) + \left(1 - t^2\right) + [p(t)q(t)]^{\frac{1}{3}} = 0, & \mathcal{B}^{\frac{12}{3}}q(t) + 1 + t + [p(t)q(t)]^{\frac{1}{3}} = 0, \ t \in (0, 1), \\ p(t) = p'(t) = p''(t) = q(t) = q'(t) = q''(t) = 0, & \text{at } t = 0, \\ p(1) = p\left(\frac{1}{4}\right), \quad q(1) = q\left(\frac{1}{3}\right). \end{cases} \tag{33}$$

Clearly, H1ð Þ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at pð Þ¼ ; q ð Þ 0; 0 , and H1ð Þ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at ð Þ¼ p; q ð Þ 1; 1 . Simple computation yields that H1, H<sup>2</sup> are nondecreasing for every t ∈ð Þ 0; 1 : Also, for <sup>τ</sup>, t<sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> , and p, q <sup>≥</sup> 0, one has max <sup>1</sup> 4 ; 1 3 � � <sup>¼</sup> <sup>1</sup> 3 , Clearly, H1ð Þ t; p; q 6¼ 0, H2ð Þ t;p; q 6¼ 0, at pð Þ¼ ; q ð Þ 0; 0 , and H1ð Þ t;p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at ð Þ¼ p; q ð Þ 1; 1 . Simple computation yields that H1, H<sup>2</sup> are nondecreasing for every t ∈ð Þ 0; 1 : Also, for <sup>τ</sup>, t<sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> , and p, q <sup>≥</sup> 0, one has max <sup>1</sup> 4 ; 1 3 � � <sup>¼</sup> <sup>1</sup> 3 ,

$$
\mathcal{H}\_1(t, \tau p, \tau q) \ge \tau^{\frac{1}{2}} \mathcal{H}\_1(t, p, q), \qquad \mathcal{H}\_2(t, \tau p, \tau q) \ge \tau^{\frac{1}{2}} \mathcal{H}\_2(t, p, q).
$$

Thus, all the conditions of Theorem 3.4 are fulfilled, so the system (32) of BVPs has unique positive solution in B<sup>f</sup> where f tðÞ¼ t 5 <sup>2</sup>; t 9 2 � �: Thus, all the conditions of Theorem 3.4 are fulfilled, so the system (32) of BVPs has unique positive solution in B<sup>f</sup> where f tðÞ¼ t 5 <sup>2</sup>; t 9 2 � �:

Example 5.2. Consider the following system of BVPs: Example 5.2. Consider the following system of BVPs:

$$\begin{cases} \mathcal{B}^{\frac{\theta}{2}}p(t) + (1+t)^2 + [p(t) + q(t)]^3 = 0, \quad \mathcal{B}^{\frac{\theta}{2}}q(t) + 1 + t + [p(t) + q(t)]^2 = 0, \quad t \in (0, 1), \\\ p^{(j)}(t) = q^{(j)}(t) = 0, \; j = 0, 1, 2, 3, \;\text{at } \; t = 0, \\\ p(1) = p\left(\frac{1}{2}\right), \; q(1) = q\left(\frac{1}{2}\right). \end{cases} \tag{34}$$

It is obvious that H1ð Þ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at pð Þ¼ ; q ð Þ 0; 0 , and H1ð Þ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at pð Þ¼ ; q ð Þ 1; 1 . Also, an easy computation yields that H1, H<sup>2</sup> are nondecreasing for each t∈ð Þ 0; 1 : Moreover, for τ, t∈ð Þ 0; 1 , and p, q ≥ 0, we see that max 3f g ; 2 ¼ 3, It is obvious that H1ð Þ t; p; q 6¼ 0, H2ð Þ t;p; q 6¼ 0, at pð Þ¼ ; q ð Þ 0; 0 , and H1ð Þ t; p; q 6¼ 0, H2ð Þ t;p; q 6¼ 0, at pð Þ¼; q ð Þ 1; 1 . Also, an easy computation yields that H1, H<sup>2</sup> are nondecreasing for each t∈ð Þ 0; 1 : Moreover, for τ, t∈ð Þ 0; 1 , and p, q ≥ 0, we see that max 3f g ;2 ¼ 3,

$$\mathcal{H}\_1(t, \tau p, \tau q) \ge \tau^3 \mathcal{H}\_1(t, p, q), \qquad \mathcal{H}\_2(t, \tau p, \tau q) \ge \tau^3 \mathcal{H}\_2(t, p, q).$$

Thus, all the assumption of Theorem 3.4 is fulfilled, so the coupled system (33) has a unique positive solution in B<sup>f</sup> where f tðÞ¼ t 3 <sup>4</sup>; t 4 3 � �: Thus, all the assumption of Theorem 3.4 is fulfilled, so the coupled system (33) has a unique positive solution in B<sup>f</sup> where f tðÞ¼ t 3 <sup>4</sup>; t 4 3 � �:

Example 5.3. Consider the following system of BVPs: Example 5.3. Consider the following system of BVPs:

8

>>>>>>>>><

>>>>>>>>>:

$$\begin{cases} \mathcal{B}^{\frac{\gamma}{2}}p(t) = \frac{t}{40} + \frac{t}{20}\cos|p(t)| + \frac{t^2}{20}\sin|q(t)| \quad t \in (0,1), \\\mathcal{B}^{\frac{\gamma}{2}}q(t) = \frac{t^2}{50} + \frac{t^2}{60}\sin|p(t)| + \frac{t}{60}\cos|q(t)| \quad t \in (0,1), \\\ p^{(j)}(t) = q^{(j)}(t) = 0, \; j = 0, 1, 2, \quad \text{at } t = 0, \\\ p(1) = p\left(\frac{1}{2}\right), \; q(1) = q\left(\frac{1}{2}\right). \end{cases} \tag{35}$$

From system (33), we see that From system (33), we see that

$$|\mathcal{H}\_1(t, p, q)| \le \frac{t}{40} + \frac{t}{20} \cos|p(t)| + \frac{t^2}{20} \sin|q(t)|$$

and

and

<sup>∥</sup>S1ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup>1ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>Ð</sup> <sup>1</sup>

<sup>∥</sup>S1ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup>1ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>Ð</sup> <sup>1</sup>

50 Differential Equations - Theory and Current Research

<sup>∥</sup>S2ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup>2ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>Ð</sup> <sup>1</sup>

<sup>∥</sup>S1ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup>1ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>Ð</sup> <sup>1</sup>

<sup>∥</sup>S2ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup>2ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>Ð</sup> <sup>1</sup>

50 Differential Equations - Theory and Current Research

<sup>∥</sup>S2ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup>2ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>Ð</sup> <sup>1</sup>

<sup>∥</sup>S1ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup>1ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>Ð</sup> <sup>1</sup>

<sup>∥</sup>S2ð Þ� <sup>p</sup>; <sup>q</sup> <sup>S</sup>2ð Þ <sup>p</sup>; <sup>q</sup> <sup>∥</sup><sup>E</sup>�<sup>E</sup> <sup>≤</sup> <sup>Ð</sup> <sup>1</sup>

Ð 1

Ð 1

Ð 1

5. Illustrative examples

Ð 1

5. Illustrative examples

<sup>2</sup>p tðÞþ 1 � t

p tðÞ¼ p 0 ðÞ¼ t p 00

pð Þ¼ 1 p

1 4 � �

<sup>2</sup>p tðÞþ 1 � t

<sup>τ</sup>, t<sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> , and p, q <sup>≥</sup> 0, one has max <sup>1</sup>

D7

8 >>>><

>>>>:

8 >>>><

>>>>:

D9

8 >>><

8 >>><

>>>:

pð Þ¼ 1 p

>>>:

D9

p tðÞ¼ p 0 ðÞ¼ t p 00

pð Þ¼ 1 p

D7

solution in B<sup>f</sup> where f tðÞ¼ t

solution in B<sup>f</sup> where f tðÞ¼ t

1 2 � �

pð Þ¼ 1 p

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

" #

" #

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

Example 5.1. Consider the given system of BVPs

1 4 � �

<sup>τ</sup>, t<sup>∈</sup> ð Þ <sup>0</sup>; <sup>1</sup> , and p, q<sup>≥</sup> 0, one has max <sup>1</sup>

<sup>2</sup> � � <sup>þ</sup> ½ � p tð Þq tð Þ <sup>1</sup>

Example 5.1. Consider the given system of BVPs

, qð Þ¼ 1 q

H1ð Þ t; τp; τq ≥ τ

Example 5.2. Consider the following system of BVPs:

<sup>2</sup>p tð Þþ ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>2</sup> <sup>þ</sup> ½ � p tð Þþ q tð Þ <sup>3</sup> <sup>¼</sup> <sup>0</sup>, <sup>D</sup><sup>9</sup>

<sup>p</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>q</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>; <sup>3</sup>, at t <sup>¼</sup> <sup>0</sup>,

, qð Þ¼ 1 q

1 2 � � :

1 2 � � :

<sup>p</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>q</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>; <sup>3</sup>, at t <sup>¼</sup> <sup>0</sup>,

<sup>2</sup>p tð Þþ ð Þ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>2</sup> <sup>þ</sup> ½ � p tð Þþ q tð Þ <sup>3</sup> <sup>¼</sup> <sup>0</sup>, <sup>D</sup><sup>9</sup>

, qð Þ¼ 1 q

1 2 � �

5 <sup>2</sup>; t 9 2 � � :

Example 5.2. Consider the following system of BVPs:

H1ð Þ t; τp; τq ≥ τ

ðÞ¼ t q tðÞ¼ q

, qð Þ¼ 1 q

<sup>2</sup> � � <sup>þ</sup> ½ � p tð Þq tð Þ <sup>1</sup>

From which we get

From which we get

Hence, we get

Hence, we get

where M ¼

where M ¼

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> <sup>∥</sup><sup>p</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup>ds <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> <sup>∥</sup><sup>p</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup>ds <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> <sup>∥</sup><sup>p</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup>ds <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> <sup>∥</sup><sup>p</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup>ds <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

<sup>∥</sup><sup>p</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup> <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

<sup>∥</sup><sup>p</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup> <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

<sup>∥</sup><sup>p</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup> <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

<sup>∥</sup><sup>p</sup> � <sup>p</sup>∥<sup>E</sup>�<sup>E</sup> <sup>þ</sup> <sup>Ð</sup> <sup>1</sup>

∥Sð Þ� p; q Sð Þ p; q ∥<sup>E</sup>�<sup>E</sup> ≤ M∥ð Þ� p; q ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>, (32)

∥Sð Þ� p; q Sð Þ p; q ∥<sup>E</sup>�<sup>E</sup> ≤ M∥ð Þ� p; q ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>, (32)

<sup>3</sup> q tð Þþ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>þ</sup> ½ � p tð Þq tð Þ <sup>1</sup>

ðÞ¼ t 0, at t ¼ 0,

<sup>3</sup> q tð Þþ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>þ</sup> ½ � p tð Þq tð Þ <sup>1</sup>

1

<sup>2</sup>q tð Þþ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>þ</sup> ½ � p tð Þþ q tð Þ <sup>2</sup> <sup>¼</sup> <sup>0</sup>, t∈ð Þ <sup>0</sup>; <sup>1</sup> ,

<sup>2</sup>q tð Þþ <sup>1</sup> <sup>þ</sup> <sup>t</sup> <sup>þ</sup> ½ � p tð Þþ q tð Þ <sup>2</sup> <sup>¼</sup> <sup>0</sup>, t∈ð Þ <sup>0</sup>; <sup>1</sup> ,

<sup>3</sup>H2ð Þ t; p; q :

<sup>3</sup>H2ð Þ t; p; q :

1

ðÞ¼ t 0, at t ¼ 0,

Clearly, H1ð Þ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at pð Þ¼ ; q ð Þ 0; 0 , and H1ð Þ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at ð Þ¼ p; q ð Þ 1; 1 . Simple computation yields that H1, H<sup>2</sup> are nondecreasing for every t ∈ð Þ 0; 1 : Also, for

Clearly, H1ð Þ t;p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at pð Þ¼ ; q ð Þ 0; 0 , and H1ð Þ t; p; q 6¼ 0, H2ð Þ t; p; q 6¼ 0, at ð Þ¼p; q ð Þ 1; 1 . Simple computation yields that H1, H<sup>2</sup> are nondecreasing for every t ∈ð Þ 0; 1 : Also, for

<sup>3</sup>H1ð Þ t; p; q , H2ð Þ t; τp; τq ≥ τ

<sup>3</sup>H1ð Þ t;p; q , H2ð Þ t; τp; τq ≥ τ

Thus, all the conditions of Theorem 3.4 are fulfilled, so the system (32) of BVPs has unique positive

Thus, all the conditions of Theorem 3.4 are fulfilled, so the system (32) of BVPs has unique positive

<sup>0</sup> G1ð Þ 1;s a1ð Þs ds� h i

<sup>0</sup> G1ð Þ 1;s a1ð Þs ds� h i

<sup>0</sup> G2ð Þ 1;s a2ð Þs ds h i

<sup>0</sup> G2ð Þ 1;s a2ð Þs ds h i

<sup>0</sup> G1ð Þ 1;s b1ð Þs ds

<sup>0</sup> G1ð Þ 1;s b1ð Þs ds

<sup>0</sup> G2ð Þ 1;s b2ð Þs ds

<sup>0</sup> G2ð Þ 1;s b2ð Þs ds

<sup>3</sup> <sup>¼</sup> <sup>0</sup>, <sup>D</sup><sup>11</sup>

0 ðÞ¼ t q 00

<sup>3</sup> <sup>¼</sup> <sup>0</sup>, <sup>D</sup><sup>11</sup>

0 ðÞ¼ t q 00

1 3 � � :

4 ; 1 3 � � <sup>¼</sup> <sup>1</sup> 3 ,

1

4 ; 1 3 � � <sup>¼</sup> <sup>1</sup> 3 ,

1 3 � � :

ðÞ¼ t q tðÞ¼ q

1

5 <sup>2</sup>; t 9 2 � � : <sup>0</sup> G1ð Þ 1;s b1ð Þs ∥q � q∥<sup>E</sup>�<sup>E</sup>ds

<sup>0</sup> G1ð Þ 1;s b1ð Þs ∥q � q∥<sup>E</sup>�<sup>E</sup>ds

<sup>0</sup> G2ð Þ 1;s b2ð Þs ∥q � q∥<sup>E</sup>�<sup>E</sup>ds:

<sup>0</sup> G2ð Þ 1;s b2ð Þs ∥q � q∥<sup>E</sup>�<sup>E</sup>ds:

<sup>0</sup> G1ð Þ 1;s b1ð Þs ds h i

<sup>0</sup> G2ð Þ 1;s b2ð Þs ds h i

∥q � q∥<sup>E</sup>�<sup>E</sup>

∥q � q∥<sup>E</sup>�<sup>E</sup>:

(31)

(31)

∥q � q∥<sup>E</sup>�<sup>E</sup>

∥q � q∥<sup>E</sup>�<sup>E</sup>:

(33)

(33)

(34)

(34)

<sup>0</sup> G1ð Þ 1;s b1ð Þs ds h i

<sup>0</sup> G2ð Þ 1;s b2ð Þs ds h i

. Hence, we received the required results.

. Hence, we received the required results.

<sup>4</sup> ¼ 0, t ∈ð Þ 0; 1 ,

<sup>4</sup> ¼ 0, t ∈ð Þ 0; 1 ,

$$|\mathcal{H}\_2(t, p, q)| \le \frac{t^2}{50} + \frac{t^2}{60} \sin |p(t)| + \frac{t}{60} \cos |q(t)|.$$

where <sup>φ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup> <sup>40</sup> , <sup>φ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup> 2 <sup>50</sup> , <sup>ψ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup> <sup>20</sup> , <sup>ψ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup> 2 <sup>60</sup> , <sup>σ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup> 2 <sup>20</sup> , <sup>σ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup> 60. Also, <sup>η</sup> <sup>¼</sup> <sup>ξ</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> , λ<sup>1</sup> ¼ λ<sup>2</sup> ¼ 0:17677: Thus, by computation, we have where <sup>φ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup> <sup>40</sup> , <sup>φ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup> 2 <sup>50</sup> , <sup>ψ</sup>1ðÞ¼ <sup>t</sup> <sup>t</sup> <sup>20</sup> , <sup>ψ</sup>2ðÞ¼<sup>t</sup> <sup>t</sup> 2 <sup>60</sup> , <sup>σ</sup>1ðÞ¼<sup>t</sup> <sup>t</sup> 2 <sup>20</sup> , <sup>σ</sup>2ðÞ¼ <sup>t</sup> <sup>t</sup> 60. Also, <sup>η</sup> <sup>ξ</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> ,λ<sup>1</sup> ¼ λ<sup>2</sup> ¼ 0:17677: Thus, by computation, we have

$$\mathbf{G}\_{j}(1,\mathbf{s}) = 6.65710 \frac{(1-\mathbf{s})^{\frac{2}{2}}}{\Gamma\left(\frac{7}{2}\right)}, \text{ for } j = 1, 2.$$

Upon computation, we get Upon computation, we get

$$\Delta\_1 = \int\_0^1 \mathbf{G}\_1(1, s)\varphi\_1(s)ds = 0.003577 < \text{\textquotedbl{}s\textquotedbl{}} \quad \Delta\_2 = \int\_0^1 \mathbf{G}\_2(1, s)\varphi\_2(s)ds = 0.000924 < \text{\textquotedbl{}s\textquotedbl{}}$$

Similarly, we can also compute. Similarly, we can also compute.

<sup>Λ</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup> <sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>1ð Þþ <sup>s</sup> <sup>σ</sup>1ð Þ<sup>s</sup> � �ds <sup>¼</sup> <sup>0</sup>:<sup>03092853</sup> <sup>&</sup>lt; <sup>1</sup>, <sup>Λ</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup> <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>2ð Þþ <sup>s</sup> <sup>σ</sup>2ð Þ<sup>s</sup> � �ds <sup>¼</sup> <sup>0</sup>:<sup>00289</sup> <sup>&</sup>lt; <sup>1</sup>: Further, we see that max 0f :007626; 0:00185g ¼ 0:007626. So, all the conditions of Theorem 3.3 are satisfied. So, the BVP (34) has at least one solution and the solution lies in <sup>Λ</sup><sup>1</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup> <sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>1ð Þþ <sup>s</sup> <sup>σ</sup>1ð Þ<sup>s</sup> � �ds <sup>¼</sup> <sup>0</sup>:<sup>03092853</sup> <sup>&</sup>lt; <sup>1</sup>, <sup>Λ</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup> <sup>1</sup> <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>ψ</sup>2ð Þþ <sup>s</sup> <sup>σ</sup>2ð Þ<sup>s</sup> � �ds <sup>¼</sup> <sup>0</sup>:<sup>00289</sup> <sup>&</sup>lt; <sup>1</sup>: Further, we see that max 0f :007626; 0:00185g ¼ 0:007626. So, all the conditions of Theorem 3.3 are satisfied. So, the BVP (34) has at least one solution and the solution lies in

$$\mathcal{E} = \{(p, q) \in \mathbf{C} : \|(p, q)\|\_{\mathbf{E} \times \mathbf{E}} < 0.007626\}.$$

#### 52 Differential Equations - Theory and Current Research 52 Differential Equations - Theory and Current Research

Example 5.4. Taking the following system of BVPs Example 5.4. Taking the following system of BVPs

$$\begin{cases} \begin{aligned} \mathscr{B}^{\frac{\mu}{2}}p(t) + \frac{[p(t) + q(t)]^2 + 1}{(15 + t^2)\delta\_1} &= 0, & \mathscr{B}^{\frac{\mu}{2}}q(t) + \frac{[p(t) + q(t)]^2 + t}{(15 + t^2)\delta\_2} &= 0, & t \in (0, 1), \\ p^{(j)}(t) = q^{(j)}(t) = 0, & j = 0, 1, 2, 3, 4, \text{ at } t = 0, & \end{aligned} \tag{36} \\ \begin{aligned} p(1) = p\left(\frac{1}{4}\right), \quad q(1) = q\left(\frac{1}{4}\right). \end{aligned} \end{cases} \tag{37}$$

1 3δ<sup>1</sup>

1

:

Acknowledgements

Conflict of interest

Research funder

Research funder

Author details

(11571378).

Kamal Shah1 and Yongjin Li<sup>2</sup>

Kamal Shah1 and Yongjin Li<sup>2</sup>

Author details

(11571378).

(36)

Pakistan

Pakistan

References

References

improved this chapter very well.

Conflict of interest

improved this chapter very well.

Acknowledgements

We are very thankful to the reviewers for his/her careful reading and suggestion which

We are very thankful to the reviewers for his/her careful reading and suggestion which

Existence Theory of Differential Equations of Arbitrary Order

http://dx.doi.org/10.5772/intechopen.75523

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

53

53

This work has been supported by the National Natural Science Foundation of China

This work has been supported by the National Natural Science Foundation of China

1 Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa,

2 Department of Mathematics, Sun Yat-sen University, Guanghou, China

1 Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa,

[1] Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Scientific; 2000 [2] Kilbas AA, Srivastava H, Trujillo J. Theory and application of fractional differential equations, North Holland Mathematics Studies. Vol. 204. Amsterdam: Elseveir; 2006 [3] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential

[1] Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Scientific; 2000 [2] Kilbas AA, Srivastava H, Trujillo J. Theory and application of fractional differential equations, North Holland Mathematics Studies. Vol. 204. Amsterdam: Elseveir; 2006 [3] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential

[5] Podlubny I. Fractional differential equations. In: Mathematics in Science and Engineering.

[5] Podlubny I. Fractional differential equations. In: Mathematics in Science and Engineering.

[6] Jalili M, Samet B. Existence of positive solutions to a coupled system of fractional differential equation. Mathematical Methods in Applied Science. 2015;38:1014-1031

[4] Podlubny I. Fractional differential equations. New York: Academic press; 1993

[6] Jalili M, Samet B. Existence of positive solutions to a coupled system of fractional differential equation. Mathematical Methods in Applied Science. 2015;38:1014-1031

[4] Podlubny I. Fractional differential equations. New York: Academic press; 1993

2 Department of Mathematics, Sun Yat-sen University, Guanghou, China

We declare the there is no conflict of interest regarding this chapter.

We declare the there is no conflict of interest regarding this chapter.

\*

\*

\*Address all correspondence to: stslyj@mail.sysu.edu.cn

\*Address all correspondence to: stslyj@mail.sysu.edu.cn

Equations. New York: Wiley; 1993

Equations. New York: Wiley; 1993

New York: Academic Press; 1999

New York: Academic Press; 1999

It is simple to check that H1,<sup>0</sup> ¼ H2,<sup>0</sup> ¼ H1,<sup>∞</sup> ¼ H2,<sup>∞</sup> ¼ ∞: Also, for any tð Þ ; p; q ∈ I � I � I, we see that It is simple to check that H1,<sup>0</sup> ¼ H2,<sup>0</sup> ¼ H1,<sup>∞</sup> ¼ H2,<sup>∞</sup> ¼ ∞: Also, for any tð Þ ;p; q ∈ I � I � I, we see that

$$\begin{array}{rcl} \mathcal{H}\_1(t, p, q) & \leq & \frac{1}{3\delta\_1} \\\\ \mathcal{H}\_2(t, p, q) & \leq & \frac{1}{3\delta\_2}. \end{array}$$

Thus, all the assumptions of Theorem 3.5 are satisfied with taking α ¼ 1, so the coupled system (35) has two solutions satisfying 0 < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < 1 < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>. Thus, all the assumptions of Theorem 3.5 are satisfied with taking α ¼ 1, so the coupled system (35) has two solutions satisfying 0 < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < 1 < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>.

Example 5.5. Consider the following coupled systems of boundary value problems: Example 5.5. Consider the following coupled systems of boundary value problems:

$$\begin{cases} \mathcal{B}^{\frac{3}{2}}p(t) + \Gamma\left(\frac{5}{2}\right) \left[ \frac{tp(t)}{16} + \frac{t^2 q(t)}{32} \right] = 0, t \in (0, 1), \\\mathcal{B}^{\frac{3}{2}}q(t) + \Gamma\left(\frac{5}{2}\right) \left[ \frac{9t^2 |\cos(p(t))|}{16\sqrt{\pi}} + \frac{9t |\cos(q(t))|}{32\sqrt{\pi}} \right] = 0, t \in (0, 1), \\\ p^{(j)}(t) = q^{(j)}(t) = 0, \ j = 0, 1, 2, \text{ at } t = 0, \\\ p(1) = p\left(\frac{1}{2}\right), \ q(1) = q\left(\frac{1}{2}\right). \end{cases} \tag{37}$$

Here, a1ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup> 2 � � <sup>t</sup> <sup>16</sup> , b1ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup> 2 � � <sup>t</sup> 2 <sup>32</sup> , a2ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup> 2 � � <sup>9</sup><sup>t</sup> 2 <sup>16</sup> ffiffi <sup>π</sup> <sup>p</sup> , b2ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup> 2 � � <sup>9</sup><sup>t</sup> <sup>32</sup> ffiffi <sup>π</sup> <sup>p</sup> . Moreover M ¼ Ð 1 <sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup> <sup>0</sup> G1ð Þ 1;s b1ð Þs ds Ð 1 <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup> <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>b</sup>2ð Þ<sup>s</sup> ds " # <sup>¼</sup> <sup>0</sup>:0460 0:<sup>0007</sup> <sup>0</sup>:0068 0:<sup>0058</sup> � �: Here, a1ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup> 2 � � <sup>t</sup> <sup>16</sup> , b1ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup> 2 � � <sup>t</sup> 2 <sup>32</sup> , a2ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup> 2 � � <sup>9</sup><sup>t</sup> 2 <sup>16</sup> ffiffi <sup>π</sup> <sup>p</sup> , b2ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup> 2 � � <sup>9</sup><sup>t</sup> <sup>32</sup> ffiffi <sup>π</sup> <sup>p</sup> . Moreover M ¼ Ð 1 <sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup> <sup>0</sup> G1ð Þ 1;s b1ð Þs ds Ð 1 <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup> <sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>b</sup>2ð Þ<sup>s</sup> ds " # <sup>¼</sup> <sup>0</sup>:0460 0:<sup>0007</sup> <sup>0</sup>:0068 0:<sup>0058</sup> � �:

Here, <sup>r</sup>ð Þ¼ <sup>M</sup> <sup>4</sup>:<sup>61</sup> � <sup>10</sup>�<sup>2</sup> <sup>&</sup>lt; <sup>1</sup>: Therefore, matrix <sup>M</sup> converges to zero, and hence the solutions of (36) are Hyers-Ulam stable by using Theorem 4.4. Here, <sup>r</sup>ð Þ¼ <sup>M</sup> <sup>4</sup>:<sup>61</sup> � <sup>10</sup>�<sup>2</sup> <sup>&</sup>lt; <sup>1</sup>: Therefore, matrix <sup>M</sup> converges to zero, and hence the solutions of (36) are Hyers-Ulam stable by using Theorem 4.4.

#### 6. Conclusion 6. Conclusion

We have developed a comprehensive theory on existence of solutions and its Hyers-Ulam stability for system of multipoint BVP of FDEs. The concerned theory has been enriched by providing suitable examples. We have developed a comprehensive theory on existence of solutions and its Hyers-Ulam stability for system of multipoint BVP of FDEs. The concerned theory has been enriched by providing suitable examples.

#### Acknowledgements Acknowledgements

Example 5.4. Taking the following system of BVPs

52 Differential Equations - Theory and Current Research

52 Differential Equations - Theory and Current Research

1 4 � �

pð Þ¼ 1 p

<sup>2</sup> p tð Þþ ½ � p tð Þþ q tð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> 15 þ t <sup>2</sup> � �δ<sup>1</sup>

> 1 4 � �

Example 5.4. Taking the following system of BVPs

<sup>p</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>q</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>4</sup>, at t <sup>¼</sup> <sup>0</sup>,

, qð Þ¼ 1 q

1 4 � � :

<sup>p</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>q</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>; <sup>3</sup>; <sup>4</sup>, at t <sup>¼</sup> <sup>0</sup>,

, qð Þ¼ 1 q

<sup>2</sup> p tð Þþ ½ � p tð Þþ q tð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup> 15 þ t <sup>2</sup> � �δ<sup>1</sup>

two solutions satisfying 0 < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> < 1 < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>.

<sup>2</sup>p tðÞþ <sup>Γ</sup> <sup>5</sup>

D5

D5

<sup>2</sup>q tð Þþ <sup>Γ</sup> <sup>5</sup>

2 � � tp tð Þ

<sup>2</sup>p tðÞþ <sup>Γ</sup> <sup>5</sup>

two solutions satisfying 0 < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup> <1 < ∥ð Þ p; q ∥<sup>E</sup>�<sup>E</sup>.

2 � � 9t

> 1 2 � �

> > 2 � � <sup>t</sup> 2

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

<sup>2</sup>q tð Þþ <sup>Γ</sup> <sup>5</sup>

1 2 � �

pð Þ¼ 1 p

2 � � <sup>t</sup> 2

<sup>16</sup> , b1ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup>

<sup>0</sup> <sup>G</sup>1ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>1ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

<sup>0</sup> <sup>G</sup>2ð Þ <sup>1</sup>;<sup>s</sup> <sup>a</sup>2ð Þ<sup>s</sup> ds <sup>Ð</sup> <sup>1</sup>

D5

8

>>>>>>>>>><

8

>>>>>>>>>><

>>>>>>>>>>:

2 � � <sup>t</sup>

Here, a1ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup>

M ¼

providing suitable examples.

providing suitable examples.

Here, a1ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup>

6. Conclusion

6. Conclusion

D5

pð Þ¼ 1 p

>>>>>>>>>>:

<sup>16</sup> , b1ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup>

Ð 1

Ð 1

are Hyers-Ulam stable by using Theorem 4.4.

Ð 1

2 � � <sup>t</sup>

Ð 1

M ¼

are Hyers-Ulam stable by using Theorem 4.4.

<sup>¼</sup> <sup>0</sup>, <sup>D</sup><sup>11</sup>

1 4 � � :

<sup>¼</sup> <sup>0</sup>, <sup>D</sup><sup>11</sup>

It is simple to check that H1,<sup>0</sup> ¼ H2,<sup>0</sup> ¼ H1,<sup>∞</sup> ¼ H2,<sup>∞</sup> ¼ ∞: Also, for any tð Þ ; p; q ∈ I � I � I, we see

H1ð Þ t; p; q ≤

It is simple to check that H1,<sup>0</sup> ¼ H2,<sup>0</sup> ¼ H1,<sup>∞</sup> ¼ H2,<sup>∞</sup> ¼ ∞: Also, for any tð Þ ; p; q ∈ I � I � I, we see

Thus, all the assumptions of Theorem 3.5 are satisfied with taking α ¼ 1, so the coupled system (35) has

Thus, all the assumptions of Theorem 3.5 are satisfied with taking α ¼ 1, so the coupled system (35) has

H2ð Þ t; p; q ≤

� �

1 2 � � :

<sup>2</sup>∣cosð Þ p tð Þ <sup>∣</sup> 16 ffiffiffi <sup>π</sup> <sup>p</sup> <sup>þ</sup>

H1ð Þ t; p; q ≤

H2ð Þ t; p; q ≤

Example 5.5. Consider the following coupled systems of boundary value problems:

16 þ t <sup>2</sup>q tð Þ 32

2 � � tp tð Þ

2 � � 9t

<sup>p</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>q</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, at t <sup>¼</sup> <sup>0</sup>,

, qð Þ¼ 1 q

<sup>32</sup> , a2ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup>

" #

, qð Þ¼ 1 q

� �

Example 5.5. Consider the following coupled systems of boundary value problems:

<sup>2</sup>∣cosð Þ p tð Þ <sup>∣</sup> 16 ffiffiffi <sup>π</sup> <sup>p</sup> <sup>þ</sup>

16 þ t <sup>2</sup>q tð Þ 32

� �

1 2 � � :

<sup>p</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>q</sup>ð Þ<sup>j</sup> ðÞ¼ <sup>t</sup> <sup>0</sup>, j <sup>¼</sup> <sup>0</sup>; <sup>1</sup>; <sup>2</sup>, at t <sup>¼</sup> <sup>0</sup>,

2 � � <sup>9</sup><sup>t</sup> 2 <sup>16</sup> ffiffi

<sup>32</sup> , a2ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup>

" #

<sup>0</sup> G1ð Þ 1;s b1ð Þs ds

2 � � <sup>9</sup><sup>t</sup> 2 <sup>16</sup> ffiffi

<sup>0</sup> G2ð Þ 1;s b2ð Þs ds

Here, <sup>r</sup>ð Þ¼ <sup>M</sup> <sup>4</sup>:<sup>61</sup> � <sup>10</sup>�<sup>2</sup> <sup>&</sup>lt; <sup>1</sup>: Therefore, matrix <sup>M</sup> converges to zero, and hence the solutions of (36)

Here, <sup>r</sup>ð Þ¼ <sup>M</sup> <sup>4</sup>:<sup>61</sup> � <sup>10</sup>�<sup>2</sup> <sup>&</sup>lt; <sup>1</sup>: Therefore, matrix <sup>M</sup> converges to zero, and hence the solutions of (36)

We have developed a comprehensive theory on existence of solutions and its Hyers-Ulam stability for system of multipoint BVP of FDEs. The concerned theory has been enriched by

We have developed a comprehensive theory on existence of solutions and its Hyers-Ulam stability for system of multipoint BVP of FDEs. The concerned theory has been enriched by

<sup>2</sup> q tð Þþ ½ � p tð Þþ q tð Þ <sup>2</sup> <sup>þ</sup> <sup>t</sup> 15 þ t <sup>2</sup> � �δ<sup>2</sup>

<sup>2</sup> q tð Þþ ½ � p tð Þþ q tð Þ <sup>2</sup> <sup>þ</sup> <sup>t</sup> 15 þ t <sup>2</sup> � �δ<sup>2</sup>

> 1 3δ<sup>1</sup>

1 3δ<sup>1</sup>

1 3δ<sup>2</sup> :

1 3δ<sup>2</sup> :

¼ 0, t∈ ð Þ 0; 1 ,

¼ 0, t∈ ð Þ 0; 1 ,

9t∣cosð Þ q tð Þ ∣ 32 ffiffiffi <sup>π</sup> <sup>p</sup>

9t∣cosð Þ q tð Þ ∣ 32 ffiffiffi <sup>π</sup> <sup>p</sup>

� �

<sup>π</sup> <sup>p</sup> , b2ðÞ¼ <sup>t</sup> <sup>Γ</sup> <sup>5</sup>

<sup>0</sup> G1ð Þ 1;s b1ð Þs ds

<sup>0</sup> G2ð Þ 1;s b2ð Þs ds

2 � � <sup>9</sup><sup>t</sup> <sup>32</sup> ffiffi

<sup>π</sup> <sup>p</sup> , b2ðÞ¼ <sup>t</sup> <sup>Γ</sup> 52

¼ 0, t ∈ð Þ 0; 1 ,

¼ 0, t ∈ð Þ 0; 1 ,

<sup>π</sup> <sup>p</sup> . Moreover

<sup>¼</sup> <sup>0</sup>:0460 0:<sup>0007</sup> 0:0068 0:0058 � �

<sup>π</sup> <sup>p</sup> . Moreover

:

:

<sup>¼</sup> <sup>0</sup>:0460 0:<sup>0007</sup> 0:0068 0:0058 � �

� � <sup>9</sup><sup>t</sup> <sup>32</sup> ffiffi

¼ 0, t∈ ð Þ 0; 1 ,

¼ 0, t∈ ð Þ 0; 1 ,

(36)

(36)

(37)

(37)

D11

8 >>>>>><

8 >>>>>><

>>>>>>:

that

that

pð Þ¼ 1 p

>>>>>>:

D11

We are very thankful to the reviewers for his/her careful reading and suggestion which improved this chapter very well. We are very thankful to the reviewers for his/her careful reading and suggestion which improved this chapter very well.

#### Conflict of interest Conflict of interest

We declare the there is no conflict of interest regarding this chapter. We declare the there is no conflict of interest regarding this chapter.

#### Research funder Research funder

This work has been supported by the National Natural Science Foundation of China (11571378). This work has been supported by the National Natural Science Foundation of China (11571378).

#### Author details Author details

Kamal Shah1 and Yongjin Li<sup>2</sup> \* Kamal Shah1 and Yongjin Li<sup>2</sup> \*

\*Address all correspondence to: stslyj@mail.sysu.edu.cn \*Address all correspondence to: stslyj@mail.sysu.edu.cn

1 Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan 1 Department of Mathematics, University of Malakand, Dir(L), Khyber Pakhtunkhwa, Pakistan

2 Department of Mathematics, Sun Yat-sen University, Guanghou, China 2 Department of Mathematics, Sun Yat-sen University, Guanghou, China

#### References References


[7] Khan RA, Shah K. Existence and uniqueness of solutions to fractional order multi-point boundary value problems. Communications on Pure and Applied Analysis. 2015;19:515-526 [7] Khan RA, Shah K. Existence and uniqueness of solutions to fractional order multi-point boundary value problems. Communications on Pure and Applied Analysis. 2015;19:515-526

[22] Obloza M. Hyers stability of the linear differential equation. Rocznik Nauk-Dydakt. Prace

[22] Obloza M. Hyers stability of the linear differential equation. Rocznik Nauk-Dydakt. Prace

[23] Wang J, Li X. Ulam-Hyers stability of fractional Langevin equations. Applied Mathemat-

[24] Stamova I. Mittag-Leffler stability of impulsive differential equations of fractional order.

[25] Kumama P, Ali A, Shah K, Khan RA. Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations. Journal of Nonlinear Science and

[26] Wang J, Lv L, Zhou Y. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electronic Journal of Qualitative Theory of Differential

[27] Haq F, Shah K, Rahman G, Shahzad M. Hyers–Ulam stability to a class of fractional differential equations with boundary conditions. International Journal of Applied Com-

[29] Zeidler E. Non Linear Functional Analysis and Its Applications. New York, USA:

[30] Agarwal R, Meehan M, Regan DO. Fixed Points Theory and Applications. Cambridge:

[31] Urs C. Coupled fixed point theorem and applications to periodic boundary value prob-

Existence Theory of Differential Equations of Arbitrary Order

http://dx.doi.org/10.5772/intechopen.75523

http://dx.doi.org/10.5772/intechopen.75523

Existence Theory of Differential Equations of Arbitrary Order

55

55

[23] Wang J, Li X. Ulam-Hyers stability of fractional Langevin equations. Applied Mathemat-

[24] Stamova I. Mittag-Leffler stability of impulsive differential equations of fractional order.

[25] Kumama P, Ali A, Shah K, Khan RA. Existence results and Hyers-Ulam stability to a class of nonlinear arbitrary order differential equations. Journal of Nonlinear Science and

[26] Wang J, Lv L, Zhou Y. Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electronic Journal of Qualitative Theory of Differential

[27] Haq F, Shah K, Rahman G, Shahzad M. Hyers–Ulam stability to a class of fractional differential equations with boundary conditions. International Journal of Applied Com-

[29] Zeidler E. Non Linear Functional Analysis and Its Applications. New York, USA:

[28] Deimling K. Nonlinear Functional Analysis. New York: Springer-Verlag; 1985

[30] Agarwal R, Meehan M, Regan DO. Fixed Points Theory and Applications. Cambridge:

[31] Urs C. Coupled fixed point theorem and applications to periodic boundary value prob-

[28] Deimling K. Nonlinear Functional Analysis. New York: Springer-Verlag; 1985

Mat. 1993;13:259-270

Mat. 1993;13:259-270

ics and Computation. 2015;258:72-83

ics and Computation. 2015;258:72-83

Applications. 2017;10:2986-2997

Equations. 2011;(63):1-10

Applications. 2017;10:2986-2997

putational Mathematics. 2017:1-13

putational Mathematics. 2017:1-13

Cambridge University Press; 2004

Cambridge University Press; 2004

lem. Miskolic Mathematical Notes. 2013;14:323-333

lem. Miskolic Mathematical Notes. 2013;14:323-333

Equations. 2011;(63):1-10

Springer; 1986

Springer; 1986

Quarterly of Applied Mathematics. 2015;73(3):525-535

Quarterly of Applied Mathematics. 2015;73(3):525-535


[22] Obloza M. Hyers stability of the linear differential equation. Rocznik Nauk-Dydakt. Prace Mat. 1993;13:259-270 [22] Obloza M. Hyers stability of the linear differential equation. Rocznik Nauk-Dydakt. Prace Mat. 1993;13:259-270

[7] Khan RA, Shah K. Existence and uniqueness of solutions to fractional order multi-point boundary value problems. Communications on Pure and Applied Analysis. 2015;19:515-526

[7] Khan RA, Shah K. Existence and uniqueness of solutions to fractional order multi-point boundary value problems. Communications on Pure and Applied Analysis. 2015;19:515-526

[8] Shah K, Khan RA. Study of solution to a toppled system of fractional differential equations with integral boundary. International Journal of Applied and Computational Math-

[9] Shah K, Khalil H, Khan RA. Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations.

[10] Ahmad B, Alsaedi A. Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. Fixed Point Theory Applications.

[11] Su X. Boundary value problem for a coupled system of nonlinear fractional differential

[12] Ahmad B, Nieto JJ. Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Computers & Mathematcs with

[13] Yang W. Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Computers & Mathematcs with Applications.

[14] Bai Z, Lü H. Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications. 2005;311:495-505 [15] Kaufmann ER, Mboumi E. Positive solutions for boundary value problem of a nonlinear fractional differential equations, Electronic Journal of Qualitative Theory of Differential

[16] Shah K, Khan RA. Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions.

[17] Shah K, Khalil H, Khan RA. Upper and lower solutions to a coupled system of nonlinear fractional differential equations. Progress Fractional Differential Applications. 2016;2(1):

[18] Zhang SQ. Positive solutions for boundary value problem problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations. 2006;2006:1-12

[19] Wang J, Xiang H, Lu Z. Positive solutions to nonzero boundary value problem for a coupled system of nonlinear fractional differential equations, International Journal of

[20] Ulam SM. Problems in Modern Mathematics. New York, USA: John Wiley and Sons; 1940

[8] Shah K, Khan RA. Study of solution to a toppled system of fractional differential equations with integral boundary. International Journal of Applied and Computational Math-

[9] Shah K, Khalil H, Khan RA. Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations.

[10] Ahmad B, Alsaedi A. Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations. Fixed Point Theory Applications.

[11] Su X. Boundary value problem for a coupled system of nonlinear fractional differential

[12] Ahmad B, Nieto JJ. Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Computers & Mathematcs with

[13] Yang W. Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations. Computers & Mathematcs with Applications.

[14] Bai Z, Lü H. Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications. 2005;311:495-505 [15] Kaufmann ER, Mboumi E. Positive solutions for boundary value problem of a nonlinear fractional differential equations, Electronic Journal of Qualitative Theory of Differential

[16] Shah K, Khan RA. Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional order differential equations with anti periodic boundary conditions.

[17] Shah K, Khalil H, Khan RA. Upper and lower solutions to a coupled system of nonlinear fractional differential equations. Progress Fractional Differential Applications. 2016;2(1):

[18] Zhang SQ. Positive solutions for boundary value problem problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations. 2006;2006:1-12

[19] Wang J, Xiang H, Lu Z. Positive solutions to nonzero boundary value problem for a coupled system of nonlinear fractional differential equations, International Journal of

[20] Ulam SM. Problems in Modern Mathematics. New York, USA: John Wiley and Sons; 1940

[21] Ulam SM. A Collection of Mathematical Problems. New York: Interscience; 1960

[21] Ulam SM. A Collection of Mathematical Problems. New York: Interscience; 1960

Journal of Chaos, Solitons and Fractals. 2015;77:240-246

Journal of Chaos, Solitons and Fractals. 2015;77:240-246

equations. Applied Mathematics Letters. 2009;22:64-69

equations. Applied Mathematics Letters. 2009;22:64-69

Differential Equations and Applications. 2015;7(2):245-262

Differential Equations and Applications. 2015;7(2):245-262

ematics. 2016;2(3) 19 pages

ematics. 2016;2(3) 19 pages

54 Differential Equations - Theory and Current Research

54 Differential Equations - Theory and Current Research

2010 Article ID 364560

2012;63:288-297

2012;63:288-297

Equations. 2008;8:1-11

Equations. 2008;8:1-11

Differential Equations. 2010. 12-pages

Differential Equations. 2010. 12-pages

1-10

1-10

Applications. 2009;58:18381843

Applications. 2009;58:18381843

2010 Article ID 364560


**Chapter 3**

Provisional chapter


N

**An Extension of Massera's Theorem for** *N***-Dimensional**

DOI: 10.5772/intechopen.73183

In this chapter, we consider a periodic SDE in the dimension n ≥ 2, and we study the existence of periodic solutions for this type of equations using the Massera principle. On the other hand, we prove an analogous result of the Massera's theorem for the SDE considered.

Keywords: stochastic differential equations, periodic solution, Markov process, Massera

The theory of stochastic differential equations is given for the first time by Itô [7] in 1942. This theory is based on the concept of stochastic integrals, a new notion of integral generalizing the

The stochastic differential equations (SDE) are applied for the first time in the problems of Kolmogorov of determining of Markov processes [8]. This type of equations was, from the first work of Itô, the subject of several investigations; the most recent include the generalization of known results for EDO, such as the existence of periodic and almost periodic solutions. It has, among others, the work of Bezandry and Diagana [1, 2], Dorogovtsev [4], Vârsan [12],

> © 2016 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.

© 2018 The Author(s). Licensee IntechOpen. 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.

**Stochastic Differential Equations**

Stochastic Differential Equations

Additional information is available at the end of the chapter

Da Prato [3], and Morozan and his collaborators [10, 11].

Additional information is available at the end of the chapter

Osmanov Hamid Ibrahim Ouglu

Osmanov Hamid Ibrahim Ouglu

http://dx.doi.org/10.5772/intechopen.73183

Abstract

theorem

1. Introduction

Lebesgue–Stieltjes one.

Boudref Mohamed Ahmed, Berboucha Ahmed and

Boudref Mohamed Ahmed, Berboucha Ahmed and

An Extension of Massera's Theorem for

#### **An Extension of Massera's Theorem for** *N***-Dimensional Stochastic Differential Equations** An Extension of Massera's Theorem for N-Dimensional Stochastic Differential Equations

DOI: 10.5772/intechopen.73183

Boudref Mohamed Ahmed, Berboucha Ahmed and Osmanov Hamid Ibrahim Ouglu Boudref Mohamed Ahmed, Berboucha Ahmed and Osmanov Hamid Ibrahim Ouglu

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/intechopen.73183

### Abstract

In this chapter, we consider a periodic SDE in the dimension n ≥ 2, and we study the existence of periodic solutions for this type of equations using the Massera principle. On the other hand, we prove an analogous result of the Massera's theorem for the SDE considered.

Keywords: stochastic differential equations, periodic solution, Markov process, Massera theorem
