Abstract

In this paper, we are going to investigate the existence and uniqueness of solutions of a coupled system of nonlinear fractional hybrid equations with nonseparated type integral boundary hybrid conditions. We are going to use Banach’s and Leray-Schauder alternative fixed point theorems to obtain the main results. Lastly, we are giving two examples to show the effectiveness of the main results.

1. Introduction

Fractional differential equations appear naturally in a number of fields bymany fields of scince such as physics, engineering, biophysics, blood flow phenomena, aerodynamics, electron-analytical chemistry, biology, and economy; for more details, we refer the readers to [1–4] and many other references therein which give an excellent account on the study of fractional differential equations.

Hybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown function hybrid with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers (see [5, 6]).

On the other hand, coupled systems of fractional differential equations are very important to study byattract the attention of many researches, because they appear naturally in many problems (see [3, 7–10]).

In [11], Sitho et al. discussed the following existence bysome results for the following hybrid fractional integrodifferential equations: where denotes the Riemann-Liouville fractional derivative of order is the Riemann-Liouville fractional integral of order , with with .

In [12], Hilal and Kajouni considered the following boundary value problems for hybrid differential equations with fractional order (BVPHDEF for short) involving Caputo differential operators of order where and are real constants with .

Dhage and Lakshmikantham [13] discussed the following first-order hybrid differential equation: where and . They established the existence, uniqueness results, and some fundamental differential inequalities for hybrid differential equations initiating the study of theory of such systems and proving the utilization of the theory of inequalities, the existence of extremal solutions, and comparison results.

Zhao et al. [5] discussed the following fractional hybrid differential equations involving Riemann-Liouville differential operators: where and . They established the existence byof solutions and some fundamental differential inequalities are also established and the existence of extremal solutions.

Benchohra et al. [14] studied the following boundary value problems for differential equations with fractional order: where is the Caputo fractional derivative, is a continuous function, and are real constants with .

Hannabou and Hilal [15] considered the boundary value problem of a class of impulsive hybrid fractional coupled differential equations: where stand for Caputo fractional derivatives of order , respectively, and are continuous functions defined by where for , and , and represent the right, left limits of at .

The present paper is a continuation of the work in [16] in order to study the existence and uniqueness of solutions for a coupled system of fractional hybrid equation of the following forme:

subject to the fractional nonseparated integral boundary hybrid conditions where with for are the Caputo fractional derivatives, , and are given continuous functions.

By a solution of the problems (7)–(8), we mean a function such that (i)the function is continuous for each ,(ii) satisfies the equations in (7)–(8).

This paper is organized as follows: in the second section, we recall some notations and several known results. In the third section, we show the existence and uniqueness of solutions of problem (7)–(8), these results can be viewed as extension of the result given in [12]. In the fourth section, we give some examples to demonstrate the application of our main results.

2. Preliminaries and Notations

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Let be a Banach space of all continuous functions defined frome endowed with norm. Then, the product space is also a Banach space equipped with the norm.

We denote by the space of Lebesgue integrable real-valued functions on equipped with the norm defined by

Definition 1 (see [17]). The fractional integral of the function of order is defined by where is the gamma function.

Definition 2 (see [17]). Let be a fonction difened on the Riemann-Liouville fractional derivative of order is defined by where and denotes the integer part of α.

Definition 3 (see [17]). Let be a function defined on the Caputo fractional derivative of order is defined by where and denotes the integer part of α.

Lemma 4 (see [2]). Let then the following relations hold:

Lemma 5 (see [2]). Let and If is a continuous function, then we have

Lemma 6 (Leray-Schauder alternative, see [18]). Let be a completely continuous operator (i.e., a map that is restricted to any bounded set in is compact). Let . Then, either the set is unbounded or has at least one fxed point.
We make the following assumption:
The function is increasing in almost everywhere for

Lemma 7. Assume that hypothesis holds. Then, for any The function is a solution of the coupled system, subject to the boundary condition (8), has a solution given by where

Proof. Using Lemma 5, we obtain where .

According to the condition we find that

Using the facts that and we have

Substituting the values of and we obtain

Analogously, we can deduce that

By a direct computation, the converse of the lemma can be easily verified.

3. Main Results

In view of Lemma 7, we define the operator by

Where,

For computational convenience, we set

Before giving the main results, we impose the following assumptions:

The functions are continuous and bounded; that is, there exist positive numbers such that for all .

and are continuous functions.

There exist positive constants such that for all and , we have

There exist positive constants such that

; ; ; , with for

3.1. First Result

In our first result, we discuss the existence of solutions for system (7)–(8) by means of the Banach fixed point theorem.

Theorem 8. Suppose that – are satisfied.
Then, there exist a unique solution for systems (7)–(8) provided that .

Proof. We put , , , , for .
Let with , where

We prove that .

For , we have

Consequently,

In the same way, we obtain that

Therefore, we have

Now, for and , we get

Analogously, one has and thus

Since by thene the operator is a contraction mapping. Hence, we deduce that systems (7)–(8) have a unique solution.

3.2. Second Result

In our second result, we discuss the existence of solutions for system (7)–(8) by means of the so-called Leray-Schauder alternative.

Theorem 9. Assume that conditions – and hold. Furthermore, it is assumed that

Then, system (7)–(8) have at least one solution.

Proof. We will show that the operator satisfes all the assumptions of Lemma 6.

In the first step, we prove that the operator is completely continuous.

Clearly, it follows by the continuity of functions and that the operator is continuous.

Let be bounded. Then, we can find positive constants and such that

Thus, for any we get

In a similar manner, we have

From the inequalities above, we deduce that the operator is uniformly bounded.

Next, we prove that is equicontinuous.

The continuity of implies that the operator is continuous. Moreover, is uniformly bounded on .

Suppose that . Then we have

Similarly, one has

which tend to independently of . This implies that the operator is equicontinuous. Thus, by the above fndings, the operator is completely continuous.

In the next step, we will prove that the set is bounded.

Let . Then, we have . Thus, for any we can write

Then,

In consequence, we have

then with

This shows that the set is bounded. Hence, all the conditions of Lemma 6 are satisfied, and consequently, the operator has at least one fxed point, which corresponds to a solution of system (7)–(8). This completes the proof.

4. Examples

4.1. Example 1

Consider the following system of fractional hybrid differentiel equation: where and

Clearly, , , , and ; furthermore, we have

Thus, by Theorem 8, system (46) has an unique solution.

4.2. Example 2

Consider the following system: where , , , , , , , , and

In this conrete application, we have

The riview of Theorem 9, problem (49) has a least one solution.

5. Conclusion

It is known that most natural phenomena are modeled by different types of fractional differential equations. This diversity in investigating complicated fractional differential equations increases our ability for exact modeling of different phenomena. This is useful in making modern software which helps us to allow for more cost-free testing and less material consumption. For our contribution in this present work, we investigate a fractional hybrid differential system with mixed integral hybrid and boundary hybrid conditions. We investigatedtwo numerical examples to illustrate our main results

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.