Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions

Abstract

This paper is motivated by some papers treating the fractional hybrid differential equations with nonlocal conditions and the system of coupled hybrid fractional differential equations; an existence theorem for fractional hybrid differential equations involving Caputo differential operators of order is proved under mixed Lipschitz and Carathéodory conditions. The existence and uniqueness result is elaborated for the system of coupled hybrid fractional differential equations.

1. Introduction

Our aim in this paper is to study the existence of solution for the boundary value problems for hybrid differential equations with fractional order and nonlocal condition (BVPHDEFNL for short) of the form where is the Caputo fractional derivative.

is a continuous function and . And exploitation of results obtained to study the existence of solutions for a system of coupled hybrid fractional differential equations is as follows:where is the Caputo fractional derivative.

, , and are continuous functions .

Fractional differential equations are a generalization of ordinary differential equations and integration to arbitrary noninteger orders. The origin of fractional calculus goes back to Newton and Leibniz in the seventeenth century. It is widely and efficiently used to describe many phenomena arising in engineering, physics, economy, and science. There are several concepts of fractional derivatives, some classical, such as Riemann-Liouville or Caputo definitions. For noteworthy papers dealing with the integral operator and the arbitrary fractional order differential operator, see [1–7].

The quadratic perturbations of nonlinear differential equations have attracted much attention. We call such fractional hybrid differential equations. There have been many works on the theory of hybrid differential equations, and we refer the readers to the articles [8–12].

Dhage and Lakshmikantham [11] 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 proved, utilizing the theory of inequalities, the existence of extremal solutions and comparison results.

Zhao et al. [13] have discussed the following fractional hybrid differential equations involving Riemann-Liouville differential operators:where and . The authors of [13] established the existence theorem for fractional hybrid differential equation and some fundamental differential inequalities. They also established the existence of extremal solutions.

Hilal and Kajouni [14] have studied boundary fractional hybrid differential equations involving Caputo differential operators of order as follows: where and , , and are real constants with . They proved the existence result for boundary fractional hybrid differential equations under mixed Lipschitz and Carathéodory conditions. Some fundamental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions.

The nonlocal condition is a condition attached to the main equation; it replaces the classic nonlocal condition in order to model physical phenomena of the fashion nearest from reality. The nonlocal condition involves the functionwhere , are given constants and .

Let us observe that Cauchy problems with nonlocal conditions were initiated by Byszewski and Lakshmikantham [2] and, since then, such problems have also attracted several authors including A. Aizicovici, K. Ezzinbi, Z. Fan, J. Liu, J. Liang, Y. Lin, T.-J. Xiao, G. N’Guérékata, E. Hernàndez, and H. Lee (see [2, 15]).

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

By we denote the Banach space of all continuous functions from into with the normAnd let denote the class of functions such that (i)the map is measurable for each ,(ii)the map is continuous for each .The class is called the Carathéodory class of functions on which are Lebesgue integrable when bounded by a Lebesgue integrable function on .

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

Definition 1. The fractional integral of the function of order is defined bywhere is the gamma function.

Definition 2. For a function given on the interval , the Caputo fractional order derivative of is defined bywhere and denotes the integer part of .

Lemma 3 (see [16]). Let . Then the fractional differential equationhas solutions

Lemma 4 (see [16]). Let . Thenfor some

Definition 5. By a solution of the BVPHDEFNL (1) we mean a function such that (i), where for each ,(ii) satisfies the equations in (1).

3. Existence Result

In this section, we prove the existence results for the boundary value problems for hybrid differential equations with fractional order (1) on the closed and bounded interval under mixed Lipschitz and Carathéodory conditions on the nonlinearities involved in it.

We defined the multiplication in by Clearly, is a Banach algebra with respect to above norm and multiplication in it.

We prove the existence of solution for the BVPHDEFNL (1) by a fixed point theorem in Banach algebra due to Dhage [10].

Lemma 6 (see [10]). Let be a nonempty, closed convex, and bounded subset of the Banach algebra and let and be two operators such that(a) is Lipschitzian with a Lipschitz constant ,(b) is completely continuous,(c) for all ,(d), where .Then the operator equation has a solution in .

We make the following assumptions:(H₀):the function is increasing in almost everywhere for .(H₁):there exists a constant such that  for all and .(H₂):there exists a function such that  for all .(H₃):there exists a constant such that , for each .As a consequence of Lemmas 3 and 4 we have the following result which is useful in what follows.

Lemma 7. Assume that hypothesis holds. Then for any , the function is a solution of the BVPHDEFNL: if and only if satisfies the hybrid integral equation

Proof. Assume that is a solution of the problem (18). Applying the Caputo fractional operator of the order , we obtain the first equation in (17). Again, substituting and in (18) we will have the second equation in (17).
Conversely, , so we getThen and , and even Thus,implies that

Theorem 8. Assume hypotheses . Further, ifthen the hybrid fractional order differential equation (1) has a solution defined on .

Proof. We defined a subset of bywhere and
It is clear that satisfies hypothesis of Lemma 6. By an application of Lemma 7, (1) is equivalent to the nonlinear hybrid integral equationDefine two operators and by Then the hybrid integral equation (25) is transformed into the operator equation as We will show that the operators and satisfy all the conditions of Lemma 6.
Claim 1. Let . Then by hypothesis , for all . Taking supremum over , we obtain for all .
Claim 2 (we show that is continuous in ). Let be a sequence in converging to a point . Then by Lebesgue dominated convergence theorem,And since is a continuous functionthenfor all . This shows that is a continuous operator on .
Claim 3 ( is compact operator on ). First, we show that is a uniformly bounded set in .
Let . Then by hypothesis , for all , Thus, , for all .
This shows that is uniformly bounded on .
Next, we show that is an equicontinuous set on
We set .
Let . Then for any ,Since is continuous on compact , it is uniformly continuous. Hence,for all and for all
This shows that is an equicontinuous set in .
Then by Arzelá-Ascoli theorem, is a continuous and compact operator on .
Claim 4 (hypothesis of Lemma 6 is satisfied). Let and be arbitrary such that . Then,and so,which impliesTaking supremum over ,Then , and hypothesis of Lemma 6 is satisfied.
Finally, we have So, Thus, all the conditions of Lemma 6 are satisfied and hence the operator equation has a solution in . As a result, BVPHDEFNL (1) has a solution defined on . This completes the proof.

4. An Example

In this section we give an example to illustrate the usefulness of our main results. Let us consider the following fractional boundary value problem: where , are given positive constants.

And , where .

We set Let and . We haveHence, condition holds with . Also we havewhere . We have Then condition holds.

Furthermore, since , then we set and we haveWe will check that condition (23) is satisfied with .

Since , then .

Thus,which is satisfied for each . Then by Theorem 8 problem (42) has a solution on

5. System of Coupled Hybrid Fractional Differential Equations

The aim of this section is to obtain the existence results, by means of Banach’s fixed point theorem, for the problem of coupled hybrid fractional differential equations for (1). Consider where is the Caputo fractional derivative.

, , and are continuous functions .

Main Results. Let denote a Banach space equipped with the norm , where . Notice that the product space with the norm , is also a Banach space.

In view of Lemma 7, we define an operator by whereIn the sequel, we need the following assumptions: :the functions are continuous and bounded; that is, there exist positive numbers such that for all .:there exist real constants and such that and for all .:there exist real constants and for each .:there exist real constants such that and .

For brevity, let us set Now we present our result for the existence and uniqueness of solutions for problem (49). This result is based on Banach’s contraction mapping principle.

Theorem 9. Suppose that conditions , , and hold and that are continuous functions. In addition, there exist positive constants , such thatIf , then problem (49) has a unique solution.

Proof. Let us set and and define a closed ball: , whereClaim 5 (we show that ). Let . We have Hence,From (57), it follows that .
Next, for and for any , we have which yieldsWorking in a similar manner, one can find thatWe deduce that In view of condition , it follows that is a contraction. So has a unique fixed point. This implies that problem (49) has a unique solution on . This completes the proof.

In our second result, we discuss the existence of solutions for problem (49) by means of Leray-Schauder alternative.

Lemma 10 (see [17]). 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 fixed point.

Theorem 11. Assume that conditions hold. Furthermore, it is assumed that and , where and are given by (52). Then the boundary value problem (49) has at least one solution.

Proof. We will show that the operator satisfies all the assumptions of Lemma 10. 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 can get which yieldsIn a similar manner,We deduce that the operator is uniformly bounded.
Now we show that the operator is equicontinuous.
We take with and obtain which tend to independently of . This implies that the operator is equicontinuous. Thus, by the above findings, the operator is completely continuous.
In the next step, it will be established that the set is bounded.
Let . Then we have . Thus, for any , we can write Then,which imply thatThus,which, in view of (55), givesThis shows that the set is bounded. Hence, all the conditions of Lemma 10 are satisfied and consequently the operator has at least one fixed point, which corresponds to a solution of problem (49). This completes the proof.

Competing Interests

The authors declare that there are no competing interests regarding the publication of this paper.