On Fractional Order Hybrid Differential Equations

Abstract

We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order . Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.

1. Introduction

Fractional calculus is a field of mathematics that deals with derivatives and integrals of arbitrary orders. Fractional differential equations have been of great interest because of their intensive development of fractional calculus and its applications [1–11].

Recently, the quadratic perturbation of nonlinear differential equations (called hybrid differential equations) has captured much attention. The importance of the investigations of hybrid differential equation lies in the fact that they include several dynamic systems as special cases. Dhage and Lakshmikantham [12] discussed the existence and uniqueness theorems of the solution to the ordinary first-order hybrid differential equation with perturbation of first type where and , where is a bounded interval in for some and with is the class of continuous functions and is called the Caratheodory class of functions which are Lebesgue integrable bounded by a Lebesgue integrable function on . Moreover (i)the map is measurable for each ;(ii)the map is continuous for each .

Dhage and Jadhav [13] discussed the existence and uniqueness theorems of the solution of the ordinary first-order hybrid differential equation with perturbation of second type

Fractional hybrid differential equations have been studied using Riemman-Liouville derivative in literature (see, e.g., [14–16] and the references therein). Ammi et al. [14] discussed a generalization of (1) by replacing the ordinary by fractional derivative in Riemann-Liouvile sense with delay in the form where and is the continuous function defined by for all . They used that is a solution to (3) if and only if it is a solution to the integral equation then they studied this integral equation. However, it is easy to see that this concept of a solution is not realistic because of neglecting the initial condition. In fact, if we consider the following trivial example with we get the function which is a solution to (3) but not to (4).

Lu et al. [15] discussed a generalization of (2) by replacing the classical differentiation by fractional derivative in the Riemann-Liouvile sense as They proved that is a solution to (5) if and only if it is a solution to the integral equation then they studied this integral equation. Again, we recall that this concept of a solution is not realistic because of neglecting the initial condition. In fact, consider the following trivial example with ; we get that the function is a solution to (5) but not for (6).

Here we discuss the existence of solutions to hybrid fractional differential equations in both types using the Caputo fractional derivative instead of the classical one in both (1) and (2).

Our paper is organized as follows. In Section 2 some basic definitions and theorems are given. Fractional hybrid differential equation of type 1 is discussed in Section 3 while in Section 4 we discuss the fractional hybrid differential equation of type 2. Our conclusion is presented in Section 5.

2. Preliminaries

Below we present some definitions and theorems used in the rest of this paper.

Definition 1 (see [1, 8–11]). If , the set of all integrable functions, and then the left Riemann-Liouville fractional integral of order , is defined by

Definition 2 (see [8–11]). For the left Riemann-Liouville fractional derivative of order is defined by where is such that and .

Definition 3 (see [8–11]). For the left Caputo fractional derivative of order is defined by where is such that and .

The following two fixed point theorems are used in our paper (see [12, 13, 17–19]).

Theorem 4. Let and be open and closed balls in a Banach algebra centered at origin of radius , for some real number , and let be two operators satisfying the following.(a) is Lipschitz with Lipschitz constant ;(b) is continuous and compact;(c), where . Then, either (i)the equation has a solution in , or(ii)there is an element such that satisfying , for some .

Definition 5. Let be a Banach space. A mapping is called -Lipschitzian if there exists a continuous and nondecreasing function such that for all , where . If is not necessarily nondecreasing and satisfies , for , the mapping is called a nonlinear contraction with a contraction function .

Theorem 6. Let be a closed convex and bounded subset of the Banach space and let and be two operators such that (a) is nonlinear contraction,(b) is continuous and compact, and(c) for all . Then the operator equation has a solution in .

3. Fractional Hybrid Differential Equation of the First Type

Consider the fractional hybrid differential equation in the form we have the following Lemma.

Lemma 7. Any function satisfies (10) with will also satisfy the integral equation In addition if the function is injective, and is an absolutely continuous function, then the converse is true.

Proof. Assume that satisfies (10). Then, is absolutely continuous that we get that exists and is Lebesgue integrable on . Applying the fractional integration to both sides of (10) we get (11).
Conversely, assume that satisfies (11) with is absolutely continuous we get that is absolutely continuous. Then by differentiating both sides and then operating by the fractional integration we get and for the initial condition substitute by in (11) we get and since is injective then .
Now consider the fractional hybrid differential equation of first type in the form where, and .

Definition 8. The function is called a mild solution of the hybrid differential equation of first type (14) if it satisfies the integral equation
The function , the space of absolutely continuous real-valued functions defined on , is called a strong solution of (14) if (a)the function is absolutely continuous for each , and(b) satisfies (14).

Theorem 9. Assume the following. (H1)There exists a constant such that for all and .(H2)There exists a function such that a.e. for all .(H3).(H4)There exists such that , where .Then (14) has a mild solution on .

Proof. Let be an open ball centered at the origin and of radius in the Banach algebra (the Banach space of continuous valued functions defined on the interval equipped with the sup-norm, , and with multiplication defined by , for ). We prove the existence of a mild solution to problem (14) by discussing the solution to the integral equation (15) which is equivalent to the operator equation where are defined by
Now we divide our proof in several steps.
Step 1. The operator is Lipschitz on .
Let and ; then by we get Taking the supremum over we get that is Lipschitz on with Lipschitz constant .
Step 2. The operator is continuous operator on .
Let be a convergent sequence in converging to . Then, by the Lebesgue dominated converging theorem, for all with prove the continuity of the operator .
Step 3. The operator is compact operator on .
Let be arbitrary in . By hypothesis (H2) and using Young’s inequality for convolutions we get which by taking the supremum over gives which proves that is a uniformly bounded set in . Now, we prove that is an equicontinuous set in . For we have Hence, for , there exists a such that This shows that is an equicontinuous set in . By the Arzela Ascoli Theorem we get that the operator is a compact operator.
Step 4. , where .
Using results in Step 3 we get which gives from the hypothesis (H3) that It remains to prove that the conclusion (ii) of Theorem 4 cannot be realizable.
Let and be such that and . It follows that Taking supremum over and using (H4) and we get which contradicts ; thus (ii) of Theorem 4 is not possible; hence the operator has a solution in . As a result problem (14) has a mild solution on which completes the proof of our theorem.

We finish this section by the following example.

Example 10. Consider the fractional hybrid differential equation It is easy to see that all hypotheses of Theorem 9 are satisfied with We conclude that hence (28) has a mild solution in .

4. Fractional Hybrid Differential Equation of Second Type

Consider the fractional hybrid differential equation in the form similar to Lemma 7 we can prove

Lemma 11. Any function satisfies (31) with will also satisfy the integral equation In addition if the function is injective, and is an absolutely continuous function then the converse is true.

Now consider the fractional hybrid differential equation of second type in the form where, and .

Definition 12. One has the following functions.(1)The function is called a mild solution of the hybrid differential equation of second type (33) if it satisfies the integral equation (2)The function is called a strong solution of (14) if(a)the function is absolutely continuous for each , and(b) satisfies (33).

Theorem 13. Assume the following. (A1)There exists constants such that for all and .(A2)There exists a function such that a.e. for all .(A3)There exists such that where .Then (33) has a mild solution on .

Proof. Set and define the set by . We prove the existence of a mild solution to problem (33) by discussing the solution to the integral equation (34) which is equivalent to the operator equation where
Now we prove our theorem by proving that the conditions of Theorem 6 are satisfied.(a)Using the hypothesis (A1) we get thus the operator is a nonlinear contraction with the function defined by .(b)Similarly, in proving Theorem 9 we can prove that is continuous and compact.(c)Let be fixed and be arbitrary such that . Then we get which proves that ; thus .
Thus the conditions of Theorem 6 are satisfied; then the operator equation has a solution in which proves the existence of a mild solution to problem (33) in .

We finish this section with the following example.

Example 14. Consider the fractional hybrid differential equation we get that and where we get that all hypotheses of Theorem 13 are satisfied with We conclude that hence (39) has a mild solution in .

5. Conclusions

In this paper we gave definitions of both strong and mild solutions to the fractional hybrid boundary value problems in two types using the Caputo fractional derivative of order and then we discussed the existence of at least one mild solution for each type. We gave examples proving the importance of taking into account of the initial conditions, therefore our results are realistic. We mention that in [14, 15] the initial conditions were neglected.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.