Existence and Continuous Dependence for Fractional Partial Hyperbolic Differential Equations

Abstract

This paper is concerned with a class of fractional hyperbolic partial differential equations with the Caputo derivative. Existence and continuous dependence results of solutions are obtained under the hypothesis of the Lipschitz condition without any restriction on the Lipschitz constant. Examples are discussed to illustrate the results.

1. Introduction

This paper is mainly concerned with the fractional order hyperbolic functional differential equation where , is the standard Caputo fractional derivative of order , is a given function, is a given continuous function, , are given absolutely continuous functions with , for each and , and is the space of continuous functions on .

Fractional differential equations have been widely and efficiently used to describe many phenomena arising in engineering, physics, economy, and science ([1–3]). For example, it was showed by Hilfer that time fractional derivatives are equivalent to infinitesimal generators of generalised time fractional evolution which arise in the transition from microscopic to macroscopic time scales; see, for example, [4, 5]. More applications of fractional derivatives can be found in [6, 7]. For this reason, the theory of differential equations of fractional order has recently received a lot of attention and now constitutes a significant branch of nonlinear analysis. Numerous research papers and monographs have appeared which are devoted to fractional differential equations. See [6, 8–12], for example, and references therein. A special issue on recent advances in fractional differential equations was introduced in [13].

In [1, 14], the authors studied a class of fractional hyperbolic functional differential equations. The existence results were obtained under the Lipschitz conditions, with some additional restrictions on the Lipschitz constants. In this paper, we continue to study fractional hyperbolic functional differential equation with finite delay (1)–(3). We give the existence of solutions under the assumption of Lipschitz condition without any restriction on the Lipschitz constant. As a corollary, existence for corresponding boundary value problem is obtained. Continuous dependence of solutions on the initial data is also studied. Some examples are presented to illustrate our results.

2. Preliminaries

In this section we collect some definitions and results needed in our further investigations.

Definition 1 (see [3, 6]). Let be a fixed number. The Riemann-Liouville fractional integral of order of the function is defined by provided that the right side is pointwise defined, where denotes the gamma function; that is, .

Definition 2 (see [3, 6]). Let be fixed and . The Caputo fractional derivative of order of at the point is defined by provided that the right side is pointwise defined, where denotes the integer part of the real number .

If , then

We list some basic properties of fractional integral and derivative without proof. For more details, we refer the readers to [3, 6].

Lemma 3 (see [3]). (1) For and , the identity holds for and some constants . If additionally , then for .
(2) If and , then holds almost everywhere for . If additionally is continuous on , then the identity holds everywhere for .
(3) If and , then (4) If and , then for , where and denotes the space of the function such that is absolutely continuous on .

Definition 4 (see [1, 14]). Let and . The partial Riemann-Liouville integral of order of with respect to is defined by the expression

Analogously, we define the integral

Definition 5 (see [1, 14]). Let and . The left-side mixed Riemann-Liouville integral of order of is defined by

In particular, where .

By we mean . Denote by the mixed second-order partial derivative.

Definition 6 (see [1, 14]). Let and . The Caputo mixed fractional order derivative of order of is defined by the expression ; that is,

The case is included and we have

In the sequel we need the following lemma to transform partial fractional differential equations into integral equations. We denote by the space of absolutely continuous functions defined on .

Lemma 7 (see [1, 14]). A function such that its mixed derivative exists and is integrable on is a solution of problem if and only if satisfies where

We also need the following generalization of Gronwall’s inequality for two independent variables and singular kernel.

Lemma 8 (see [1, 15]). Let be a real function and a nonnegative, locally integrable function on . If there are constants and such that then there exists a constant such that

3. Main Results

We begin with the definition of solutions to problem (1)–(3).

Definition 9. A function is said to be a solution to (1)–(3), if its mixed derivative exists and is integrable, (2), (3) are satisfied, and for , where is the function given by (20).

We first give an existence and uniqueness result based on the Banach contraction principle.

Theorem 10. Assume that is continuous and is Lipschitz continuous with respect to the last variable; that is, there is such that for any and . Then there exists a unique solution to (1)–(3) on .

Proof. Let and , where denote the integer part of a real number . Then , and for .
We first focus on . Let us define a subset which contains those functions that for and the restriction of on is continuous. It is easily seen that is closed in , and can also be considered as a closed subset in . We define an operator by One can easily verify that is well-defined and maps into itself. Further, for any and every , we haveNoting that for and , we have It follows that Take the supremum for ; we get that From (25) we know that is a contraction on . An application of Banach contraction principle yields a unique fixed point of in , which is the unique solution of (1)–(3) on . We denote by the restriction of this unique solution on .
Next we consider the area . We define a closed subset which contains those functions that for , for and the restriction of on is continuous. Let us define an operator, also denoted by , from into itself as where Since for is known, is a known function. Using the same arguments as above we obtain that there is a unique fixed point of on . We denote by the restriction of this unique function on .
Take the next area and, repeating this process, we obtain defined on , respectively. Define a function as It is easy to check that is the unique solution of (1)–(3) on , and this completes the proof.

Remark 11. In Theorem 10 there is no restriction to the Lipschitz constant, from which it seems that the assumption in [1, Theorem  3.2] may not be necessary.

Using the same argument as in Theorem 10, we can prove an existence result for the following initial value problem without delay: where is a given function and , are given absolutely continuous functions with .

Theorem 12. Assume that is continuous and is Lipschitz continuous with respect to the last variable; that is, there is such that for any and . Then there exists a unique solution to (34) on .

Next we prove a continuous dependent result with respect to the order of the fractional derivative. For simplicity, we consider the case that tends to monotonically, that is, and simultaneously. We denote this case by .

Theorem 13. Let be the solution of problem (1)–(3) and the solution of the problem Suppose that the hypotheses of Theorem 10 hold. Then

Proof. Existence and uniqueness of the solutions can be deduced by Theorem 10. Let It follows from the continuity of , and on closed bounded domains that . By Definition 9, and can be expressed as the form of (23). Let be given and . Then, for , we have Noting that for , it follows that Since is fixed and is sufficiently small, we can find such that . Based on the fact that gamma function is continuously differentiable on and is bounded on , we can find a constant such that Further, we can find constant such that It then follows from (39)–(42) that where Define a function as Take supremun on both sides of (43); we get that An application of Lemma 7 yields that where is a constant dependent on and . This concludes the proof.

4. Examples

In this section we discuss some examples as applications of our results. We first consider the partial hyperbolic functional differential equations of the form

and we set Then for each and , we have that is, is Lipschitz continuous with respect to the last variable with Lipschitz constant . We derive from Theorem 10 that problem (48) has a unique solution defined on . A similar example is discussed on page 32 in [1]. We indicate that the restriction on the Lipschitz constant there is not necessary.

We now consider the boundary value problem where can be any given continuous function defined on . The solution to (51) has the form Since a continuous function is bounded on a closed bounded region, it is easy to see that the assumption of Theorem 12 holds. Consequently, Theorem 12 implies that problem (51) has a unique solution on .

Conflict of Interests

The authors declare that they have no competing interests.

Acknowledgments

The authors are grateful to the reviewers for a painstaking reading of this paper and for their valuable suggestions on improving this paper. This work was supported by the National Natural Science Foundation of China (11271316, 11571300, and 11201410) and the Natural Science Foundation of Jiangsu Province (BK2012260).