On a System of Volterra Type Hadamard Fractional Integral Equations in Fréchet Spaces

Abstract

In this article we present some results concerning the existence of solutions for a system of Hadamard integral equations. Our investigation is conducted with an application of an extension of the fixed point theorem of Burton-Kirk in Fréchet spaces.

1. Introduction

In recent years, fractional differential and integral equations are emerging as a useful tool in modeling of the dynamics of many physical systems and electrical phenomena, which has been demonstrated by many researchers in the fields of mathematics, science, and engineering; see monographs [1–7] and a series of papers [8–18] and the references cited therein. Some properties of the Hadamard fractional calculus were investigated by Butzer et al. [19, 20]. Additional papers can be found in [7, 21] and the references therein. Recently, some interesting results on the existence of the solutions of some classes of integral equations have been obtained by Abbas et al. [22–24], Banaś et al. [25, 26], and Pachpatte [27, 28] and the references therein.

This paper deals with the existence of solutions of the following system of Hadamard fractional integral equations of the form That is,where , , , , is the Hadamard integral of order , are given continuous functions, , and is the (Euler’s) gamma function defined by

2. Preliminaries

Let ; for , be the space of Lebesgue-integrable functions with the standard norm

Definition 1 (see [4]). The Hadamard fractional integral of order for a function is defined as

Example 2. The Hadamard fractional integral of order for the function , defined by with , is

Definition 3. Let , , , and For , define the Hadamard partial fractional integral of order by the expression

Let be a Fréchet space with a family of seminorms such that A subset of is bounded if, for every , there exists such that We associate a sequence of Banach spaces to as follows: For every , we consider the equivalence relation defined by if and only if for We denote by the quotient space, the completion of with respect to . To every , we associate a sequence of subsets as follows: For every , we denote by the equivalence class of of subset and we defined We denote by , , and , respectively, the closure, the interior, and the boundary of with respect to in For more details, we refer the reader to [29].

Definition 4. Let be a Fréchet space. A function is said to be a contraction if for each there exists such that

We need the following extension of the Burton-Kirk fixed point theorem in the case of a Fréchet space.

Theorem 5 (see [30]). Let be a Fréchet space and let be two operators such that (a) is a compact operator(b) is a contraction operator with respect to a family of seminorms (c)the set is bounded Then the operator equation has a solution in

3. Existence Results

Set

Definition 6. A function is said to be a solution of (2) if satisfies (2) on

For each we consider the set , and we define in the seminorms by Then is a Fréchet space with the family of seminorms

Also, the product space is a Fréchet space with the family of seminorms

Now, we are concerned with the existence of solutions for system (2). Let us introduce the following hypotheses:There exist continuous functions , with such that  for each and each There exist continuous functions ; , , and nondecreasing functions such that  for , ; , and  Moreover, assume that for each with

For any , set and

Theorem 7. Assume that hypotheses and hold. Ifthen system (2) has at least one solution in the space

Proof. Define the operators ; , by and consider the continuous operators defined byWe shall show that operators and satisfied all the conditions of Theorem 5.
The proof will be given in several steps.
Step 1 ( is compact). To this aim, we must prove that is continuous and it transforms every bounded set into a relatively compact set. Recall that is bounded if and only if and is relatively compact if and only if, , the family is equicontinuous and uniformly bounded on The proof will be given in several claims.
Claim 1 ( is continuous). Let be a sequence in such that in Then, for each and , we haveSince and is continuous, then for each ; , and , (27) gives Claim 2 ( maps bounded sets into bounded sets in ). Let be the bounded set in as in Claim 1. Then, for each , there exists , such that for all we have
For arbitrarily fixed and each , we have whereHenceClaim 3 ( maps bounded sets into equicontinuous sets in C). Let , , , and let ; thus for each , we have From the continuity of functions and as and , the right-hand side of the above inequality tends to zero. As a consequence of Claims 1–3 and from the Arzelá-Ascoli theorem, we can conclude that is continuous and compact.
Step 2 ( is a contraction). Consider Then, by , for any and each and , we have Thus, Hence, Inequality (21) implies that is a contraction operator.
Step 3 (the set is bounded). Let , such that , for some Then, for any and each and , we have where Thus, Hence, the set is bounded. As a consequence of Steps 1–3 and from Theorem 5, we deduce that has a fixed point which is a solution of system (2).

4. An Example

Consider the following coupled system of Hadamard fractional integral equations of the formwhere ; , , and and Set The functions ; are continuous and satisfy assumption , with Also, the functions are continuous and satisfy assumption , with Then, Finally, we shall show that condition (21) holds with Indeed, for each , we get Hence by Theorem 7, system (39) has a solution defined on .

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.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (No. 11671339).