Existence and Stability of Solutions for Hadamard-Stieltjes Fractional Integral Equations

Abstract

We give some existence results and Ulam stability results for a class of Hadamard-Stieltjes integral equations. We present two results: the first one is an existence result based on Schauder’s fixed point theorem and the second one is about the generalized Ulam-Hyers-Rassias stability.

1. Introduction

Fractional differential and integral equations have recently been applied in various areas of engineering, mathematics, physics, bioengineering, and other applied sciences [1, 2]. There has been a significant development in ordinary and partial fractional differential and integral equations in recent years; see the excellent classical monograph of Kilbas et al. [3] or the recent monograph of Abbas et al. [4].

The stability of functional equations was originally raised by Ulam in 1940 in a talk given at Wisconsin University. The problem posed by Ulam was the following: under what conditions does there exist an additive mapping near an approximately additive mapping? (for more details see [5]). The first answer to Ulam’s question was given by Hyers in 1941 in the case of Banach spaces in [6]. Thereafter, this type of stability is called the Ulam-Hyers stability. In 1978, Rassias [7] provided a remarkable generalization of the Ulam-Hyers stability of mappings by considering variables. The concept of stability for a functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus, the stability question of functional equations is how do the solutions of the inequality differ from those of the given functional equation? Considerable attention has been given to the study of the Ulam-Hyers and Ulam-Hyers-Rassias stability of all kinds of functional equations; one can see the monographs of [8, 9]. Bota-Boriceanu and Petrusel [10], Petru et al. [11], and Rus [12, 13] discussed the Ulam-Hyers stability for operatorial equations and inclusions. Castro and Ramos [14], and Jung [15] considered the Hyers-Ulam-Rassias stability for a class of Volterra integral equations. More details from historical point of view and recent developments of such stabilities are reported in [12, 16].

In [17], Butzer et al. investigate properties of the Hadamard fractional integral and the derivative. In [18], they obtained the Mellin transforms of the Hadamard fractional integral and differential operators and in [19], Pooseh et al. obtained expansion formulas of the Hadamard operators in terms of integer order derivatives. Many other interesting properties of those operators are summarized in [20] and the references therein.

This paper deals with the existence of the Ulam stability of solutions to the following Hadamard-Stieltjes fractional integral equation: where , , and , , , are given continuous functions, and is the Euler gamma function.

Our investigations are conducted with an application of Schauder’s fixed point theorem for the existence of solutions of the integral equation (1). Also, we obtain some results about the generalized Ulam-Hyers-Rassias stability of solutions of (1). Finally, we present an example illustrating the applicability of the imposed conditions.

This paper initiates the study of the existence and the Ulam stability of such class of integral equations.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Denote by the Banach space of functions that are Lebesgue integrable with norm Let be the Banach space of all continuous functions with the norm

Definition 1 (see [3, 21]). The Hadamard fractional integral of order for a function is defined as

Definition 2. Let , , and . For , define the Hadamard partial fractional integral of order by the expression

If is a real function defined on the interval , then the symbol denotes the variation of on . We say that is of bounded variation on the interval whenever is finite. If , then the symbol indicates the variation of the function on the interval , where is arbitrarily fixed in . In the same way we define . For the properties of functions of bounded variation we refer to [22].

If and are two real functions defined on the interval , then under some conditions (see [22]) we can define the Stieltjes integral (in the Riemann-Stieltjes sense) of the function with respect to In this case we say that is Stieltjes integrable on with respect to . Several conditions are known guaranteeing Stieltjes integrability [22]. One of the most frequently used requirements are that is continuous and is of bounded variation on .

In what follows we use the following properties of the Stieltjes integral ([23], section 8.13).

If is Stieltjes integrable on the interval with respect to a function of bounded variation, then If and are Stieltjes integrable functions on the interval with respect to a nondecreasing function such that for , then

In the sequel we consider Stieltjes integrals of the form and Hadamard-Stieltjes integrals of fractional order of the form where ,  , and the symbol indicates the integration with respect to .

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

Now, we consider the Ulam stability for the integral equation (1). Consider the operator defined by Clearly, the fixed points of the operator are solution of the integral equation (1). Let and be a continuous function. We consider the following inequalities:

Definition 4 (see [12, 24]). Equation (1) is Ulam-Hyers stable if there exists a real number such that for each and for each solution of the inequality (13) there exists a solution of (1) with

Definition 5 (see [12, 24]). Equation (1) is generalized Ulam-Hyers stable if there exists with such that for each and for each solution of the inequality (13) there exists a solution of (1) with

Definition 6 (see [12, 24]). Equation (1) is Ulam-Hyers-Rassias stable with respect to if there exists a real number such that for each and for each solution of the inequality (15) there exists a solution of (1) with

Definition 7 (see [12, 24]). Equation (1) is generalized Ulam-Hyers-Rassias stable with respect to if there exists a real number such that for each solution of the inequality (14) there exists a solution of (1) with .

Remark 8. It is clear that (i) Definition 4   Definition 5, (ii) Definition 6   Definition 7, and (iii) Definition 6 for    Definition 4.

One can have similar remarks for the inequalities (13) and (15).

3. Existence and Ulam Stabilities Results

In this section, we discuss the existence of solutions and we present conditions for the Ulam stability for the Hadamard integral equation (1).

The following hypotheses will be used in the sequel.()There exist functions such that, for any and ,  with ()For all such that , the function is nondecreasing on . Also, for all such that , the function is nondecreasing on .()The functions and are nondecreasing on or , respectively.()The functions and are continuous on for each fixed or , respectively. Also, the functions and are continuous on for each fixed or , respectively.()There exists such that, for each , we have Set

Theorem 9. Assume that the hypotheses hold. Then the integral equation (1) has a solution defined on

Proof. Let be a constant such that We will use Schauder’s theorem [25], to prove that the operator defined in (12) has a fixed point. The proof will be given in four steps.
Step 1 ( transforms the ball into itself). For any and each , we have Thus, . This implies that transforms the ball into itself.
Step 2 ( is continuous). Let be a sequence such that in . Then From Lebesgue’s dominated convergence theorem and the continuity of the function , we get Step 3 ( is bounded). This is clear since and is bounded.
Step 4 ( is equicontinuous). Let ,  ,  . Then Thus, we obtain Hence, we get As and , the right-hand side of the above inequality tends to zero.
As a consequence of Steps 1 to 4 together with the Arzelá-Ascoli theorem, we can conclude that is continuous and compact. From an application of Schauder’s theorem [25], we deduce that has a fixed point which is a solution of the integral equation (1).

Now, we are concerned with the stability of solutions for the integral equation (1).

Theorem 10. Assume that hold. Furthermore, suppose that there exists , such that, for each , we have Then the integral equation (1) is generalized Ulam-Hyers-Rassias stable.

Proof. Let be a solution of the inequality (14). By Theorem 9, there exists which is a solution of the integral equation (1). Hence By the inequality (14) for each , we have Set For each , we have Hence the integral equation (1) is generalized Ulam-Hyers-Rassias stable.

4. An Example

As an application of our results we consider the following Hadamard-Stieltjes integral equation where The condition is satisfied with and . We can see that the functions and satisfy . Consequently Theorem 9 implies that the Hadamard integral equation (35) has a solution defined on .

Also, the hypothesis is satisfied with Indeed, for each we get Consequently, Theorem 10 implies that (35) is generalized Ulam-Hyers-Rassias stable.

Conflict of Interests

The authors declare no conflict of interests.

Acknowledgments

The work of J. J. Nieto has been partially supported by the Ministerio de Economia y Competitividad of Spain under Grant MTM2013–43014–P and Xunta de Galicia under Grant R2014/002 and cofinanced by the European Community Fund FEDER.