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

1Laboratory of Mathematics, University of Saı̈da, P.O. Box 138, 20000 Saı̈da, Algeria 2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia 3Laboratory of Mathematics, University of Sidi Bel-Abbès, P.O. Box 89, 22000 Sidi Bel-Abbès, Algeria 4Department of Mathematical Analysis, Faculty of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain


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 R are given continuous functions, and Γ(⋅) 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.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.Denote by  1 (, R) the Banach space of functions  :  → R that are Lebesgue integrable with norm Let  := (, R) be the Banach space of all continuous functions  :  → R with the norm Definition 1 (see [3,21]).The Hadamard fractional integral of order  > 0 for a function  ∈  1 ([1, ], R) is defined as Definition 2. Let  1 ,  2 ≥ 0,  = (1, 1), and  = ( 1 ,  2 ).For  ∈  1 (, R), define the Hadamard partial fractional integral of order  by the expression 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) If  and V are Stieltjes integrable functions on the interval [, ] with respect to a nondecreasing function  such that 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 .
, and  = ( 1 ,  2 ).For  ∈  1 (, R), 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 Definition 4 (see [12,24]).Equation ( 1) is Ulam-Hyers stable if there exists a real number   > 0 such that for each  > 0 and for each solution  ∈  of the inequality (13) there exists a solution V ∈  of (1) with Definition 5 (see [12,24]).Equation ( 1) is generalized Ulam-Hyers stable if there exists   : ([0, ∞), [0, ∞)) with   (0) = 0 such that for each  > 0 and for each solution  ∈ C of the inequality ( 13) there exists a solution V ∈  of (1) with Definition 6 (see [12,24]).Equation ( 1) is Ulam-Hyers-Rassias stable with respect to Φ if there exists a real number  ,Φ > 0 such that for each  > 0 and for each solution  ∈  of the inequality ( 15) there exists a solution V ∈  of (1) with Definition 7 (see [12,24]).Equation ( 1) is generalized Ulam-Hyers-Rassias stable with respect to Φ if there exists a real number  ,Φ > 0 such that for each solution  ∈  of the inequality ( 14 One can have similar remarks for the inequalities ( 13) and (15).

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.
Proof.Let  > 0 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 From Lebesgue's dominated convergence theorem and the continuity of the function , we get     (  ) (, ) − () (, )     → 0 as  → ∞. (26) Step 3 ((  ) is bounded).This is clear since (  ) ⊂   and   is bounded.
Step 4 Thus, we obtain 1 ( As  1 →  2 and  1 →  2 , 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).
Proof.Let  be a solution of the inequality (14).By Theorem 9, there exists V which is a solution of the integral equation (1).Hence Hence the integral equation ( 1) is generalized Ulam-Hyers-Rassias stable.

An Example
As an application of our results we consider the following where Consequently, Theorem 10 implies that (35) is generalized Ulam-Hyers-Rassias stable.