Ulam stabilities for partial Hadamard fractional integral equations

This paper deals with some existence and Ulam stability results for a class of partial integral equations via Hadamard’s fractional integral, by applying Schauder’s fixed-point theorem.

Many other interesting properties of those operators and others are summarized in [26] and the references therein.
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 [28]). The first answer to Ulam's question was given by Hyers in 1941 in the case of Banach spaces in [14]. Thereafter, this type of stability is called the Ulam-Hyers stability. In 1978, Rassias [23] 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 [15,16]. Bota-Boriceanu and Petrusel [7], Petru et al. [21], and Rus [24,25] discussed the Ulam-Hyers stability for operatorial equations and inclusions. Castro and Ramos [10] and Jung [18] 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 [17,24]. This paper deals with the existence for the Ulam stability of solutions to the following Hadamard partial fractional integral equation of the form u(x, y) = μ(x, y) where We present two results for the integral equation (1). The first one is based on Banach's contraction principle and the second one on the nonlinear alternative of Leray-Schauder type.
The present paper initiates the Ulam stability for integral equations involving the Hadamard fractional integral.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper. Denote by C := C(J, R) the Banach space of continuous functions u : J → R with the norm R) the Banach space of functions u : J → R that are Lebesgue integrable with norm Definition 2.1 [12,19] The Hadamard fractional integral of order q > 0 for a function g ∈ L 1 ([1, a], R), is defined as where (·) is the Euler gamma function. Definition 2.2 Let r 1 , r 2 ≥ 0, σ = (1, 1) and r = (r 1 , r 2 ). For w ∈ L 1 (J, R), define the Hadamard partial fractional integral of order r by the expression Now, we consider the Ulam stability for the integral equation (1). Consider the operator N : C → C defined by: Clearly, the fixed points of the operator N are solution of the integral equation (1). Let > 0 and : J → [0, ∞) be a continuous function. We consider the following inequalities (1) is Ulam-Hyers stable if there exists a real number c N > 0 such that for each > 0 and for each solution u ∈ C of the inequality (3) there exists a solution v ∈ C of Eq. (1) with

Definition 2.3 [3,24] Equation
Definition 2.4 [3,24] Equation (1) is generalized Ulam-Hyers stable if there exists c N : Definition 2.5 [3,24] Equation (1) is Ulam-Hyers-Rassias stable with respect to if there exists a real number c N , > 0 such that for each > 0 and for each solution u ∈ C of the inequality (5) there exists a solution v ∈ C of Eq. (1) with Definition 2.6 [3,24] Equation (1) is generalized Ulam-Hyers-Rassias stable with respect to if there exists a real number c N , > 0 such that for each solution u ∈ C of the inequality (4) (3) and (5).

Existence and Ulam stabilities results
In this section, we discuss the existence of solutions and present conditions for the Ulam stability for the Hadamard integral equation (1).
The following hypotheses will be used in the sequel. (H 1 ) There exist functions p 1 , p 2 ∈ C(J, R + ) such that for any u ∈ R and (x, y) ∈ J, (H 2 ) There exists λ > 0 such that for each (x, y) ∈ J, we have then the integral equation (1) has a solution defined on J.
Proof Let ρ > 0 be a constant such that We shall use Schauder's theorem [11], to prove that the operator N defined in (2) has a fixed point. The proof will be given in four steps.
Step 1: N transforms the ball B ρ := {u ∈ C : u C ≤ ρ} into itself. For any u ∈ B ρ and each (x, y) ∈ J, we have Thus, by (6) and the definition of ρ we get (N u) C ≤ ρ. This implies that N transforms the ball B ρ into itself.
From Lebesgue's dominated convergence theorem and the continuity of the function f, we get Step 3: N (B ρ ) is bounded. This is clear since N (B ρ ) ⊂ B ρ and B ρ is bounded.
Step 4: t, u(s, t))| st dtds t, u(s, t))| st dtds As x 1 → x 2 and y 1 → y 2 , the right-hand side of the above inequality tends to zero.
As a consequence of steps 1-4 together with the Arzelá-Ascoli theorem, we can conclude that N is continuous and compact. From an application of Schauder's theorem [11], we deduce that N has a fixed point u which is a solution of the integral equation (1). Now, we are concerned with the stability of solutions for the integral equation (1).

Theorem 3.2
Assume that (H 1 ), (H 2 ) and the condition (6) hold. Furthermore, suppose that there exist q i ∈ C(J, R + ); i = 1, 2 such that for each (x, y) ∈ J we have

Then, the integral equation (1) is generalized Ulam-Hyers-Rassias stable.
Proof Let u be a solution of the inequality (4). By Theorem 3.1, there exists v which is a solution of the integral equation (1). Hence By the inequality (4) for each (x, y) ∈ J, we have Set For each (x, y) ∈ J, we have N , (x, y).