Hyers-Ulam and Hyers-Ulam-Rassias Stability for a Class of Integro-Differential Equations∗

The concept of stability for functional, differential, integral and integro-differential equations has been studied in a quite extensive way during the last six decades and have earned particular interest due to their great number of applications (see [1, 3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 26] and the references therein). This occurs with particular emphasis in the case of Hyers-Ulam and HyersUlam-Rassias stabilities. These stabilities were originated from a famous question raised by S. M. Ulam at the University of Wisconsin in 1940: “When a solution of an equation differing slightly from a given one must be somehow near to the solution of the given equation?” A first partial answer to this question was given by D. H. Hyers, for Banach spaces, in the case of an additive Cauchy equation. This is why the obtained result is nowadays called the Hyers-Ulam stability. Different generalizations of that initial answer of D. H. Hyers were obtained by T. Aoki [2], Z. Gajda [17] and Th. M. Rassias [25]. The interested reader can obtain a detailed description of these advances in [4]. Afterwards, new directions were introduced by Th. M. Rassias, see [24], introducing therefore the so-called Hyers-Ulam-Rassias stability. In this paper, we study the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability for the following class of Volterra integro-differential equation,


Introduction
The concept of stability for functional, differential, integral and integro-differential equations has been studied in a quite extensive way during the last six decades and have earned particular interest due to their great number of applications (see [1,3,5,6,8,9,10,11,12,13,14,15,16,18,19,20,21,22,23,26] and the references therein). This occurs with particular emphasis in the case of Hyers-Ulam and Hyers-Ulam-Rassias stabilities. These stabilities were originated from a famous question raised by S. M. Ulam at the University of Wisconsin in 1940: "When a solution of an equation differing slightly from a given one must be somehow near to the solution of the given equation?" A first partial answer to this question was given by D. H. Hyers, for Banach spaces, in the case of an additive Cauchy equation. This is why the obtained result is nowadays called the Hyers-Ulam stability. Different generalizations of that initial answer of D. H. Hyers were obtained by T. Aoki [2], Z. Gajda [17] and Th. M. Rassias [25]. The interested reader can obtain a detailed description of these advances in [4]. Afterwards, new directions were introduced by Th. M. Rassias, see [24], introducing therefore the so-called Hyers-Ulam-Rassias stability.
In this paper, we study the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability for the following class of Volterra integro-differential equation, y (x) = f x, y(x), x a k(x, τ, y(τ), y(α(τ)))dτ , y(a) = c ∈ R, (1.1) with y ∈ C 1 ([a, b]), for x ∈ [a, b] where, for starting, a and b are fixed real numbers, The formal definition of the above mentioned stabilities are now introduced for our integro-differential equation (1.1).
If for each function y satisfying where θ ≥ 0, there is a solution y 0 of the integro-differential equation and a constant C > 0 independent of y and y 0 such that for all x ∈ [a, b], then we say that the integro-differential equation (1.1) has the Hyers-Ulam stability. If for each function y satisfying where σ is a non-negative function, there is a solution y 0 of the integro-differential equation and a constant C > 0 independent of y and y 0 such that for all x ∈ [a, b], then we say that the integro-differential equation (1.1) has the Hyers-Ulam-Rassias stability. Some of the present techniques to study the stability of functional equations use a combination of fixed point results with a generalized metric in appropriate settings. In view of this, and just for the sake of completeness, let us recall the definition of a generalized metric and the corresponding Banach Fixed Point Theorem. Definition 1. Let X a nonempty set. We say that a function d : X × X → [0, +∞] is a generalized metric on X if: Theorem 1. Let (X, d) be a generalized complete metric space and T : X → X a strictly contractive operator with a Lipschitz constant L < 1. If there exists a nonnegative integer k such that d(T k+1 x, T k x) < ∞ for some x ∈ X, then the following three propositions hold true: 1. the sequence (T n x) n∈N converges to a fixed point x * of T ; 2. x * is the unique fixed point of T in X * = {y ∈ X : d(T k x, y) < ∞}; 3. if y ∈ X * , then (1.6)

Hyers-Ulam-Rassias Stability in the Finite Interval Case
In this section we will present sufficient conditions for the Hyers-Ulam-Rassias stability of the integro-differential equation (1.1), where x ∈ [a, b], for some fixed real numbers a and b.
with M > 0 and the kernel k : and This means that under the above conditions, the integro-differential equation (1.1) has the Hyers-Ulam-Rassias stability.
Proof. By integration we have that is equivalent to So, we will consider the operator T : Note that for any continuous function u, Tu is also continuous. Indeed, Under the present conditions, we will deduce that the operator T is strictly contractive with respect to the metric (1.7). Indeed, for all u, v ∈ C 1 ([a, b]), we have, Due to the fact that M β + Lβ 2 < 1 it follows that T is strictly contractive. Thus, we can apply the above mentioned Banach Fixed Point Theorem, which ensures that we have the Hyers-Ulam-Rassias stability for the integro-differential equation (1.1). Additionally, we can apply again the Banach Fixed Point Theorem, which guarantees us that and consequently (1.13) holds.

Hyers-Ulam Stability in the Finite Interval Case
In this section we will present sufficient conditions for the Hyers-Ulam stability of the integro-differential equation (1.1).
This result can be obtained by using an analogous procedure as in the previous theorem (and so the full details of its proof are here omitted). In particular, we may consider the operator T :

Hyers-Ulam-Rassias Stability in the Infinite Interval Case
In this section, we analyse the Hyers-Ulam-Rassias stability of the integro-differential equation (1.1) but, instead of considering a finite interval [a, b] (with a, b ∈ R), we will consider the infinite interval [a, ∞), for some fixed a ∈ R.
Thus, we will now be dealing with the integro-differential equation is a continuous delay function which therefore fulfills α(τ) ≤ τ for all τ ∈ [a, ∞).
Our strategy will be based on a recurrence procedure due to the already obtained result for the corresponding finite interval case.
Proof. For any n ∈ N, we will define I n = [a, a + n]. By Theorem 2, there exists a unique bounded differentiable function y 0,n : I n → C such that for all x ∈ I n . The uniqueness of y 0,n implies that if x ∈ I n then y 0,n (x) = y 0,n+1 (x) = y 0,n+2 (x) = · · · . (1.41) For any x ∈ [a, ∞), let us define n(x) ∈ N as n(x) = min{n ∈ N : x ∈ I n }. We also define a function y 0 : [a, ∞) → C by For any x 1 ∈ [a, ∞), let n 1 = n(x 1 ). Then x 1 ∈ Int I n 1 +1 and there exists an ε > 0 such that y 0 (x) = y 0,n 1 +1 (x) for all x ∈ (x 1 − ε, x 1 + ε). By Theorem 2, y 0,n 1 +1 is continuous at x 1 , and so it is y 0 . Now, we will prove that y 0 satisfies y 0 (x) = c + Finally, we will prove the uniqueness of y 0 . Let us consider another bounded differentiable function y 1 which satisfies (1.37) and (1.38), for all x ∈ [a, ∞). By the uniqueness of the solution on I n(x) for any n(x) ∈ N we have that y 0|I n(x) = y 0,n(x) and y 1|I n(x) satisfies (1.37) and (1.38) for all x ∈ I n(x) , so
Moreover, by using the exact solution y 0 (x) = e x 2 we realize that for all x ∈ 0, 2 5 . Let us now turn to a second example in which the Hyers-Ulam stability is illustrated.
In particular, having in mind the exact solution y 0 (x) = e x of (1.55), it follows that for all x ∈ [0, 1].