Semi - Hyers - Ulam - Rassias stability for an integro - di ﬀ erential equation of order (cid:2)

: The Laplace transform method is applied in this article to study the semi - Hyers - Ulam - Rassias stability of a Volterra integro - di ﬀ erential equation of order n, with convolution - type kernel. This kind of stability extends the original Hyers - Ulam stability whose study originated in 1940. A general integral equation is formulated ﬁ rst, and then some particular cases ( polynomial function and exponential func - tion ) for the function from the kernel are considered.


Introduction
An important aspect from the qualitative theory of differential and integral equations is the stability of solutions.It is well known that this concept has various meanings in the literature, one of them being the Hyers-Ulam stability.This notion appeared first in connection with homomorphisms, in an open problem formulated by Ulam (see the book [1]), and answered by Hyers in [2].A vast field of research developed afterward, with many authors generalizing the initial definition of stability (Hyers-Ulam-Rassias stability, generalized Hyers-Ulam-Rassias stability, semi-Hyers-Ulam-Rassias stability, and Mittag-Leffler-Hyers-Ulam stability) and obtaining results for various classes of functional equations (see, for instance, [3][4][5][6][7] and the references therein).Systematic approaches of the subject are given in [8,9].
The present work is about a class of Volterra integro-differential equations.The study of Hyers-Ulam stability for integral or integro-differential equations is not so extended as in the case of differential equations, but we can mention several papers approaching this problem, by various methods: [26][27][28][29][30].
In this article, we use the Laplace transform method.In the context of Hyers-Ulam stability, this method appeared first for linear differential equations, in the article by Rezaei et al. [31].Then, the Laplace transform was used to obtain stability results in several other articles: [32] for linear differential equations, [33] for Laguerre differential equation and Bessel differential equation, [34] for the Mittag-Leffler-Hyers-Ulam stability of a linear differential equation of first order, [35] for fractional differential equations, and [24] for the convection partial differential equation.
In the following, inspired by [31] and continuing the research from [36], we will study a Volterra integro-differential equation of order n with a convolution type kernel: In the previous article [36], we obtained stability results for the integro-differential equation (1.1) of order I.Those results will be extended here for a class of equations of order n.The outline of this article is the following: In Section 2, we present the stability notion, properties of the Laplace transform, and some auxiliary results.The main results are given in Section 3 and concern the semi-stability of the integrodifferential equation (1.1), for some particular cases of the function g .

Preliminary notions and results
In the rest of the article, we denote by s R( ) the real part of a complex number s and by the real field or the complex field .Throughout the work, we assume that the functions f g x , , : 0, ( ) ∞ → are contin- uous and of the exponential order.
The Laplace transform of the function f is denoted by f ( ) and is defined by , where σ f is the abscissa of convergence of the function f .The inverse Laplace transform of a function F is denoted by F 1 ( ) − .In the next section of the article, we will use the following auxiliary results, proved in [31]., are nonnegative integers with m n < and a b , Then there exists an infinitely differentiable function g : 0, where σ s P s max : 0 Lemma 2.2.[31] Given an integer n 1 > , let f : 0, ( ) ∞ → be a continuous function and let P s ( ) be a complex polynomial of degree n.Then there exists an n times differentiable function h : 0, ( ) ∞ → such that where σ s P s max : 0 and σ f is the abscissa of convergence for f .It holds that h 0 0 k ( ) In the rest of the article, we write x x x 0 , 0 , , 0 x 0 n ( ) ( ) + , respectively.Let ε 0 > and consider the inequality ∞ .We say, as in [27], that equation (1.1) is semi-Hyers-Ulam-Rassias stable, if there exists a real number c 0 > and a function k : 0, 0, ∞ such that for each x that verifies the inequality (2.1), there exists a solution x 0 of the equation (1.1) with

Stability results
We prove the stability results for the solution of equation (1.1) in some particular cases for the function g.
Then, for every function x : 0, ( ) ∞ → satisfying the inequality (2.1), for all t 0, ( ) ∈ ∞ and some ε 0 > , there exists a solution x : 0, 0 ( ) , where The Laplace transform of the function p is we can write We obtain Semi-Hyers-Ulam-Rassias stability for an integro-differential equation  3 , and let Let σ f be the abscissa of convergence for f .By using Lemma 2.1, it follows that there exists an infinitely differentiable function h i such that for every s with s σ max 0, R( ) { } > , and By Lemma 2.1, there exists an infinitely differentiable function z such that for every s with s σ max 0, , R( and from here f 0 0. 0 ( ) = Also and from here, f 0 0. 0 ( ) and from here, for every s with s σσ max 0, , f R( ) { } > .We obtain Since the Laplace transform of the function z : 0, we obtain Further on, from the formula for the inverse Laplace transform follows Semi-Hyers-Ulam-Rassias stability for an integro-differential equation  5 Let us consider m 0 = , n 2 = and the following equation: Consider also ε 0 > and the inequality By applying Theorem 3.1, we obtain that for every function x : 0, ( ) ∞ → satisfying the inequality (3.4), there exists a solution x : 0, 0 ( ) , for all max 0, 0, As in [31], a Corollary of Theorem 3.1 can be formulated.
As a function of α, the expression and σ is defined in (3.5).
Proof.In the same way and with the same notation as in Theorem 3.1, we obtain ,0 Using again Lemma 2.1, there exists an infinitely differentiable function z such that z s ω s ω P s In the same way as in Theorem 3.1, we have that f to Corollary 3.1 can be also formulated in this case.
When the function g is an exponential function, we obtain the following result.