Stabilities for a class of higher order integro-differential equations

This work is devoted to analyse different kinds of stabilities for higher order integro-differential equations within appropriate metric spaces. We will consider the σ-semi-Hyers-Ulam stability which is a new kind of stability somehow between the Hyers-Ulam and the Hyers-Ulam-Rassias stabilities. Sufficient conditions are obtained in view to guarantee Hyers-Ulam, σ-semiHyers-Ulam and Hyers-Ulam-Rassias stabilities for such a class of integro-differential equations. We will be considering finite and infinite intervals as integration domains. Among the used techniques, we have fixed point arguments and generalizations of the Bielecki metric. INTRODUCTION The study of properties associated with stability for functional, differential, integral and integro-differential has been widely diffused and has been a subject of great interest in the last seven decades earning particular interest due to their great number of applications in elasticity, semiconductors, heat conduction, fluid flow, scattering theory, chemical reactions and population dynamic, and others (see [1, 3, 6, 8, 9, 10, 11, 12, 16, 17, 18, 20, 21, 22, 23, 24]). In part, this is because of their potential applicability in mathematical models where we cannot expect to easily obtain the exact solution of the problem but we may expect to obtain an approximate solution which should be stable in a certain specific sense. Hyers-Ulam and Hyers-Ulam-Rassias are among the most studied types of stabilities. These two stabilities were originated from a famous question raised by S. M. Ulam [28] at the University of Wisconsin in 1940, about to know 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 parcial answer to this question was given by D. H. Hyers, for Banach spaces, in the case of the additive Cauchy equation f (x+y) = f (x)+ f (y). It is for this reason that the obtained result is nowadays called the Hyers-Ulam stability. After that preliminary answer, other approaches emerged, and new orientations were introduced by Th. M. Rassias, see [25], introducing therefore the so-called Hyers-Ulam-Rassias stability. Different generalizations were obtained by other researchers, by considering the possibility of using different involved norms and others types of equations, in particular by Aoki [2], Gajda [19], and Rassias [26]. The interested reader can obtain a detailed description of these advances in [4]. In this paper, we study the Hyers-Ulam stability, the σ-semi-Hyers-Ulam stability and the Hyers-Ulam-Rassias stability for the following class of higher order integro-differential equation, y(n)(x) = f ( x, y(x), ∫ x a k(x, τ, y(τ), y(α(τ)))dτ ) , y( j)(a) = 0, j = 0, . . . , n − 1 (1) with y ∈ Cn([a, b]), for x ∈ [a, b] where, for starting, a and b are fixed real numbers, f : [a, b] × C × C → C and k : [a, b] × [a, b] × C × C → C are continuous functions, and α : [a, b] → [a, b] is a continuous delay function which fulfills α(τ) ≤ τ for all τ ∈ [a, b]. ICNPAA 2018 World Congress AIP Conf. Proc. 2046, 020012-1–020012-10; https://doi.org/10.1063/1.5081532 Published by AIP Publishing. 978-0-7354-1772-4/$30.00 020012-1 The formal definition of the above mentioned stabilities are now introduced for the higher order integro-differential equation (1). If for each continuously differentiable function y satisfying ∣∣∣∣y(n)(x) − f (x, y(x), ∫ x a k(x, τ, y(τ), y(α(τ)))dτ)∣∣∣∣ ≤ σ(x), x ∈ [a, b], where σ is a non-negative function, there is a solution y0 of the higher order integro-differential equation and a constant C > 0 independent of y and y0 such that |y(x) − y0(x)| ≤ C σ(x), for all x ∈ [a, b], then we say that the higher order integro-differential equation (1) has the Hyers-Ulam-Rassias stability. If for each continuously differentiable function y satisfying ∣∣∣∣y(n)(x) − f (x, y(x), ∫ x a k(x, τ, y(τ), y(α(τ)))dτ)∣∣∣∣ ≤ θ, x ∈ [a, b], where θ ≥ 0, there is a solution y0 of the higher order integro-differential equation and a constant C > 0 independent of y and y0 such that |y(x) − y0(x)| ≤ C θ, for all x ∈ [a, b], then we say that the higher order integrodifferential equation has the Hyers-Ulam stability. We also use a stability in-between the two just mentioned stabilities of Hyers-Ulam-Rassias and Hyers-Ulam, introduced in [14], in the following way: Definition 1 Let σ a non-decreasing function defined on [a, b]. If for each continuously differentiable function y satisfying ∣∣∣∣y(n)(x) − f (x, y(x), ∫ x a k(x, τ, y(τ), y(α(τ)))dτ)∣∣∣∣ ≤ θ, x ∈ [a, b], where θ ≥ 0, there is a solution y0 of the higher order integro-differential equation (1) and a constant C > 0 independent of y and y0 such that |y(x) − y0(x)| ≤ C σ(x), x ∈ [a, b], then we say that the higher order integro-differential equation (1) has the σ-semi-Hyers-Ulam stability. Some of the usual techniques to study the stability of functional, integral and integro-differential equations are a combination of fixed point results with a generalized metric in appropriate settings (cf. [5, 8, 9, 10, 11, 12, 13, 15]). 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 2 Let X a nonempty set. A function d : X × X → [0,+∞] is called a generalized metric on X if and only if d satisfies: (i) d(x, y) = 0 if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; (iii) d(x, z) ≤ d(x, y) + d(y, z) for all x, y, z ∈ X. Theorem 3 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+1x,T k x) < ∞ for some x ∈ X, then the following three propositions hold true: (i) the sequence (T x)n∈N converges to a fixed point x∗ of T; (ii) x∗ is the unique fixed point of T in X∗ = {y ∈ X : d(T k x, y) < ∞}; (iii) if y ∈ X∗, then d(y, x∗) ≤ 1 1 − L d(Ty, y). (2) 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 higher order integrodifferential equation (1), where x ∈ [a, b], for some fixed real numbers a and b. We will consider the space of continuously differentiable functions Cn([a, b]) on [a, b] endowed with a generalization of the Bielecki metric d(u, v) = sup x∈[a,b] |u(x) − v(x)| σ(x) , (3) where σ is a non-decreasing continuous function σ : [a, b]→ (0,∞). If instead of σ in (3) we have ep( · −a) with p > 0 then we have the well-known Bielecki metric. We recall that (Cn([a, b]), d) is a complete metric space (cf., [7], [27]). Theorem 4 Let α : [a, b] → [a, b] be a continuous delay function with α(t) ≤ t for all t ∈ [a, b] and σ : [a, b] → (0,∞) a non-decreasing continuous function. In addition, suppose that there is β ∈ R such that ∫ x a σ(τ)dτ ≤ βσ(x), (4)


INTRODUCTION
The study of properties associated with stability for functional, differential, integral and integro-differential has been widely diffused and has been a subject of great interest in the last seven decades earning particular interest due to their great number of applications in elasticity, semiconductors, heat conduction, fluid flow, scattering theory, chemical reactions and population dynamic, and others (see [1,3,6,8,9,10,11,12,16,17,18,20,21,22,23,24]). In part, this is because of their potential applicability in mathematical models where we cannot expect to easily obtain the exact solution of the problem but we may expect to obtain an approximate solution which should be stable in a certain specific sense. Hyers-Ulam and Hyers-Ulam-Rassias are among the most studied types of stabilities. These two stabilities were originated from a famous question raised by S. M. Ulam [28] at the University of Wisconsin in 1940, about to know 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 parcial answer to this question was given by D. H. Hyers, for Banach spaces, in the case of the additive Cauchy equation f (x +y) = f (x)+ f (y). It is for this reason that the obtained result is nowadays called the Hyers-Ulam stability. After that preliminary answer, other approaches emerged, and new orientations were introduced by Th. M. Rassias, see [25], introducing therefore the so-called Hyers-Ulam-Rassias stability. Different generalizations were obtained by other researchers, by considering the possibility of using different involved norms and others types of equations, in particular by Aoki [2], Gajda [19], and Rassias [26]. The interested reader can obtain a detailed description of these advances in [4].
In this paper, we study the Hyers-Ulam stability, the σ-semi-Hyers-Ulam stability and the Hyers-Ulam-Rassias stability for the following class of higher order integro-differential equation, y (n) (x) = f x, y(x), x a k(x, τ, y(τ), y(α(τ)))dτ , The formal definition of the above mentioned stabilities are now introduced for the higher order integro-differential equation (1). If for each continuously differentiable function y satisfying y (n) (x) − f x, y(x), x a k(x, τ, y(τ), y(α(τ)))dτ ≤ σ(x), x ∈ [a, b], where σ is a non-negative function, there is a solution y 0 of the higher order integro-differential equation and a constant C > 0 independent of y and y 0 such that |y(x) − y 0 (x)| ≤ C σ(x), for all x ∈ [a, b], then we say that the higher order integro-differential equation (1) has the Hyers-Ulam-Rassias stability.
If for each continuously differentiable function y satisfying y (n) (x) − f x, y(x), x a k(x, τ, y(τ), y(α(τ)))dτ ≤ θ, x ∈ [a, b], where θ ≥ 0, there is a solution y 0 of the higher order integro-differential equation and a constant C > 0 independent of y and y 0 such that |y(x) − y 0 (x)| ≤ C θ, for all x ∈ [a, b], then we say that the higher order integrodifferential equation has the Hyers-Ulam stability.
We also use a stability in-between the two just mentioned stabilities of Hyers-Ulam-Rassias and Hyers-Ulam, introduced in [14], in the following way:
Some of the usual techniques to study the stability of functional, integral and integro-differential equations are a combination of fixed point results with a generalized metric in appropriate settings (cf. [5,8,9,10,11,12,13,15]). 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 2
Let X a nonempty set. A function d : X × X → [0, +∞] is called a generalized metric on X if and only if d satisfies: for all x, y, z ∈ X.

Theorem 3
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: (i) the sequence (T n x) n∈N converges to a fixed point x * of T ; (ii) x * is the unique fixed point of T in X * = {y ∈ X : d(T k x, y) < ∞}; (iii) if y ∈ X * , then

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 higher order integrodifferential equation (1), where x ∈ [a, b], for some fixed real numbers a and b.
We will consider the space of continuously differentiable functions C n ([a, b]) on [a, b] endowed with a generalization of the Bielecki metric where σ is a non-decreasing continuous function σ : [a, b] → (0, ∞). If instead of σ in (3) we have e p( · −a) with p > 0 then we have the well-known Bielecki metric. We recall that (C n ([a, b]), d) is a complete metric space (cf., [7], [27]).
s 1 a k(s 1 , τ, y(τ), y(α(τ)))dτ ds 1 . . . ds n−2 ds n−1 ds n where we are using the conditions y ( j) (a) = 0, j = 0, . . . , n − 1. So, we will consider the operator T : for all Note that for any continuous function u, T u is also continuous. Indeed, Note that by the hypothesis (4) we have    Under the present conditions, we will deduce that the operator T is strictly contractive with respect to the metric (3). Indeed, for all u, v ∈ C n ([a, b]), we have, Due to the fact that M β n + Lβ n+1 < 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 higher order integro-differential equation.
Additionally, (6) follows from (2) and (5). Indeed, from (5), we have and so, using integration, we obtain Therefore, having in mind (7) and (9), we conclude that Now we can apply again the Banach Fixed Point Theorem and from (2) we have that From the definition of the metric d and by (10) and consequently (6) holds.

σ-SEMI-HYERS-ULAM AND HYERS-ULAM STABILITIES IN THE FINITE INTERVAL CASE
In this section we will present sufficient conditions for the σ-semi-Hyers-Ulam stability and for the Hyers-Ulam stability of the higher order integro-differential equation (1).

Proof.
We will consider the operator T : for all x ∈ [a, b] and u ∈ C n ([a, b]). By the same procedure as above, we have that T is strictly contractive with respect to the metric (3) due to the fact that M β n + Lβ n+1 < 1. Thus, we can apply the Banach Fixed Point Theorem, which ensures that we have the σ-semi-Hyers-Ulam stability for the higher order integro-differential equation (1).
On the other hand, from (14) and using integration we obtain y(x) − Therefore, having in mind (16) and (17), we conclude that From (2), the definition of the metric d and by (18) and consequently by the definition of σ we have that (15) holds.
for all x ∈ [a, b]. Moreover, suppose that f : [a, b] × C × C → C is a continuous function satisfying the Lipschitz

HYERS-ULAM-RASSIAS STABILITY IN THE INFINITE INTERVAL CASE
In this section, we analyse the Hyers-Ulam-Rassias stability of the higher order integro-differential equation (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 higher order integro-differential equations, x a k(x, τ, y(τ), y(α(τ)))dτ , y ( j) (a) = 0, j = 0, . . . , n − 1, with y ∈ C n ([a, ∞)), Our strategy will be based on a recurrence procedure due to the already obtained result for the corresponding finite interval case.
To prove (28), by (32) and (30), we have that for all x ∈ [a, ∞), |y(x) − y 0 ( Finally, we will prove the uniqueness of y 0 . Let us consider another bounded differentiable function y 1 which satisfies (27) and (28), for all x ∈ [a, ∞). By the uniqueness of the solution on I m(x) for any m(x) ∈ N we have that y 0|I m(x) = y 0,m(x) and y 1|I m(x) satisfies (27) and (28) for all x ∈ I m(x) , so y 0 (x) = y 0 | I m(x) (x) = y 1 | I m(x) (x) = y 1 (x).

Remark 8
With the necessary adaptations, Theorem 7 also holds true for infinite intervals of the type (−∞, b], with b ∈ R, as well as for (−∞, ∞).

Illustrative Examples
In this section we will present some examples simply to illustrate that the conditions of the above results are possible to attain.