New Sufﬁcient Conditions to Ulam Stabilities for a Class of Higher Order Integro-Differential Equations

: In this work, we present sufﬁcient conditions in order to establish different types of Ulam stabilities for a class of higher order integro-differential equations. In particular, we consider a new kind of stability, the σ -semi-Hyers-Ulam stability, which is in some sense between the Hyers–Ulam and the Hyers–Ulam–Rassias stabilities. These new sufﬁcient conditions result from the application of the Banach Fixed Point Theorem, and by applying a speciﬁc generalization of the Bielecki metric.


Introduction
In 1940, S. M. Ulam [1] proposed the well-known Ulam stability problem. The difficulty of this problem lies in the conditions to be imposed to guarantee the existence of a linear mapping near an approximately linear mapping. It is known that most of the time it is not possible to obtain exact solutions for some integro-differential equations. Therefore, special techniques are applied, allowing us to obtain approximate solutions. In this case, it is crucial to find error bounds to the approximations when replacing the exact solutions in practical problems.
In 1941, D. H. Hyers [2] gave a partial answer to the problem under the assumption that the groups are Banach spaces, considering the additive Cauchy equation f (x + y) = f (x) + f (y). This contribution originated the naming Hyers-Ulam stability. Meanwhile some other approaches came to light, and later in 1978, new directions were introduced by Th. M. Rassias [3] aiming to solve the Ulam stability problem, which gave origin to the concept of Hyers-Ulam-Rassias stability. Furthermore, new developments were carried out involving different norms and other types of equations. We refer in particular to the works presented by T. Aoki [4], Z. Gajda [5] and Th. M. Rassias [6]. For more details on the subject, we refer to [7,8] and the references therein.
The work initiated by S. M. Ulam in the 1940s had relevant consequences in the field of applications as, for example, in chemical reactions, elasticity, fluid flows, semiconductors and population dynamics (see [9][10][11][12]). The study of problems involving differential, functional, integro-differential and integral equations, in particular their stability issues, has suffered greatly from the growing engagement over the years with a spread of interest among researchers, for example, see [2,[13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. Their applicability in mathematical models for which we cannot easily find exact solutions, namely those involving symmetry issues, the study of the stability of the approximate solutions is an open field of research. In particular, this work may be relevant in the study of the stability of the symmetrical flow of Newtonian and non-Newtonian fluids related to one-dimensional models obtained by Cosserat Theory associated with fluid dynamics (see [28,29]).

Notations and Preliminaries
Let us consider the higher order integro-differential equation defined by with initial conditions where n ∈ N, ϕ ∈ C n ([a, b]) and x ∈ [a, b], with fixed real numbers a and b. Moreover, we consider two continuous functions F : Next, we introduce several stability concepts related to problem (1) and (2).

Definition 1 (Hyers-Ulam stability).
If for each continuously differentiable function ϕ satisfying with x ∈ [a, b] and β ≥ 0, the higher order integro-differential equation has a solution ϕ 0 and there is a constant C > 0, independent of ϕ 0 and ϕ, such that for all x ∈ [a, b], then we say that the given problem (1) and (2) with x ∈ [a, b], the higher order integro-differential equation has a solution ϕ 0 and there is a constant C > 0, independent of ϕ 0 and ϕ, such that for all x ∈ [a, b], then we say that the given problem (1) and (2) has the Hyers-Ulam-Rassias stability.
Now, we will introduce a new kind of stability which was presented in [30].
where x ∈ [a, b] and β ≥ 0, the higher order integro-differential equation has a solution ϕ 0 and there is a constant C > 0, independent of ϕ 0 and ϕ, such that for all x ∈ [a, b], then we say that the given problem (1) and (2) has the θ-semi-Hyers-Ulam stability.
Theorem 1 (Banach fixed point theorem). Let (X, d) be a generalized complete metric space and let T : X → X a strictly contractive operator with a Lipschitz constant L < 1. If there exists a non-negative 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 In the following, we consider the space of continuously differentiable functions in the interval [a, b], C n ([a, b]), endowed with a generalization of the Bielecki metric, given by with θ a non-decreasing continuous function θ : [a, b] → (0, ∞) and (C n ([a, b]), d) is a complete metric space (see [39,40]).

Hyers-Ulam-Rassias Stability
In the following theorem we will present sufficient conditions for the Hyers-Ulam-Rassias stability relating to problem (1) and (2).
for all x ∈ [a, b], where α ∈ R. Suppose also that the continuous function F : where M > 0 is a Lipschitz constant, and the continuous kernel G : where x ∈ [a, b], and M α n + Lα n+1 < 1, then there is a unique function ϕ 0 ∈ C n ([a, b]), solution of problem (1) and (2), such that for all x ∈ [a, b].
Let us define the continuous operator T : . Indeed, for any continuous function ϕ, we have . . dr n−2 dr n−1 dr n −→ 0 when x → x 0 . Therefore, using condition (11), we have Consequently, using conditions (17) and (18), we will prove that the operator T is strictly contractive to the Bielecki metric (10). In order to prove that, we have for all Therefore, by the fact that M α n + Lα n+1 < 1 we have that the operator T is strictly contractive. Thus, we can apply Theorem 1, which ensures that we have the Hyers-Ulam-Rassias stability for problem (1) and (2). Additionally, from (14), we have where x ∈ [a, b]. Therefore, using integration, we obtain Now, using conditions (16) and (20), we have Moreover, from (9) follows Finally, from the definition of the metric d and (21), we obtain and consequently condition (15) holds.
for all x ∈ [a, b], where α ∈ R. Suppose also that the continuous function F : where M > 0 is a Lipschitz constant, and the continuous kernel G : where x ∈ [a, b], β ≥ 0, and M α n + Lα n+1 < 1, then there exists a unique function ϕ 0 ∈ C n ([a, b]), solution of the problem (1) and (2), such that for all x ∈ [a, b].
Following the same ideas as in the proof of Theorem 2, we can prove that T is strictly contractive to the metric (10) due to the fact that M α n + Lα n+1 < 1.
for all x ∈ [a, b], where α ∈ R. Suppose also that the continuous function F : [a, b] × C × C → C satisfies the condition where M > 0 is a Lipschitz constant, and the continuous kernel G : where x ∈ [a, b], β ≥ 0, and M α n + Lα n+1 < 1, then there exists a unique function ϕ 0 ∈ C n ([a, b]), solution of the problem (1)-(2), such that for all x ∈ [a, b].

Examples
In this section, we present some examples in order to illustrate the results obtained throughout the work.

First Example: 2-Differentiable Function
Let us define the space We consider the 2-differentiable functions ϕ : 0, π 2 → R on the space D, and the integro-differential equation given by with x ∈ 0, π 2 . We also consider the continuous function θ : 0, π 2 → (0, ∞) defined by which fulfills the inequality with α ∈ 10 22 1 − e − 22 20 π , ∞ . Thus, considering these assumptions all the conditions of Theorem 2 are satisfied. Now, considering the continuous function F : 0, π 2 × C × C → C defined by we have with M = 2.

Third Example: 3-Differentiable Function and a Bigger Perturbation
We will consider the integro-differential Equation (44) but with another function θ and a bigger perturbation of the solution. Considering the continuous function θ : 0, 1 we have with α ∈ 81 164 , ∞ .

Conclusions
In this work, we presented new sufficient conditions for the Hyers-Ulam-Rassias, the Hyers-Ulam and the σ-semi-Hyers-Ulam stabilities for a general higher order integrodifferential equation by using the Banach fixed point theorem and a generalization of the Bielecki metric thus enabling the study of the stability of an expanding number of particular equations. Some examples were presented to illustrate the theoretical results.