Existence and Uniqueness of the Solution for an Integral Equation with Supremum, via w -Distances

: Following the idea of T. Wongyat and W. Sintunavarat, we obtain some existence and uniqueness results for the solution of an integral equation with supremum. The paper ends with the study of Gronwall-type theorems, comparison theorems and a result regarding a Ulam–Hyers stability result for the corresponding ﬁxed point problem.


Introduction
The object of investigation of this paper is the qualitative theory of integral equations with supremum. These equations arise naturally when solving real-world problems, for example in the study of systems with automatic regulation and automatic control, problems in control theory. These types of equations are characterized by the fact that the maximum values of some regulated state parameters depend on certain time intervals, see for example [1] and the references therein. Recently, the interest in differential equations with supremum has an intensive development (see [2][3][4]). The aim of this paper focuses on two aspects: one is to prove existence and uniqueness results using w-weak generalized contractions theorem; the other is to prove a Gronwall-type theorem and comparison theorems. Using this theory symmetry is important in determining the qualitative properties of the solution of the integral equation.
We consider the following class of integral equation with supremum [α,s] x(θ))ds, t ∈ [α, β] with α, β real and α < β, the functions To prove our results, we shall use the w-weak generalized contractions theorem due to T. Wongyat and W. Sintunavarat [5] and we obtain an existence and uniqueness result for the solutions of this equation.
We recall that each metric on the nonempty set T is a w-distance on T . Definition 2 ([5]). We say that the function ψ :

Definition 4 ([5]
). Let (T , d) be a metric space. We say that a w-distance q is a ceiling distance of d if and only if q(x, y) ≥ d(x, y), for all x, y ∈ T .

Definition 5 ([5])
. We consider q a w-distance on the metric space (T , d), the altering distance function If the below inequality holds we say that the operator A : T → T is a w-generalized weak contraction mapping where m(x, y) := max q(x, y), q (x, A(y)) + q (A(x), y) 2 . (3) If q = d, then we say that A is a generalized weak contraction mapping.
Now we consider (T , d) a complete metric space. The following fixed point result of the equation A(x) = x, x ∈ T via w-distances represents the motivation of our work. Theorem 1 ([5]). We consider q : T × T → [0, ∞) a continuous w-distance on T and a ceiling distance of d, the altering distance function ψ : [0, ∞) → [0, ∞), and the continuous function φ Then, A has a unique fixed point in T and the sequence of successive approximations {x n } n∈N , defined by x n = A n (x 0 ), for each x 0 ∈ T , for all n ∈ N, converges to the unique fixed point of A.
In this paper, we emphasize some connection between w-generalized weak contraction mapping and the Picard operator theory.
In the sequel, we recall the following results (see [12][13][14]). Let (T , d) be a metric space. We say that the operator A : T → T is weakly a Picard operator (WPO) if the successive approximations sequence {A n (x)} n∈N , converges for all x ∈ T and its limit (which generally depend on x) is a fixed point of A. If an operator A is WPO with F A = {x * }, then, we say that the operator A is a Picard operator (PO).
Definition 6. Let A be a weakly Picard operator and c > 0. We say that the operator A is a c-weakly Picard If T is a nonempty set, then (T , d, ≤) is an ordered metric space, where ≤ is a partial order relation on T . Now we present some properties regarding WPOs and POs.
Theorem 3 ([13]). (Abstract Gronwall Theorem) Let (T , d, ≤) be an ordered metric space and we consider the operator A : T → T . We suppose Then the below conclusions hold: Then, for x, y, z ∈ T , We present now the concept of Hyers-Ulam stability in the setting of metric spaces given by I.A. Rus in [15]. Definition 7. Let (T , d) be a metric space and we consider the operator A : T → T . Then, we say that the fixed point equation is Ulam-Hyers stable if there exists c A ∈ R * + such that: for any ε > 0 and for each solution y * ∈ T of (5), i.e., d(y * , A(y * )) ≤ ε, there exists a solution x * of (5) such that We recall the following abstract result of the Ulam-Hyers stability of the fixed point Equation (5).
Proof. Let T = C([α, β], R) and we consider the metric d : T × T → [0, ∞) defined as below It is clear that (T , d) is a complete metric space. We consider the function q : |y(t)| , for all x, y ∈ C([α, β], R). (8) We get that q is a w-distance on T and also a ceiling distance of d.
We will show that A satisfies the contraction condition (4).
Therefore the condition (4) holds and thus we may conclude that A has a unique fixed point. So there exists a unique solution for the integral equation with supremum (1).
From the above theorem, the operator A defined in (6) is a PO. Now we establish a Gronwall-type theorem for Equation (1).
By the proof of Theorem 6, it follows that A is a Picard operator. The conclusion of the theorem follows from Theorem 3.

Theorem 9.
We consider the integral equation with supremum (1) and we suppose that all the conditions of Theorem 6 are satisfied. Then, the integral Equation (1) is Ulam-Hyers stable.
Proof. Applying Theorem 6 and Theorem 5 we get the conclusion of the theorem.

Conclusions
The purpose of this paper is to establish some fixed point results for generalized contraction operators with respect to w-distances. The operators considered here contain a supremum over a certain time interval. Section 3 begins with an existence and uniqueness theorem proved using the method of w-distances. Adding to the hypotheses that sustain the existence and uniqueness of the solution, the fact that f is an increasing function, we obtain Gronwall-type and comparison theorems. In the last part of the paper we study the Ulam-Hyers stability using Picard operators techniques. We define a fixed point equation from the integral equation with supremum. If the defined operator is c-weakly Picard we have Ulam-Hyers stability of the corresponding fixed point problem.

Conflicts of Interest:
The authors declare no conflict of interest.