Existence of Solutions for Functional Integral Equation Involving the Henstock-Kurzweil-Stieltjes Integral

Abstract In this paper, we apply the method associated with the technique of measure of noncompactness and some generalizations of Darbo fixed points theorem to study the existence of solutions for a class of integral equation involving the HenstockKurzweil-Stieltjes integral. Meanwhile, an example is provided to illustrate our results.


Introduction
Existence theorems of coupled fixed points have been considered by several authors (Chang & Cho, 1996;Roshan, 2017).In (Chang & Cho, 1996), the authors proved the existence of coupled fixed points for a class of integral operator: where ] is the space of all real valued continuous functions on [0, L].
In this paper we establish the existence of solutions for the following integral equation involving the Henstock-Kurzweil-Stieltjes integral: (x, y)(t) = h (t, x(t), y(t)) + ϕ ( t, ∫ t 0 f (t, s, x(s), y(s))dg(s) where h, ϕ, f are continuous functions, g : [0, L] → R is of boundary variation.dg can be identified with a Stieltjes measure and will have the effect of suddenly changing the state of the system at the points of discontinuity of g, that is, the system could be controlled by some impulsive force.The Henstock-Kurzweil-Stieltjes integral, which is a generalization of the Lebesgue-Stieltjes integral (Krejčí, 2006;Kurzweil, 1957;Lee, 1989;Schwabik & Ye, 2005), has been proved useful in the study of ordinary differential equations (Chew, 1988;Chew & Flordeliza, 1991;Heikkilä & Ye, 2012;Ye & Liu, 2016).
To achieve our goal, the approach associated with the technique of measure of noncompactness and some generalizations of Darbo fixed points theorem will be used.
This paper is organized as follows.In Section 2, we recall some basic concepts of the Henstock-Kurzweil-Stieltjes integrals and measure of noncompactness.In Section 3, we verify the existence of solutions for (2) by a coupled fixed point theorem.In Section 4, we give an example to illustrate Theorem 3.2 in this paper.

Preliminaries
In this section, so that the paper is self-contained, we provide preliminary material with respect to the Henstock-Kurzweil-Stieltjes integral and measure of noncompactness.
For given functions f, g : [a, b] → R and a δ-fine partition D, we define where Definition 2.1.(Krejčí, 2006) Let f, g : [a, b] → R be given.We say that J ∈ R is the Henstock-Kurzweil-Stieltjes integral (HKS ) over [a, b] of f with respect to g and denote if for every ε > 0, there exists positive function δ > 0, such that for every δ-fine D, we have exist and are finite with the convention  and Var [a,b] g n ≤ C independently of n, then

Measure of Noncompactness
In this subsection, we recall some fundamental facts concerning measure of noncompactness (see [Banaś & Goebel(1980)]).Let (E, ∥ • ∥) be a real Banach space with zero element 0 and B(x, r) denote the closed ball in E centered at x with radius r.The symbol B r stands for the ball B(0, r).Denote by X, convX the closure and the closed convex hull of a nonempty subset X of E singly.Finally, let us denote by m E the family of all nonempty and bounded subsets of E and by n E its subfamilies consisting of all relatively compact subsets.
Definition 2.4.(Mursaleen, 2017) Let (E, d) be a metric space and X a bounded subset of E. The Hausdorff measure of noncompactness (µ-measure or ball measure of noncompactness) of the set X, denoted by µ(X) is defined to be the infimum of the set of all reals ε > 0 such that X can be covered by a finite number of balls of radii < ε, that is, The function µ is called the Hausdorff measure of noncompactness.
Definition 2.7.(Chang & Cho, 1996) An element (x, y) ∈ X×X is called a coupled fixed point of a mapping T : X×X → X if T (x, y) = x and T (y, x) = y.
Lemma 2.8.(Banaś & Goebel, 1980, Theorem 2) Let Ω be a nonempty, bounded, closed and convex subset of a Banach space E and let T : Ω → Ω be a continuous mapping.Assume that there exists a constant k ∈ [0, 1) such that for any X ⊂ Ω.Then T has a fixed point.
The following generalization of Darbo fixed point theorem will be needed in Section 3.
Lemma 2.9.(Roshan, 2017, Theorem 3.7) Let Ω be a nonempty, bounded, closed and convex subset of a Banach space E, µ be an arbitrary measure of noncompactness.Moreover assume that T : Ω × Ω → Ω be a continuous function satisfying for all X 1 , X 2 ⊆ Ω × Ω, where φ ∈ Φ.Then T has at least a coupled fixed point.

Main Results
In this section, we shall prove the existence of solutions of Eq. ( 2).
Firstly, we give the following assumptions: and and and there exist functions and M 3 = sup{m 1 (t) ∫ t 0 m 2 (s)dg(s)}, for any t, s ∈ [0, L] such that s ≤ t, and for each x ∈ R; (D 4 ) There exists r > 0 such that Let C[0, L] × C[0, L] be equipped with the norm ∥(x, y)∥ = ∥x∥ + ∥y∥.We define an operator ) . (5) Then we have the following statement.
Theorem 3.1.Under the assumptions (D 1 ) − (D 4 ), the operator F given in ( 5) has at least one coupled fixed point in the space C and ∥x∥, ∥y∥ ≤ r, This implies that F maps the space B r × B r into B r , where B r = {x, y ∈ C[0, L] : ∥x∥ ≤ r, ∥y∥ ≤ r}, r is a constant arising in assumption (D 4 ).
(ii) We prove that the operator F is continuous on B r × B r .
Thus, taking into account the property of the function φ and linking (6), for each t ∈ [0, L] we get Hence, the operator F is continuous on B r × B r .
(iii) Taking arbitrary nonempty subsets X 1 , X 2 of the ball B r .Fix ε > 0, choose arbitrarily Without loss of generality, assuming that t 2 < t 1 .Then, for arbitrary (x, y) ∈ X 1 × X 2 , we get where Moreover, since the functions m 1 (t), m 2 (s) are continuous, we have Since (x, y) is an arbitrary element of X 1 × X 2 in (8), we obtain It follows from ( 9) and Theorem 2.5 that According to Lemma 2.9, F has at least a coupled fixed point in the space B r × B r .The proof is therefore complete.
It is obvious that Eq. ( 11) is a exception of Eq. ( 2) with Now we show that all the conditions of Theorem 3.2 are satisfied for Eq. ( 11).
(i) Obviously, h and ϕ are continuous.
(ii) Clearly, the function |h(t, 0, 0 Moreover we can get (vi) It is easy to check that for each number r ≥ 5, we have the following inequality Consequently, all the conditions of Theorem 3.2 are satisfied and Eq. ( 11) has at least one solution in the space C[0, 1] × C[0, 1].
Remark 4.2.In Example 4.1, the Cantor-Lebesgue function g ∈ C[0, 1] ∩ BV[0, 1], but g is not absolutely continuous on [0, 1].Therefore, the methods used to deal with integral equations involving the Lebesgue (or Riemann) integral (Chang & Cho, 1996;Roshan, 2017) are no longer applicable in this case.This means our existence result Theorem 3.2 is more general.

Conclusions
In this research, by using the approach associated with the technique of measure of noncompactness and some generalizations of Darbo fixed points theorem, we studied the existence of solutions for a class of integral equation involving the Henstock-Kurzweil-Stieltjes integral, and we obtained the existence of at least one solution for the functional integral equation we considered.
Denote by G[a, b] the space of all real valued regulated functions on [a, b].Obviously, both the space C[a, b] of all real valued continuous functions on [a, b] and the space BV[a, b] of all functions of bounded variation on [a, b] are subsets of G[a, b].Lemma 2.3.(Krejčí, 2006) If f ∈ G[a, b], g ∈ BV[a, b], then both ∫ (v) Further, notice that the function f is continuous, and we have| f (t, s, x, y)| ≤ e −t 2 for t, s ∈ [0, 1] and x, y ∈ R. If we put m 1 (t) = e −t 2 , m 2 (s) = 1, then we have M 3 = sup { e −t 2 ∫ t 0 dg(s) : t ∈ [0