The Technique of Quadruple Fixed Points for Solving Functional Integral Equations under a Measure of Noncompactness

Hasanen A Hammad,Amal A Khalil

doi:10.3390/math8122130

Abstract

Under the idea of a measure of noncompactness, some fixed point results are proposed and a generalization of Darbo’s fixed point theorem is given in this manuscript. Furthermore, some novel quadruple fixed points results via a measure of noncompactness for a general class of functions are presented. Ultimately, the solutions to a system of non-linear functional integral equations by the fixed point results obtained are discussed, and non-trivial examples to illustrate the validity of our study are derived.

Highlights

In the nonlinear analysis field, the fixed point technique is used to solve many mathematical problems as it is involved in differential and integral equations, integro-differential equations, fractional calculus, and other disciplines of science and technology; see [1,2,3,4,5,6,7].In the finite-dimensional spaces, this technique was created by Brouwer [8] and is known as “Brouwer’s fixed point theorem (FPT)”
In 1922, the existence and uniqueness of this technique were studied in a contraction mapping via a complete metric space, known as the “Banach contraction principle [9]”
To discuss the existence of fixed point theorems (FPTs) via an measure of noncompactness (MNC) for condensing operators, a nice paper was written by Darbo [12]

Summary

Introduction

In the nonlinear analysis field, the fixed point technique is used to solve many mathematical problems as it is involved in differential and integral equations, integro-differential equations, fractional calculus, and other disciplines of science and technology; see [1,2,3,4,5,6,7]. To discuss the existence of fixed point theorems (FPTs) via an MNC for condensing operators, a nice paper was written by Darbo [12]. His results are a generalization of the classical Banach and Schauder FPTs, and he used the theoretical study to present the solutions to differential and integral. Ξ is a nonempty, closed, bounded, and convex (NCBC) subset of a BS Z In this manuscript, the existence of the solution of nonlinear functional integral equations in the Banach space under the technique of a measure of noncompactness is obtained.

Preliminaries

Results

Supportive Application

Illustrative Example

Conclusions