Existence of Solution for Non-Linear Functional Integral Equations of Two Variables in Banach Algebra

Hari M Srivastava,Anupam Das,Bipan Hazarika,S A Mohiuddine

doi:10.3390/sym11050674

Abstract

The aim of this article is to establish the existence of the solution of non-linear functional integral equations x ( l , h ) = U ( l , h , x ( l , h ) ) + F l , h , ∫ 0 l ∫ 0 h P ( l , h , r , u , x ( r , u ) ) d r d u , x ( l , h ) × G l , h , ∫ 0 a ∫ 0 a Q l , h , r , u , x ( r , u ) d r d u , x ( l , h ) of two variables, which is of the form of two operators in the setting of Banach algebra C [ 0 , a ] × [ 0 , a ] , a > 0 . Our methodology relies upon the measure of noncompactness related to the fixed point hypothesis. We have used the measure of noncompactness on C [ 0 , a ] × [ 0 , a ] and a fixed point theorem, which is a generalization of Darbo’s fixed point theorem for the product of operators. We additionally illustrate our outcome with the help of an interesting example.

Highlights

Many real-life problems in which we go over the investigation of various branches of mathematical physics, for example, gas kinetic theory, radiation, and neutron transportation, can be depicted and demonstrated by methods of non-linear functional integral equations
Dhage [6,7] used the concept of the fixed point theorem to find the solution of functional integral equations in Banach algebra
Deepmala and Pathak [9] studied the existence of the solution of nonlinear functional integral equations of a single variable in Banach algebra C [ a, b] of all real-valued continuous functions on the interval [ a, b] equipped with the maximum norm

Summary

Introduction

Many real-life problems in which we go over the investigation of various branches of mathematical physics, for example, gas kinetic theory, radiation, and neutron transportation, can be depicted and demonstrated by methods of non-linear functional integral equations (for example, we refer to [1,2,3,4]). Dhage [6,7] used the concept of the fixed point theorem to find the solution of functional integral equations in Banach algebra. Banaś and Olszowy [8] used the class of measures of noncompactness to obtain the existence of solutions of nonlinear integral equations in Banach algebra. The authors of [9] used the measure of noncompactness to obtain the existence of the solution of the integral Equation (1) in Banach algebra C [0, a] with the help of the fixed point theorem. We used a fixed point theorem associated with Darbo’s condition of the measure of noncompactness in Banach algebra of continuous functions in [0, a] × [0, a] to establish the solvability of Equation (2). We used the modified homotopy perturbation analytic method to find the solution of Equation (2)

Preliminaries

Main Result

An Illustrative Example

Z 1 Z 1

Conclusions