Some New Generalization of Darbo’s Fixed Point Theorem and Its Application on Integral Equations

Anupam Das,Poom Kumam,Bipan Hazarika

doi:10.3390/math7030214

Abstract

In this article, we propose some new fixed point theorem involving measure of noncompactness and control function. Further, we prove the existence of a solution of functional integral equations in two variables by using this fixed point theorem in Banach Algebra, and also illustrate the results with the help of an example.

Highlights

Integral equations play a significant role in real-world problems
In this article using the concept of control function and measure of noncompactness we have proved some new fixed point theorems
We have applied this theorem to study the existence of solution of functional integral equations in Banach algebra and with the help of an example we have verified our results

Summary

Introduction

Integral equations play a significant role in real-world problems. Fixed point theory and measure of noncompactness are useful tools in solving different types of integral equations which we come across in different real life situations. We refer (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]) for application of fixed point theorems and measure of noncompactness for solving differential and integral equations. Let M Ē denote the family of all nonempty and bounded subsets of Ē and N Ē its subfamily consisting of all relatively compact sets. Let Z be a nonempty, bounded, closed and convex subset of a Banach space Ē. We denote Ẑ be the class of functions η : R+ × R+ → R satisfying the following conditions:. Let F be the class of all functions G : R+ × R+ → R+ satisfying the following conditions: max { a, b} ≤ G ( a, b) for a, b ≥ 0.

Main Result

Application