A Solution of Fredholm Integral Equation by Using the Cyclic η s q -Rational Contractive Mappings Technique in b-Metric-Like Spaces

Hasanen A Hammad,Manuel De La Sen

doi:10.3390/sym11091184

Abstract

In this article, the notion of cyclic η s q -rational contractive mappings is discussed and some fixed point theorems in the context of complete b-metric-like spaces are showed. Here, the obtained consequences unify, extend and generalize various comparable known results. Furthermore, new common fixed point outcomes in a directed graph are demonstrated. Moreover, some useful examples are discussed to justify our theoretical results and finding a solution of Fredholm integral equation was discussed as enforcement.

Highlights

Problems in the nonlinear analysis are solved by a popular tool called the Banach contraction principle, which appeared in Banach’s thesis [1] and was used to find a solution for some integral equations; this stated: consider (Ω, d) is a metric space and Γ is a nonlinear self-mapping defined on it; the mapping Γ is called a Banach contraction if there exists
The major drawback of this principle is how we apply this contraction in case of discontinuity. This problem was overcome in the past by Kannan [2] where it proved a fixed point result without continuity, while, recently many authors attempted to solve this problem
The analytical solution of nonlinear integral equations by using a cyclic contractive mapping is an important application in fixed point theory, where they have attracted the interest of a lot of authors in academic research

Summary

Introduction

Problems in the nonlinear analysis are solved by a popular tool called the Banach contraction principle, which appeared in Banach’s thesis [1] and was used to find a solution (existence and uniqueness) for some integral equations; this stated: consider (Ω, d) is a metric space and Γ is a nonlinear self-mapping defined on it; the mapping Γ is called a Banach contraction if there exists. The major drawback of this principle is how we apply this contraction in case of discontinuity This problem was overcome in the past by Kannan [2] where it proved a fixed point result without continuity, while, recently many authors attempted to solve this problem (see, for example, [3,4,5,6]). In 2010, the concept of b-metric-like was initiated by Alghamdi et al [17] as an extension of a b-metric They studied some important fixed point consequences concerned with this space.

Preliminaries and Known Results

Cyclic ηs -Rational Contractive Mappings q

Some Related Fixed Point on a Directed Graph

The Analytical Solution of Fredholm Integral Equation of the Second Kind

Conclusions