Existence of Best Proximity Point with an Application to Nonlinear Integral Equations

Shagun Sharma,Sumit Chandok

doi:10.1155/2021/3886659

Abstract

Using the idea of modified ϱ -proximal admissible mappings, we derive some new best proximity point results for ϱ − ϑ -contraction mappings in metric spaces. We also provide some illustrations to back up our work. As a result of our findings, several fixed-point results for such mappings are also found. We obtain the existence of a solution for nonlinear integral equations as an application.

Highlights

Introduction and PreliminariesProblems originating in several disciplines of mathematical analysis, such as obtaining the existence of a solution for integral and differential equations, are solved using fixedpoint theory. e study of fixed points for nonself-mapping, on the contrary, is fascinating
Where ℘(u′, Iu′) is the minimum value dist(ζ, E) and u′ is an approximate solution of equation Iu′ u′ with least possible error, such a solution is known as the best proximity point of mapping I
Various academics have discovered a number of best proximity point theorems and associated fixed-point results in metric or normed linear spaces

Summary

Introduction and Preliminaries

Problems originating in several disciplines of mathematical analysis, such as obtaining the existence of a solution for integral and differential equations, are solved using fixedpoint theory. e study of fixed points for nonself-mapping, on the contrary, is fascinating. A nonself-contraction I: ζ ⟶ E does not necessarily have a fixed point for two given nonempty closed subsets ζ and E of a complete metric space א. In this case, it is important to identify a point u′ ∈ ζ such that ℘(u′, Iu′) is minimum. E goal of this work is to use the modified ρ-proximal admissible mappings concept to obtain some best proximity point outcomes for ρ − θ-contraction mappings in metric space. On a metric space enriched with an arbitrary binary relation, some optimal proximity point results are proved For such a class of mappings, we obtain certain fixedpoint findings.

Main Results

Consequences and Related Results

Examples

Application to Integral Equations

Conclusion