Coincidence Best Proximity Point Results in Branciari Metric Spaces with Applications

Naeem Saleem,Sadia Farooq,Manuel De La Sen,Nawab Hussain

doi:10.1155/2020/4126025

Abstract

This paper is aimed at studying the uniqueness of coincidence best proximity point forϑ,α+,g-proximal contractions in complete Branciari metric space. Throughout this article, discontinuity of the Branciari metric space is used and we obtained the desired results without assuming it as a continuous. Some examples are provided to validate the results proved herein. As an application, we derive the best proximity point results in the setup of complete Branciari metric space endowed with graph. Further, our results extend and generalize the existing ones in literature.

Highlights

Introduction and PreliminariesLet F : X ⟶ X be a mapping, where X be any nonempty set
Let α : Q × Q ⟶ −∞, ∞), ðF, gÞ be a pair of mappings satisfying ðθ, α+, gÞ-proximal contraction, where F : Q ⟶ P be a triangular proximal α+-admissible and g : Q ⟶ Q be a one to one expansive mapping satisfying αR property
Let α : Q × Q ⟶ −∞, ∞), ðF, gÞ be a pair of mappings satisfying ðθ, α+, gÞ-proximal contraction, where F : Q ⟶ P be a triangular proximal α+-admissible and g : Q ⟶ Q be a one to one isometry mapping satisfying αR property

Summary

Introduction

Introduction and PreliminariesLet F : X ⟶ X be a mapping, where X be any nonempty set. “An element q∗ ∈ X is a fixed point of F if q∗ satisfies the equation Fq∗ = q∗ (known as a fixed point equation) or dðq∗, Fq∗Þ = 0:} A collection of all “fixed points” of F will be represented as FðXÞ, that is, FðXÞ = fq∗ ∈ X : dðq∗, Fq∗Þ = 0g: ð1Þ In this direction, Banach [1] gives the existence and uniqueness of the “fixed point” of the self mapping F, if mapping F is a contraction and ðX, dÞ is a complete, but it becomes more interesting, if F is a nonself mapping it is not necessary that the operator equation Fq∗ = q∗ has a solution. Several authors studied the results dealing with “approximate fixed points” in different spaces (for detail, see [3,4,5,6,7,8,9,10,11,12,13,14])

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Results

Conclusion