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Best Proximity Point Results for Perimetric Contractions

Hiranmoy Garai, Evgeniy Petrov, Pratikshan Mondal, Lakshmi Kanta Dey

TL;DR

Addresses the problem of best proximity points for non-self mappings by introducing perimetric proximal contractions of the first and second kind and a key Condition $\Lambda$. The main contributions are two theorems (Theorems t1 and t2) establishing existence of a best proximity point and guaranteeing that the number of such points is at most two under natural completeness/compactness/injectivity hypotheses, with sequences in $A_0$ or $B_0$ leading to limits satisfying $d(u,Tu)=d(A,B)$. The paper supports claims with several illustrative examples, including cases with exactly one or two best proximity points. Overall, it extends proximal contraction theory to perimeter-contractive maps, showing finite multiplicity results for best proximity points in non-self contexts.

Abstract

This paper has two aims, first one is to introduce special kind of proximal contractions guaranteeing a finite number of best proximity points, and second one is to derive best proximity point results for perimetric contractions. To meet these two aims, we introduce two new proximal contractions: perimetric proximal contractions of the first and the second kind, and derive best proximity point results for these mappings. We establish that for these particular mappings, best proximity points are not necessarily unique; however, we provide an upper bound, proving that at most two such points can exist. To establish the validity of our results, we provide illustrative examples demonstrating that these newly defined mappings can possess unique or exactly two best proximity points.

Best Proximity Point Results for Perimetric Contractions

TL;DR

Addresses the problem of best proximity points for non-self mappings by introducing perimetric proximal contractions of the first and second kind and a key Condition . The main contributions are two theorems (Theorems t1 and t2) establishing existence of a best proximity point and guaranteeing that the number of such points is at most two under natural completeness/compactness/injectivity hypotheses, with sequences in or leading to limits satisfying . The paper supports claims with several illustrative examples, including cases with exactly one or two best proximity points. Overall, it extends proximal contraction theory to perimeter-contractive maps, showing finite multiplicity results for best proximity points in non-self contexts.

Abstract

This paper has two aims, first one is to introduce special kind of proximal contractions guaranteeing a finite number of best proximity points, and second one is to derive best proximity point results for perimetric contractions. To meet these two aims, we introduce two new proximal contractions: perimetric proximal contractions of the first and the second kind, and derive best proximity point results for these mappings. We establish that for these particular mappings, best proximity points are not necessarily unique; however, we provide an upper bound, proving that at most two such points can exist. To establish the validity of our results, we provide illustrative examples demonstrating that these newly defined mappings can possess unique or exactly two best proximity points.
Paper Structure (2 sections, 7 theorems, 38 equations)

This paper contains 2 sections, 7 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Perimetric proximal contractions

Key Result

Theorem 1.3

( B10) If $(X,d)$ is complete; $A,B$ are closed, $A_0, B_0$ are non-empty; $T(A_0)\subseteq B_0$ and $T$ is a continuous proximal contraction of the first kind, then $T$ has a unique best proximity point in $A$.

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Theorem 2.4
  • ...and 11 more