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g-approximate best proximity pairs in metric space with a directed graph

Mohsenialhosseini, Saheli

Abstract

Let(X,d) be a metric space that has a directed graph G such that the sets V(G) and E(G) are respectively vertices and edges corresponding to X. We obtain sufficient conditions for the existence of an G-approximate best proximity pair of the mapping T in the metric space X endowed with a graph G such that the set V(G) of vertices of G coincides with X.

g-approximate best proximity pairs in metric space with a directed graph

Abstract

Let(X,d) be a metric space that has a directed graph G such that the sets V(G) and E(G) are respectively vertices and edges corresponding to X. We obtain sufficient conditions for the existence of an G-approximate best proximity pair of the mapping T in the metric space X endowed with a graph G such that the set V(G) of vertices of G coincides with X.

Paper Structure

This paper contains 4 sections, 10 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Conclusions

Key Result

Theorem 2.3

Moh1Let $(X,\|.\|)$ be a complete norm space, $T:X\rightarrow X,$$x_0\in X$ and $\epsilon>0$ . If $\|T^{n}(x_0)-T^{n+k}(x_0)\|\rightarrow 0$ as $n\rightarrow\infty$ for some $k>0,$ then $T^k$ has an $\epsilon-$ fixed point.

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Definition 2.4
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Proposition 3.4
  • Definition 3.5
  • Theorem 3.6
  • ...and 14 more