Mizoguchi-Takahashi local contractions to Feng-Liu contractions

TL;DR

This paper shows that Mizoguchi-Takahashi local contractions are equivalent to Feng-Liu contractions in complete, metrically convex spaces, and explores their relationships and implications in compact spaces and approximation theory.

Contribution

It proves the equivalence of Mizoguchi-Takahashi and Feng-Liu contractions in complete, metrically convex spaces and analyzes their relationships in compact spaces.

Findings

Mizoguchi-Takahashi local contractions are set-valued Feng-Liu contractions in complete, metrically convex spaces.

In compact spaces, Mizoguchi-Takahashi and Nadler local contractions are equivalent.

An invariant best approximation result is established.

Abstract

In this article, we establish that any uniformly local Mizoguchi-Takahashi contraction is actually a set-valued contraction due to Feng and Liu on a metrically convex complete metric space. Through an example, we demonstrate that this result need not hold on any arbitrary metric space. Furthermore, when the metric space is compact, we derive that any Mizoguchi-Takahashi local contraction and Nadler local contraction are equivalent. Moreover, a result related to invariant best approximation is established.