Menger Convexity and Hausdorff Metric

TL;DR

This paper extends intersection properties and fixed point results from Takahashi convexity to Menger convexity within metric spaces, broadening the understanding of convexity and fixed point theory.

Contribution

It establishes a nonempty intersection property for Menger convex spaces using the Hausdorff metric and introduces a generalized hybrid mapping with fixed point results.

Findings

Nonempty intersection property for Menger convex spaces.

Reflexive Menger convex spaces identified.

Fixed point results for generalized hybrid mappings.

Abstract

Shimizu and Takahashi have shown that every decreasing sequence of nonempty, bounded, closed, convex subsets of a complete, uniformly Takahashi convex metric space has nonempty intersection. It is well known that the Menger convexity is a generalization of the Takahashi convexity. In this article, we acquire a nonempty intersection property, in terms of the Hausdorff metric, for Menger convex metric spaces, that also provides a class of reflexive Menger convex spaces. We introduce a generalization of $(\alpha, \beta)-$generalized hybrid mapping, and using the obtained nonempty intersection property we derive the fixed point results for this generalized mapping defined on Menger convex spaces.