Metric constructions and fixed point theorems in product spaces

TL;DR

This paper introduces a unified method for constructing metrics on product spaces using convex functions, and applies it to fixed point theory, extending existing results and providing new insights into product space properties.

Contribution

It develops a general metric construction framework for product spaces and demonstrates its applications to fixed point properties and geometric structures.

Findings

Unified metric construction encompasses conventional and new metrics.

Fixed point results follow from the new framework.

Examples illustrate the applicability to length and geodesic spaces.

Abstract

The paper studies a general scheme for constructing metrics on a product of metric spaces by means of a family of continuous convex functions. This construction includes the conventional $p$-metrics and generates metrics that are topologically equivalent to the conventional ones. As an application, we study fixed point and approximate fixed point properties for nonexpansive maps on a product space equipped with the constructed metric. We show that existing fixed point results of this type are consequences of our framework. Examples are provided to illustrate the established results. The construction machinery is also used to study products of length and geodesic spaces. The obtained results encompass existing ones and provide a background for potential studies of fixed point properties on these product spaces.