On the fixed points of nonexpansive mappings in direct sums of Banach spaces

TL;DR

This paper proves that the weak fixed point property is preserved in the direct sum of Banach spaces under certain conditions, extending known results even when one component is finite-dimensional.

Contribution

It establishes new conditions under which the weak fixed point property holds for direct sums of Banach spaces, including cases with finite-dimensional Y.

Findings

The direct sum of a space with the weak fixed point property and another with specific geometric properties retains the property.

The result applies even when Y is finite-dimensional, broadening previous understanding.

The paper introduces new conditions involving the generalized Gossez-Lami Dozo property and uniform convexity.

Abstract

We show that if a Banach space X has the weak fixed point property for nonexpansive mappings and Y has the generalized Gossez-Lami Dozo property or is uniformly convex in every direction, then the direct sum of X and Y with a strictly monotone norm has the weak fixed point property. The result is new even if Y is finite-dimensional.