The fixed point property in direct sums and modulus R(a,X)

TL;DR

This paper proves that the direct sum of Banach spaces with certain properties has the fixed point property for nonexpansive mappings, especially when the spaces are uniformly nonsquare, extending to asymptotically nonexpansive mappings.

Contribution

It establishes the fixed point property for direct sums of Banach spaces with strictly monotone norms under new conditions, including uniformly nonsquare spaces and asymptotically nonexpansive mappings.

Findings

Direct sums with strictly monotone norms have FPP if each space has M(X_i)>1.

Uniformly nonsquare spaces' sums enjoy FPP under any monotone norm.

Results extend to asymptotically nonexpansive mappings.

Abstract

We show that the direct sum of Banach spaces $X_{1},..., X_{r}$ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M(X_{i})>1$ for each $i=1,...,r$. In particular, $(X_{1} \oplus ... \oplus X_{r})_{\psi}$ enjoys the fixed point property if Banach spaces $X_{i}$ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for r=2: the direct sum of uniformly nonsquare spaces $X_{1}, X_{2}$ with any monotone norm has FPP. Our results are extended for asymptotically nonexpansive mappings in the intermediate sense.