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

Authors

  • A. WiÅ›nicki Institute of Mathematics Maria Curie-SkÅ‚odowska University

Keywords:

fixed point theory, nonexpansive mappings

Abstract

We show that the direct sum \((X_{1}\oplus ...\oplus X_{r})_{\psi }\) 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 \(r=2\): a direct sum \(% X_{1}\oplus _{\psi }X_{2}\) of uniformly nonsquare spaces with any monotone norm has the fixed point property. Our results are extended for asymptotically nonexpansive mappings in the intermediate sense. 10.1017/S0004972713000440

Published

2013-12-02

Issue

Section

Articles