Asymptotic normal structure and fixed points of nonexpansive mappings

Baillon, J.-B.; Schöneberg, R.

doi:10.1090/S0002-9939-1981-0593469-1

Asymptotic normal structure and fixed points of nonexpansive mappings
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by J.-B. Baillon and R. Schöneberg PDF

Proc. Amer. Math. Soc. 81 (1981), 257-264 Request permission

Abstract:

A mapping $f$ defined on a subset $X$ of a Banach space $E$ and taking values in $E$ is said to be nonexpansive if $\left | {f(x) - f(y)} \right | \leqslant \left | {x - y} \right |$ for all $x,y \in X$. In this paper we introduce a promising new geometric property of Banach spaces and show that it yields via a minor modification of known arguments a new fixed point theorem for nonexpansive mappings which includes Kirk’s famous result as well as a recent result of Karlovitz. We also discuss in detail a situation not covered by this result.

References

An example of a reflexive Banach space lacking asymptotic normal structure

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