The structure of fixed-point sets of uniformly lipschitzian semigroups

Abstract

In this paper, by asymptotic center techniques, we shown that the set of fixed points of a uniformly k-lipschitzian semigroup (one-parameter or left reversible semitopological) in a uniformly convex Banach space is a retract of the domain if k is close to 1. The results presented in this paper includes (among others, in the discrete situation) many known results as special cases.

Hölder continuous retractions and amenable semigroups of uniformly Lipschitzian mappings in Hilbert spaces

“…Recently, Górnicki (cf. [9,Cor. 14]) proved that if S is left reversible and S = {T t : t ∈ S} is a uniformly k-Lipschitzian semigroup on C, then the set of fixed points of S is a retract of C. It is not clear how to extend Theorem 2 to left reversible semigroups.…”

“…It was shown in [17] that under the assumptions of Theorem 1, the fixedpoint set Fix T is a (continuous) retract of C. Recently, Górnicki (cf. [8,9]) proved several structural results concerning uniformly Lipschitzian mappings but many questions remain open. In [16], Pérez García and Fetter Nathansky gave conditions under which Fix T is a Hölder continuous retract and applied them to the study of n-periodic mappings in Hilbert spaces.…”

“…The aim of this note is to give a qualitative complement to the above results in the case of left amenable (semitopological) semigroups which partially extends a result of Górnicki (cf. [9,Cor. 14]).…”