A common fixed point theorem for a commuting family of weak^*continuous nonexpansive mappings

Abstract

It is shown that if S is a commuting family of weak * continuous nonexpansive mappings acting on a weak * compact convex subset C of the dual Banach space E, then the set of common fixed points of S is a nonempty nonexpansive retract of C. This partially solves a long-standing open problem in metric fixed point theory in the case of commutative semigroups.

Existence and structure of the common fixed points based on TVS

“…For a dual Banach space E satisfying the weak* fpp, it is still unknown whether E has the weak* fpp for commutative semigroups. Very recently, Borzdynski and Wisnicki [3] proved that if S is a commuting family of weak* continuous nonexpansive mappings acting on a weak* compact convex subset C of the dual Banach space E, then the set of common fixed points of S is a nonempty nonexpansive retract of C. This partially answers a long-standing open problem posed by Lau in [15] (see also [17]). Examples of Banach spaces with the weak* fpp for commutative semigroups include 1 , trace class operators on a Hilbert space, Hardy space H 1 and the Fourier algebra of a compact group (see [16,[19][20][21][22]).…”

Applications of uniform asymptotic regularity to fixed point theorems

“…Lau and Takahashi [18,Theorem 5.3] proved that the answer is affirmative if C is separable. Recently, the authors of the present paper were able to prove that commutative semigroups have (F * ) property (see [5,Theorem 3.6]). We show in Section 3 that a jointly continuous left amenable or left reversible semigroup generated by firmly nonexpansive mappings acting on a bounded τ -compact subset of a Banach space has a common fixed point.…”

Fixed point properties for semigroups of non-expansive mappings in conjugate Banach spaces

“…Remark 2. In our best knowledge up to now, the answer of this question is affirmative for commutative semigroups [3]. Unfortunately, the proof for the commutative case does not use the fact that such semigroups are left amenable, but strongly the abelian property.…”