Fixed point properties of semigroups of non-expansive mappings

Abstract

In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T . In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976.

Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces

“…X is amenable if X is left and right amenable. The semigroup S is called amenable if B.S / has an invariant mean (see, [27][28][29]). Moreover, S is amenable when S is a commutative semigroup or a solvable group.…”

Fixed point properties for semigroups of nonlinear mappings and amenability

“…It is also well-known that if AP(S), the space of continuous almost periodic functions on S, has a left invariant mean, C is compact, convex and S is a nonexpansive representation of S on C, then F (S) = ∅ (see [24,27,32]). …”

Fixed point property for Banach algebras associated to locally compact groups

“…In particular, if S is left amenable as a discrete semigroup, then S is left reversible. Left reversible semigroups have played an important role in the study of common fixed point theorems and ergodic type theorems for semigroups of nonexpansive mappings (see [18,22,23,[30][31][32][34][35][36]). …”