Fixed point properties for semigroups of nonlinear mappings and amenability

Abstract

In this paper we study fixed point properties for semitopological semigroup of nonexpansive mappings on a bounded closed convex subset of a Banach space. We also study a Schauder fixed point property for a semitopological semigroup of continuous mappings on a compact convex subset of a separated locally convex space. Such semigroups properly include the class of extremely left amenable semitopological semigroups, the free commutative semigroup on one generator and the bicyclic semigroup S 1 = a, b: ab = 1 .

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 on unbounded sets

“…Indeed, in this case, for each finite subset σ of S there is s σ ∈ S such that ss σ = s σ for all s ∈ σ by a theorem of Granirer's [10] (see also [23,Theorem 4.2] for a short proof). Consider the net {s σ c}.…”

Algebraic and analytic properties of semigroups related to fixed point properties of non-expansive mappings

“…We wonder whether one can remove the separability condition on S in Theorems 2.1, 2.2 and 2.4. When n = 1 we can answer this question for Theorem 2.2 affirmatively (see [56,Theorem 5.3]). The bicyclic semigroup is the semigroup generated by a unit e and two more elements p and q subject to the relation pq = e. We denote it by S 1 = e, p, q | pq = e .…”