Abstract
In this paper, we prove the existence of common fixed points for a generalized asymptotically nonexpansive semigroup { T s : s ∈ S } Open image in new window in CAT(0) spaces, when S is a left reversible semitopological semigroup. We also prove Δ- and strong convergence of such a semigroup when S is a right reversible semitopological semigroup. Our results improve and extend the corresponding results existing in the literature.
Highlights
1 Introduction Let S be a semitopological semigroup, i.e., S is a semigroup with a Hausdorff topology such that for each s ∈ S, the mappings s → ts and s → st from S to S are continuous, and let BC(S) be the Banach space of all bounded continuous real-valued functions with supremum norm
If S is left reversible, (S, ) is a directed system when the binary relation ‘ ’ on S is defined by t s if and only if {t} ∪ tS ⊆ {s} ∪ sS, for t, s ∈ S
Left reversible semitopological semigroups include all commutative semigroups and all semitopological semigroups which are left amenable as discrete semigroups; see [ ]
Summary
Let S be a semitopological semigroup, i.e., S is a semigroup with a Hausdorff topology such that for each s ∈ S, the mappings s → ts and s → st from S to S are continuous, and let BC(S) be the Banach space of all bounded continuous real-valued functions with supremum norm. We introduce a new semigroup for a left (or right) reversible semitopological semigroup on metric spaces, called a generalized asymptotically nonexpansive semigroup, and prove the existence and convergence theorems for this semigroup in CAT( ) spaces. A semigroup T = {Ts : s ∈ S} of mappings of C into itself is said to be (i) nonexpansive if d(Tsx, Tsy) ≤ d(x, y) for all x, y ∈ C and s ∈ S; (ii) asymptotically nonexpansive if there exists a nonnegative real number ks ≥ with lims ks = such that d(Tsx, Tsy) ≤ ( + ks)d(x, y) for each x, y ∈ C and s ∈ S. Let S be a left reversible semitopological semigroup, C be a nonempty closed convex subset of a complete CAT( ) space X, and T = {Ts : s ∈ S} be a generalized asymptotically nonexpansive semigroup of C into itself.
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