Convergence of ergodic means of orbits of semigroups of nonexpansive mappings in sets with the image-Opial property Original Research Article

In this paper we generalize the Suzuki result about convergence of ergodic means of orbits of semigroups of nonexpansive mappings. Namely, we prove the following theorem. Let (X,‖⋅‖)(X,‖⋅‖) be a Banach space and let ΓΓ be a norming set for XX. Let CC be a bounded and convex subset of XX and suppose CC is compact and sequentially compact in the ΓΓ-topology. Let CC have the ΓΓ-Opial property and let View the MathML source{f(t):t≥0} be a nonexpansive semigroup on CC. Then for z∈Cz∈C the following two conditions are equivalent: (i) zz is a common fixed point of View the MathML source{f(t):t≥0}; (ii) there exists a subnet {M(τβ,z)}{M(τβ,z)} of a net {M(t,z)}{M(t,z)} which is convergent to zz in the ΓΓ-topology. Next, we apply this result to construct a nonexpansive ergodic retraction onto a common fixed point set of a nonexpansive semigroup.