Convergence of ergodic means of orbits of semigroups of nonexpansive mappings in sets with the ΓΓ-Opial property

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 {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 {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.