Convergence of almost orbits of semigroups

Abstract

In this paper we establish the following general theorem. Let S be a semitopological reversible semigroup and let Y be a linear subspace of $${\ell }^\infty (S)$$ which contains the constants and is left translation invariant. Suppose that Y has a left invariant mean. Let $$(E, \Vert \cdot \Vert _E)$$ be a uniformly convex Banach space and let C be a nonempty, bounded, closed and convex subset of E. Assume that C has nonempty interior, is locally uniformly rotund and $$\mathcal {T}=\left\{ T\left( s\right) \right\} _{s \in S}$$ is an asymptotically nonexpansive semigroup which acts on C. Assume also that $$\mathcal {T}$$ has a unique fixed point $$\tilde{x}$$ and that, in addition, this point $$\tilde{x}$$ lies on the boundary $$\partial C$$ of C. Then $$\{u(s) \}_{s \in S}$$ converges strongly to $$\tilde{x}$$ for each almost orbit $$u \in AO_Y (\mathcal {T})$$ .