Let C be a nonempty closed convex subset of a uniformly convex Banach space E with a Fréchet differentiable norm, G a right reversible semitopological semigroup, and 𝒮 = { S ( t ) : t ∈ G } a continuous representation of G as mappings of asymptotically nonexpansive type of C into itself. The weak convergence of an almost-orbit { u ( t ) : t ∈ G } of 𝒮 = { S ( t ) : t ∈ G } on C is established. Furthermore, it is shown that if P is the metric projection of E onto set F ( S ) of all common fixed points of 𝒮 = { S ( t ) : t ∈ G } , then the strong limit of the net { P u ( t ) : t ∈ G } exists.