Convergence of a hybrid algorithm for a reversible semigroup of nonlinear operators in Banach spaces

The purpose of this paper is to study hybrid iterative schemes of Halpern types for a semigroup ℑ={T(s):s∈S}ℑ={T(s):s∈S} of relatively nonexpansive mappings on a closed and convex subset CC of a Banach space with respect to a sequence {μn}{μn} of asymptotically left invariant means defined on an appropriate invariant subspace of l∞(S)l∞(S). We prove that given a certain sequence {αn}{αn} in [0,1][0,1], x∈Cx∈C, we can generate an iterative sequence {xn}{xn} which converges strongly to ΠF(ℑ)xΠF(ℑ)x where ΠF(ℑ)xΠF(ℑ)x is the generalized projection from CC onto the fixed point set F(ℑ)F(ℑ). Our main result is even new for the case of a Hilbert space.