On convergence of iteration processes for nonexpansive semigroups in uniformly convex and uniformly smooth Banach spaces

Let C be a closed, bounded and convex subset of a uniformly convex and uniformly smooth Banach space. Let {T-t}(t >= 0) be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equations x(k+1) = c(k)T(tk+1) (x(k+1)) + (1 - c(k))x(k). We prove that, under certain conditions, the sequence {x(k)} converges weakly to a common fixed point of the semigroup {T-t}(t >= 0). There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property. However, many important spaces like L-p for 1 <= p not equal 2 do not possess the Opial property. In this paper, we do not assume the Opial property. We do assume instead that X is uniformly convex and uniformly smooth. L-p for p > 1 are prime examples of such spaces. (C) 2015 Elsevier Inc. All rights reserved.