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

Abstract

Let C be a closed, bounded and convex subset of a uniformly convex and uniformly smooth Banach space. Let {Tt}t≥0 be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equationsxk+1=ckTtk+1(xk+1)+(1−ck)xk. We prove that, under certain conditions, the sequence {xk} converges weakly to a common fixed point of the semigroup {Tt}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 Lp for 1≤p≠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. Lp for p>1 are prime examples of such spaces.