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 {Tt}t≥0{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.xk+1=ckTtk+1(xk+1)+(1−ck)xk. We prove that, under certain conditions, the sequence {xk}{xk} converges weakly to a common fixed point of the semigroup {Tt}t≥0{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 LpLp for 1≤p≠21≤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. LpLp for p>1p>1 are prime examples of such spaces.