Convergence theorems of iterative algorithms for a family of finite nonexpansive mappings

Let E be a Banach space, C a nonempty closed convex subset of E, f : C -> C a contraction, and Ti : C -> C a nonexpansive mapping with nonempty F := boolean AND(N)(i=1) Fix(Ti), where N >= 1 is an integer and is the set of fixed points of Ti. Let {x(n)} be the sequence defined by x(t)(n) = tf (x(t)(n)) + (1 0- t)Tn+NTn+N-1 center dot center dot center dot T(n+1)x(t)(n) (0 < t < 1). First, it is shown that as t -> 0, the sequence {x(t)(n)} converges strongly to a solution in F of certain variational inequality provided E is reflexive and has a weakly sequentially continuous duality mapping. Then it is proved that the iterative algorithm x(n+1) = lambda(n+1)f (x(n)) + (1 - lambda(n+1))T(n+1)x(n) (n >= 0) converges strongly to a solution in F of certain variational inequality in the same Banach space provided the sequence f An I satisfies certain conditions and the sequence {x(n)} is weakly asymptotically regular. Applications to the convex feasibility problem are included.