Convergence theorems for common fixed points for finite families of nonexpansive mappings in reflexive Banach spaces

摘要

Let E be a real reflexive Banach space with a uniformly Gateaux differentiable norm. Let K be a nonempty closed convex subset of E. Suppose that every nonempty closed convex bounded subset of K has the fixed point property for nonexpansive mappings. Let T-1, T-2, ..., T-N be a family of nonexpansive self-mappings of K, with F :=boolean AND(N)(i=1) Fix(T-i) not equal 0, F = Fix(TNTN-1 ... T-1) Fix(T-1 T-N ... T-2) = ... = Fix(TN-1TN-2 ... T1TN). Let {lambda(n)} be a sequence in (0, 1) satisfying the following conditions: C-1 : lim lambda(n) = 0; C2 : Sigma lambda(n) = infinity.For a fixed delta is an element of (0,1),define S-n : K --> K by S(n)x := (1 -delta)x + delta T(n)x for all x is an element of K where T-n = T-n mod N. For an arbitrary fixed u, x(0) is an element of K, let B := {x is an element of K : T-N TN-1 ... T(1)x = gamma x + (1-gamma)u, for some gamma > 1} be bounded, and let the sequence {x(n)) be defined iteratively by Xn+1 = lambda(n+1) u + (1-lambda(n+1))Sn+1 x(n), for n >= 0. Assume that lim(n-->infinity) parallel to T(n)x(n) - Tn+1 x(n)parallel to = 0. Then, {x(n)} converges strongly to a common fixed point of the family T-1, T-2, ... T-N. This convergence theorem is also proved for non-self maps. (C) 2007 Elsevier Ltd. All rights reserved.