Some results on k-strictly pseudo-contractive mappings in Hilbert spaces

Let K be a closed convex subset of a real Hilbert space H and assume that T-i : K -> H is a k(i)-strictly pseudo-contractive mapping for some 0 <= k(i) < 1 such that boolean AND(N)(i=1) F(T-i) = {x is an element of K : x = T(i)x, i = 1.2 .... N) not equal empty set. Consider the following iterative algorithm in K given by {x(1) is an element of K, x(n+1) = alpha(n)gamma f(x(n)) + (1-alpha(n)A)P-K Sx(n), for all n >= 1. where I denotes the identity mapping on K, S : K -> H is a mapping defined by Sx = kx + (1 - k) Sigma(N)(i=1) eta(i)T(i)x, P-K is the metric projection of H onto K, A is a bounded linear strong positive operator on K, f is a contraction on K. It is proved that the sequence {x(n)} generated by the above iterative algorithm converges strongly to a common fixed point of {T-i}(i=1)(N), which solves a variational inequality related to the linear operator A. Our results improve and extend the results announced by [B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957-961; R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486-491; A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl. 241 (2000) 46-55; G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318 (2006) 43-52] and many others. (C) 2008 Elsevier Ltd. All rights reserved.