Convergence theorems of fixed points for κ-strict pseudo-contractions in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H and assume that T : C -> H is a kappa-strict pseudo-contraction such that F(T) = {x is an element of C : x = Tx} not equal theta. Consider the normal Mann's iterative algorithm given by for all x(1) is an element of C, x(n) + 1 = beta(n)x(n) + (1 - beta(n)) P-C Sx(n), n >= 1, where S : C -> H is defined by Sx = kappa x + (1 - kappa)Tx, P-C is the metric projection of H onto C and beta(n) = alpha(n)-kappa/1-kappa for all n >= 1. It is proved that if the control parameter sequence {alpha(n)} is chosen so that kappa <= alpha(n) <= 1 and Sigma(infinity)(n=1)(alpha(n)-kappa)(1-alpha(n)) = infinity, then {x(n)} converges weakly to a fixed point of T. In order to get a strong convergence theorem, we modify the normal Mann's iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. The results presented in this article respectively improve and extend the recent results of Marino and Xu [G. Marino, H.K. Xu, Weak and strong convergence theorems for K-strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336-349] from K-strictly pseudo-contractive self-mappings to nonself-mappings and of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51-60] from nonexpansive mappings to K-strict pseudo-contractions. (C) 2007 Elsevier Ltd. All rights reserved.