Convergence theorems of fixed points for [formula omitted]-strict pseudo-contractions in Hilbert spaces

Abstract

Let C be a closed convex subset of a real Hilbert space H and assume that T : C → H is a κ -strict pseudo-contraction such that F ( T ) = { x ∈ C : x = T x } ≠ 0̸ . Consider the normal Mann’s iterative algorithm given by ∀ x 1 ∈ C , x n + 1 = β n x n + ( 1 − β n ) P C S x n , n ≥ 1 , where S : C → H is defined by S x = κ x + ( 1 − κ ) T x , P C is the metric projection of H onto C and β n = α n − κ 1 − κ for all n ≥ 1 . It is proved that if the control parameter sequence { α n } is chosen so that κ ≤ α n ≤ 1 and ∑ n = 1 ∞ ( α n − κ ) ( 1 − α n ) = ∞ , 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 κ -strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349] from κ -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 κ -strict pseudo-contractions.