Convergence theorems of fixed points for image-strict pseudo-contractions in Hilbert spaces Original Research Article

Let CC be a closed convex subset of a real Hilbert space HH and assume that T:C→HT:C→H is a κκ-strict pseudo-contraction such that F(T)={x∈C:x=Tx}≠0̸F(T)={x∈C:x=Tx}≠0̸. Consider the normal Mann’s iterative algorithm given by View the MathML source∀x1∈C,xn+1=βnxn+(1−βn)PCSxn,n≥1, Turn MathJax on where S:C→HS:C→H is defined by Sx=κx+(1−κ)TxSx=κx+(1−κ)Tx, PCPC is the metric projection of HH onto CC and View the MathML sourceβn=αn−κ1−κ for all n≥1n≥1. It is proved that if the control parameter sequence {αn}{αn} is chosen so that κ≤αn≤1κ≤αn≤1 and View the MathML source∑n=1∞(αn−κ)(1−αn)=∞, then {xn}{xn} converges weakly to a fixed point of TT. 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 CC. 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 κκ-strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007) 336–349] from κκ-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 κκ-strict pseudo-contractions.