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

Let KK be a closed convex subset of a real Hilbert space HH and assume that Ti:K→HTi:K→H is a kiki-strictly pseudo-contractive mapping for some 0≤ki<10≤ki<1 such that View the MathML source∩i=1NF(Ti)={x∈K:x=Tix,i=1,2…,N}≠0̸. Consider the following iterative algorithm in KK given by View the MathML source{x1∈K,xn+1=αnγf(xn)+(I−αnA)PKSxn,∀n≥1, Turn MathJax on where II denotes the identity mapping on KK, S:K→HS:K→H is a mapping defined by View the MathML sourceSx=kx+(1−k)∑i=1NηiTix, PKPK is the metric projection of HH onto KK, AA is a bounded linear strong positive operator on KK, ff is a contraction on KK. It is proved that the sequence {xn}{xn} generated by the above iterative algorithm converges strongly to a common fixed point of View the MathML source{Ti}i=1N, which solves a variational inequality related to the linear operator AA. 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.