Weak and Strong Convergence Theorems for k-Strictly Pseudo-Contractive in Hilbert Space

LetK be a nonempty closed convex subset of a real Hilbert space H , and assume that Ti : K → H, i = 1, 2...N be a finite family of ki-strictly pseudo-contractive mappings for some 0 ≤ ki ≤ 1 such that ⋂N i=1 F (Ti) = {x ∈ K : x = Tix, i = 1, 2...N} = ∅. For the following iterative algorithm in K, for x1, x ′ 1 ∈ K and u ∈ K, { yn = PK [kxn + (1− k)Σi=1λiTixn] xn+1 = βnxn + (1− βn)yn and { y′ n = PK [α ′ nx ′ n + (1− α′ n)Σi=1λiTixn] xn+1 = β ′ nu+ (1− β ′ n)y n PK is the metric projection of H onto K, {α′ n} and {β ′ n} are sequences in (0,1) satisfying appropriate conditions, we proved that {xn} and {xn} respectively converges strongly to a common fixed point of {Ti}i=1. Our results improve and extend the results announced by Genaro L.A.and H.K.Xu [Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonl.Anal.67(2007) 2258-2271], T.H.Kim and H.K.Xu [Strong convergence of modified Mann iterations, Nonlinear Anal.61(2005)51-60] and G.Marino and H.K.Xu [Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J.Math.Anal.Appl.329(2007)336-346]. This work is supported by the National Science Foundation of China, Grant 10771050. 2856 Qinwei Fan, Minggang Yan, Yanrong Yu and Rudong Chen