Iterative Schemes for a Family of Finite Asymptotically Pseudocontractive Mappings in Banach Spaces

Let E be a real Banach space and K be a nonempty closed convex and bounded subset of E. Let Ti ： K→ K, i=1, 2,... ,N, be N uniformly L-Lipschitzian, uniformly asymptotically regular with sequences {ε^（i）n} and asymptotically pseudocontractive mappings with sequences {κ^（i）n}, where {κ^（i）n} and {ε^（i）n}, i = 1, 2,... ,N, satisfy certain mild conditions. Let a sequence {xn} be generated from x1 ∈ K by zn：= （1-μn）xn＋μnT^nnxn, xn＋1 ：= λnθnx1＋ [1 - λn（1 ＋ θn）]xn ＋ λnT^nnzn for all integer n ≥ 1, where Tn = Tn（mod N）, and {λn}, {θn} and {μn} are three real sequences in [0, 1] satisfying appropriate conditions. Then ｜｜xn- Tixn｜｜→ 0 as n→∞ for each l ∈ {1, 2,..., N}. The results presented in this paper generalize and improve the corresponding results of Chidume and Zegeye, Reinermann, Rhoades and Schu.

Let E be a real Banach space and K be a nonempty closed convex and bounded subset of E.Let Ti:K → K,i = 1,2,...,N,be N uniformly L-Lipschitzian,uniformly asymptotically regular with sequences {ε(ni)} and asymptotically pseudocontractive mappings with sequences {kn(i)},where {kn(i)} and {εn(i)},i = 1,2,...,N,satisfy certain mild conditions.Let a sequence {xn} be generated from x1 ∈ K by zn:=(1-μn)xn+μnTnn xn,xn+1:= λnθnx1+ [1-λn(1 + θn)]xn + λnTnn zn for all integer n 1,where Tn = Tn(modN),and {λn},{θn} and {μn} are three real sequences in [0,1] satisfying appropriate conditions.Then ||xn-Tlxn|| → 0 as n →∞ for each l ∈ {1,2,...,N}.The results presented in this paper generalize and improve the corresponding results of Chidume and Zegeye[1],Reinermann[10],Rhoades[11] and Schu[13].