Approximation of Fixed Points of a Generalized Asymptotically Quasi-nonexpansive Type Mapping
Liang Tian-juan · Chongqing Shifan Daxue xuebao. Ziran kexue ban · 2008
The purpose of this paper is to study the strong convergence of a generalized asymptotically quasi-nonexpansive type mapping in the non-empty closed and convex subset in Banach spaces,and to give some necessary and sufficient conditions for the modified Ishikawa iterative sequence with errors to converge strongly to a fixed point of the generalized asymptotically quasi-nonexpansive type mapping.Let E is a Banach space,C is a nonempty closed convex subset of E,and T∶C→C is a generalized asymptotically quasi-nonexpansive type mapping with asymptotically Coefficient kn such that ∑∞n=1(kn-1)∞.Let F(T) is bounded,and T stongly converges to a point of F(T) again.Define the modified Ishikawa iterative sequence {xn} with error as given in(1) with {un},{vn} bounded sequences in C and {αn},{βn},{γn},{ξn},{ηn},{δn} sequences in satisfying αn+βn+γn=1,ξn+ηn+δn=1,∑∞n=1βn+∞,∑∞n=1γn+∞.Then,{xn} converges to a fixed point of T.If and only if limn→∞ inf d(xn,F(T))=0,where d(x,A) denotes the distance of x to set A.The results presented in this paper extend and improve the corresponding results of refs.