Strong Convergence Theorem in Viscosity Approximation Sequence of Asymptotically Nonexpansive Mappings
Peiyu Li · Chongqing Shifan Daxue xuebao. Ziran kexue ban · 2008
Suppose that E is a real Banach space with uniformly Gteaux differentiable norm,D is a nonempty closed convex subset of E,and f ∶D→D is a contraction mapping and T ∶D→D is an asymptotically nonexpansive mapping.Let {xn} be the viscosity approximaton sequence defined by xn+1=αnf(yn)+(1-αn)Tnyn,yn=βnxn+(1-βn)Tnxn(n≥0),where αn∈,βn∈.This paper gives a necessary and sufficient condition for {xn} converges strongly to a fixed point of T.Let {αn} satisfy the following conditions:limn→∞αn=0,∑∞n=0αn=∞,define a sequence of contractive mappings Sn by Sn(z)=(1-dn)f(z)+dnTnz,z∈D,where dn=tn-αkn-α,tn∈(α,1)(n=1,2,…),limn→∞tn=1 and k2n-1≤(1-dn)2,n≥n0,let zn∈D be the unique fixed point of Sn,i.e.,zn=Sn(zn)=(1-dn)f(zn)+dnTnzn,n≥1,if limn→∞‖xn-Txn‖=0 and {zn} converges strongly to some z*∈F(T),then the sequence {xn} converges strongly to the fixed z*∈F(T),if and only if {yn} is bounded.My main result is as follows: The result presented in the paper not only gives an affirmative partial ansewer to Reich's open question,but also extends and improves the corresponding results of Reich,Shioji and Takahashi[3] and S.S.Chang[4].