A New Result About Uniformly Lipschitzian Mappings in Banach Space
Qilin Wang · Chongqing Shifan Daxue xuebao. Ziran kexue ban · 2009
Let E be a real Banach space,K be a nonempty closed convex subset of E,Ti:K→K,i=1,2,3 be uniformly Lipschitzian mappings.Let kn[1,∞),kn→1.Let {αn},{βn},{δn}∈,be two sequences in satisfying the following conditions:(i)δn→1(n→∞);(ii)∑∞n=0αn=∞,∑∞n=0βn=∞;(iii)∑∞n=0α2n∞,∑∞n=0αnβn∞;(iv)∑∞n=0αn(kn-1)∞.For any x0∈K,let {xn} be the iterative sequence defined by xn+1=(1-αn)xn+αnT n1ynyn=(1-βn)xn+βnT n2znzn=(1-δn)xn+δnT n3xn.If there exists a strict increasing function φ:[0,∞)→[0,∞) with φ(0)=0 such that 〈Tnix-x*,j(x-x*)〉≤knx-x*-φ(x-x*) for j(x+y)∈J(x+y),x∈K(i=1,2,3),then {xn}converges strongly to x.The main result in the paper improves the main results in references[1,8]and references[6-7].