Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings

Abstract

Suppose thatEis a real normed linear space,Cis a nonempty convex subset ofE,T:C→Cis a Lipschitzian mapping, andx*∈Cis a fixed point ofT. For givenx0∈C, suppose that the sequence{xn}⊂Cis the Mann iterative sequence defined byxn+1=(1-αn)xn+αnTxn,n≥0, where{αn}is a sequence in [0, 1],∑n=0∞αn2<∞,∑n=0∞αn=∞. We prove that the sequence{xn}strongly converges tox*if and only if there exists a strictly increasing functionΦ:[0,∞)→[0,∞)withΦ(0)=0such thatlimsup n→∞inf j(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}≤0.