Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings - Chang-He Xiang; Jiang-Hua Zhang; Zhe Chen (Journal of Applied Mathematics)
Suppose that E is a real normed linear space, C is a nonempty convex subset of E, T:C→C is a Lipschitzian mapping, and x*∈C is a fixed point of T. For given x0∈C, suppose that the sequence {xn}⊂C is the Mann iterative sequence defined by xn+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 to x* if and only if there exists a strictly increasing function Φ:[0,∞)→[0,∞) with Φ(0)=0 such that limsup n→∞inf j(xn-x*)∈J(xn-x*){〈Txn-x*,j(xn-x*)〉-∥xn-x*∥2+Φ(∥xn-x*∥)}≤0.