Necessary and sufficient condition for mann iteration converges to a fixed point of Lipschitzian mappings

摘要

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 {x_n}?C is the Mann iterative sequence defined by x_(n+1)=(1-α_n)x _n+α_nTx_n,n≥0, where {α_n} is a sequence in [0, 1], ∑_(n=0) ~∞α_n ~2<∞, ∑_(n=0) ~∞α_n=∞. We prove that the sequence {x_n} strongly converges to x if and only if there exists a strictly increasing function Φ:[0,∞)→[0,∞) with Φ(0)=0 such that lim sup_(n→∞)inf_(j(xn-x)) ∈J(x_(n-x)){〈Tx_n-x,j(x_n-x)〉-∥x_n-x ∥~2+Φ(∥x_n-x ∥)}≤0.