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

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 α 2 n < ∞, ∑∞ 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 lim supn→∞infj xn−x∗ ∈J xn−x∗ {〈Txn − x∗, j xn − x∗ 〉 − ‖xn − x∗‖2 Φ ‖xn − x∗‖ } ≤ 0.