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

Abstract

Suppose that $E$ is a real normed linear space, $C$ is a nonempty convex subset of $E$, $T:C\to C$ is a Lipschitzian mapping, and $x^{*}\in C$ is a fixed point of $T$. For given $x_0\in C$, suppose that the sequence $\{x_n\}\subset C$ is the Mann iterative sequence defined by $x_{n+1}=(1-{\alpha }_n)x_n+{\alpha }_{n}T{x}_n,n\ge 0$, where $\{{\alpha }_n\}$ is a sequence in [0, 1], , ${\sum }_{n=0}^{\infty }{\alpha }_n=\infty$. We prove that the sequence $\{x_n\}$ strongly converges to $x^{*}$ if and only if there exists a strictly increasing function $\mathrm{\Phi }:[0,\infty )\to [0,\infty )$ with $\mathrm{\Phi }(0)=0$ such that ${\mathrm{limsup}\hspace{0.17em}}_{n\to \infty }{\inf \hspace{0.17em}}_{j(x_n-x^{*})\in J(x_n-x^{*})}\{\langle T{x}_n-x^{*},j(x_n-x^{*})\rangle -\parallel x_n-x^{*}{\parallel }^2+\mathrm{\Phi }(\parallel x_n-x^{*}\parallel )\}\le 0$.