Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Abstract:

Let $E$ be a $q$-uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., $\ell _p, \ 1<p<\infty$). Let $T$ be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset $K$ of $E$ and let $\omega \in K$ be arbitrary. Then the iteration sequence $\{z_n\}$ defined by $z_0\in K, \ \ z_{n+1}=(1-\mu _{n+ 1})\omega + \mu _{n+1}y_n; \ \ y_n = (1-\alpha _n)z_n+\alpha _nTz_n$, converges strongly to a fixed point of $T$, provided that $\{\mu _n\}$ and $\{\alpha _n\}$ have certain properties. If $E$ is a Hilbert space, then $\{z_n\}$ converges strongly to the unique fixed point of $T$ closest to $\omega$.