Iterative approximation of fixed points of Lipschitz pseudocontractive maps

Abstract

Let E E be a q q -uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g., ℓ p , 1 > p > ∞ \ell _p, \ 1>p>\infty ). Let T T be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset K K of E E and let ω ∈ K \omega \in K be arbitrary. Then the iteration sequence { z n } \{z_n\} defined by z 0 ∈ K , z n + 1 = ( 1 − μ n + 1 ) ω + μ n + 1 y n ; y n = ( 1 − α n ) z n + α n T z n 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 T , provided that { μ n } \{\mu _n\} and { α n } \{\alpha _n\} have certain properties. If E E is a Hilbert space, then { z n } \{z_n\} converges strongly to the unique fixed point of T T closest to ω \omega .