MR2410211 (2009b:47107) Păcurar, Mădălina Viscosity approximation of fixed points with $\phi$-contractions. Carpathian J. Math. 24 (2008), no. 1, 88--93. (Reviewer: Diana Caponetti) 47H10 (47H09)

Abstract

Let T be a nonexpansive self-mapping of a closed bounded convex subset Y of a Hilbert space. For l in (0, 1), the author considers the iteration xl = lf(xl)+(1−l)Txl, where f from Y to Y is a $\phi$-contraction. Then, the author proves that (xl)l converges strongly as l goes to 0 to the unique fixed point of the $\phi$-contraction Pof, where P is the metric projection of Y onto the set FT of fixed points of T. The viscosity approximation method of the paper is obtained from the method proposed by A. Moudafi [J. Math. Anal. Appl. 241 (2000), no. 1, 46–55; MR1738332 (2000k:47085)] for mappings in Hilbert spaces, and by H. K. Xu [J. Math. Anal. Appl. 298 (2004), no. 1, 279–291; MR2086546 (2005e:47133)] for mappings in uniformly smooth Banach spaces, by replacing usual contractions by $\phi$-contractions.