Iterative Algorithms for Pseudocontraction Semigroups in Banach Spaces

Rabian Wangkeeree, Pakkapon Preechasilp

Abstract

In this paper, a Moudafi’s viscosity approximation method with continuous strong pseudocontractions for a pseudocontraction semigroup is considered. A strong convergence theorem is established in the framework of a reflexive and strictly convex Banach space with a uniformly Gˆateaux differentiable norm. We also propose the modified implicit iteration for a pseudocontraction semigroup and prove the strong convergence theorem. The results presented in this paper mainly improved and extended the corresponding results announced by Song and Xu [Y. Song, S. Xu, Strong convergence theorems for nonexpansive semigroup in Banach spaces, J. Math. Anal. Appl. 338 (2008) 152–161] and Song and Chen [Y. Song, R. Chen, Convergence theorems of iterative algorithms for continuous pseudocontractive mappings, Nonlinear Anal. 67 (2007) 486–497].