Strong convergence theorems for common fixed points of uniformly L-Lipschitzian pseudocontractive semi-groups

Abstract

Let K be a nonempty closed convex subset of a uniformly convex real Banach space E which has uniformly Gâteaux differentiable norm. Let be a strongly continuous uniformly L-Lipschitzian semi-group of pseudocontractive mappings from K into E satisfying the weakly inward condition with a nonempty common fixed point set. Then, for a given u∈K, there exists a unique point u_n in K satisfying , where α_n∈[0,1) and t_n  > 0 are real sequences satisfying appropriate conditions. Furthermore, {u_n } converges strongly to a fixed point of . Moreover, explicit iteration procedures which converge strongly to a fixed point of are constructed. A corollary of this result gives an affirmative answer to a recent question posed in Suzuki (2003, Proceedings of the American Mathematical Society, 131, 2133–2136).

Keywords:

• Nonexpansive maps
• pseudocontractive maps
• strongly continuous semigroups
• uniformly Gâteaux differentiable norm
• uniformly L-Lipschitzian

Keywords:

• 2000 Mathematics Subject Classifications
• 47H05
• 47H09
• 47H10
• 47H20

Acknowledgements

The second author undertook this work when he was visiting the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, as a postdoctoral fellow.