Strong convergence for asymptotical pseudocontractions with the demiclosedness principle in banach spaces

Yuan-Heng Wang,Yong-Hui Xia

doi:10.1186/1687-1812-2012-45

Abstract

The aim of this article is to give an answer to an interesting question proposed in Zhou. At the end of his article, he remarked that it was of great interest to extend his results to certain Banach spaces. So in this article, we extend the demiclosedness principle from Hilbert spaces to Banach spaces. A strong convergence theorem for asymptotical pseudo-contractions in Banach spaces is established. The approaches are based on the extended demiclosedness principle, and the generalized projective operator, and the hybrid method in mathematical programming. Our results extend the previous known results from Hilbert spaces to Banach spaces. MSC: 47H10; 47H09; 47H05.

Highlights

1 Introduction Zhou [1] proposed an interesting problem at the end of his article. He remarked that it was of great interest to extend his results to certain Banach spaces
The main aim is to establish a strong convergence theorem for asymptotical pseudo-contractions in Banach spaces
In 1972, Goebel and Kirk [2] introduced the concept of asymptotically nonexpansive mappings in the Hilbert space

Summary

Introduction

Zhou [1] proposed an interesting problem at the end of his article. He remarked that it was of great interest to extend his results to certain Banach spaces. Theorem 1.1 [3]Let H be a Hilbert space; ≠ K ⊂ H closed bounded convex; L > 0;T : K ® K completely continuous, uniformly L-Lipschitzian and asymptotically pseudo-contractive with sequence {kn} ⊂ [1, ∞); qn = 2kn- 1 for all n ≥ 1 ; Zhou [1] extended Schu’s results by establishing a fixed point theorem for asymptotically pseudo-contraction without any compact assumption on the mappings.

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Conclusion