Strong convergence of iterative algorithms with variable coefficients for asymptotically strict pseudocontractive mappings in the intermediate sense and monotone mappings

Ci-Shui Ge

doi:10.1186/1687-1812-2012-68

Abstract

In this article, we propose some iterative algorithms with variable coefficients for finding a common element of the set of fixed points of a uniformly continuous asymptotically κ-strict pseudocon-tractive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. Some strong convergence theorems of these iterative algorithms are obtained without some boundedness assumptions and without some convergence condition. The results of the article improve and extend the recent results of Ceng and Yao, Nadezhkina and Takahashi, and several others. Mathematics Subject Classification (2000): 47H09; 47J20.

Highlights

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥ · ∥, respectively
S : C ® C is called to be asymptotically nonexpansive in the intermediate sense [2], if it is continuous and the following inequality holds: lim sup sup ( Snx − Sny − x − y ) ≤ 0
In order to solve these problems, motivated and inspired by Ceng and Yao [10], Nadezhkina and Takahashi [9], and Ge et al [13], we introduce some new algorithms with variable coefficients based on the hybrid-type method and extragradient-type method for finding a common element of the set of fixed points of a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense and the set of solutions of the variational inequality problem for a monotone, Lipschitzcontinuous mapping in real Hilbert spaces

Summary

Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥ · ∥, respectively. S : C ® C is called to be asymptotically nonexpansive in the intermediate sense [2], if it is continuous and the following inequality holds: lim sup sup ( Snx − Sny − x − y ) ≤ 0.

Results

Conclusion