## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2012, Special Issue (2012), Article ID 641479, 19 pages.

### Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces

#### Abstract

Let $X$ be a uniformly convex Banach space and $\mathcal{S}=\{T(s):0\le s<\infty \}$ be a nonexpansive semigroup such that $F(\mathcal{S})={\bigcap }_{s>0}F(T(s))\ne \varnothing$. Consider the iterative method that generates the sequence $\{{x}_{n}\}$ by the algorithm ${x}_{n+1}={\alpha }_{n}f({x}_{n})+{\beta }_{n}{x}_{n}+(1-{\alpha }_{n}-{\beta }_{n})(1/{s}_{n}){\int }_{0}^{{s}_{n}}T(s){x}_{n}ds,n\ge 0$, where $\{{\alpha }_{n}\}$, $\{{\beta }_{n}\}$, and $\{{s}_{n}\}$ are three sequences satisfying certain conditions, $f:C\to C$ is a contraction mapping. Strong convergence of the algorithm $\{{x}_{n}\}$ is proved assuming $X$ either has a weakly continuous duality map or has a uniformly Gâteaux differentiable norm.

#### Article information

Source
J. Appl. Math., Volume 2012, Special Issue (2012), Article ID 641479, 19 pages.

Dates
First available in Project Euclid: 3 January 2013

https://projecteuclid.org/euclid.jam/1357180273

Digital Object Identifier
doi:10.1155/2012/641479

Mathematical Reviews number (MathSciNet)
MR2889103

Zentralblatt MATH identifier
1235.49028

#### Citation

Wang, Haiqing; Su, Yongfu; Zhang, Hong. Strong Convergence Theorems for Nonexpansive Semigroups and Variational Inequalities in Banach Spaces. J. Appl. Math. 2012, Special Issue (2012), Article ID 641479, 19 pages. doi:10.1155/2012/641479. https://projecteuclid.org/euclid.jam/1357180273

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