Annals of Functional Analysis

A general iterative algorithm for nonexpansive mappings in Banach spaces

Bashir Ali, Yekini Shehu, and ‎Godwin C‎. ‎Ugwunnadi

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Abstract

Let $E$ be a real $q$-uniformly smooth Banach space whose duality map is weakly sequentially continuous. Let $T:E\to E$ be a nonexpansive mapping with $F(T)\neq\emptyset.$ Let $A:E\to E$ be an $\eta$-strongly accretive map which is also $\kappa$-Lipschitzian. Let $f:E\to E$ be a contraction map with coefficient $0\lt \alpha\lt1.$ Let a sequence $\{y_{n}\}$ be defined iteratively by $y_{0}\in E,~~ y_{n+1}=\alpha_n\gamma f(y_n)+(I-\alpha_n\mu A)Ty_n,n\geq0,$ where $\{\alpha_n\},~~\gamma$ and $\mu$ satisfy some appropriate conditions. Then, we prove that $\{y_{n}\}$ converges strongly to the unique solution $x^{*} \in F(T)$ of the variational inequality $\langle(\gamma f-\mu A)x^{*},j(y-x^{*})\rangle\leq0,~\forall~y\in F(T).$ Convergence of the correspondent implicit scheme is also proved without the assumption that $E$ has weakly sequentially continuous duality map. Our results are applicable in $l_{p}$ spaces, $1\lt p \lt \infty$.

Article information

Source
Ann. Funct. Anal., Volume 2, Number 2 (2011), 10- 21.

Dates
First available in Project Euclid: 12 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.afa/1399900190

Digital Object Identifier
doi:10.15352/afa/1399900190

Mathematical Reviews number (MathSciNet)
MR2855282

Zentralblatt MATH identifier
1251.47055

Subjects
Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.
Secondary: 47H10‎ ‎47J20

Keywords
$\eta-$strongly accretive maps ‎$\kappa-$Lipschitzian ‎maps ‎nonexpansive maps ‎$q-$uniformly smooth Banach spaces

Citation

Ali, Bashir; ‎Ugwunnadi, ‎Godwin C‎.; Shehu, Yekini. A general iterative algorithm for nonexpansive mappings in Banach spaces. Ann. Funct. Anal. 2 (2011), no. 2, 10-- 21. doi:10.15352/afa/1399900190. https://projecteuclid.org/euclid.afa/1399900190


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