Abstract and Applied Analysis

The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

Abstract

Let $E$ be an arbitrary uniformly smooth real Banach space, let $D$ be a nonempty closed convex subset of $E$, and let $T:D\to D$ be a uniformly generalized Lipschitz generalized asymptotically $\mathrm{\Phi }$-strongly pseudocontractive mapping with $q\in F(T)\ne \varnothing$. Let $\{{a}_{n}\},\{{b}_{n}\},\{{c}_{n}\},\{{d}_{n}\}$ be four real sequences in $[0,1]$ and satisfy the conditions: (i) ${a}_{n}+{c}_{n}\le 1$, ${b}_{n}+{d}_{n}\le 1$; (ii) ${a}_{n},{b}_{n},{d}_{n}\to 0$ as $n\to \infty$ and ${c}_{n}=o({a}_{n})$; (iii) ${\Sigma }_{n=0}^{\infty }{a}_{n}=\infty$. For some ${x}_{0},{z}_{0}\in D$, let $\{{u}_{n}\},\{{v}_{n}\},\{{w}_{n}\}$ be any bounded sequences in $D$, and let $\{{x}_{n}\},\{{z}_{n}\}$ be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of $\{{x}_{n}\}$ is equivalent to that of $\{{z}_{n}\}$.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 909187, 15 pages.

Dates
First available in Project Euclid: 28 March 2013

https://projecteuclid.org/euclid.aaa/1364475837

Digital Object Identifier
doi:10.1155/2012/909187

Mathematical Reviews number (MathSciNet)
MR2975304

Zentralblatt MATH identifier
06116470

Citation

Xue, Zhiqun; Wang, Yaning; Zhou, Haiyun. The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption. Abstr. Appl. Anal. 2012 (2012), Article ID 909187, 15 pages. doi:10.1155/2012/909187. https://projecteuclid.org/euclid.aaa/1364475837

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