Abstract and Applied Analysis

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

Zhiqun Xue, Yaning Wang, and Haiyun Zhou

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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 D be a uniformly generalized Lipschitz generalized asymptotically Φ -strongly pseudocontractive mapping with q F ( T ) . 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 1 , b n + d n 1 ; (ii) a n , b n , d n 0 as n and c n = o ( a n ) ; (iii) Σ n = 0 a n = . For some x 0 , z 0 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

Permanent link to this document
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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