Abstract and Applied Analysis

Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities

Xionghua Wu, Yeong-Cheng Liou, Zhitao Wu, and Pei-Xia Yang

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Abstract

Let { t n } ( 0,1 ) be such that t n 1 as n , let α and β be two positive numbers such that α + β = 1 , and let f be a contraction. If T be a continuous asymptotically pseudocontractive self-mapping of a nonempty bounded closed convex subset K of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence { t n } , we show the existence of a sequence { x n } n satisfying the relation x n = ( 1 - t n / k n ) f ( x n ) + ( t n / k n ) T n x n and prove that { x n } converges strongly to the fixed point of T , which solves some variational inequality provided T is uniformly asymptotically regular. As an application, if T be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subset K of a real Banach space with a uniformly Gateaux differentiable norm and which possesses uniform normal structure, we prove that the iterative process defined by z 0 K ,  z n + 1 = ( 1 - t n / k n ) f ( z n ) + ( α t n / k n ) T n z n + ( β t n / k n ) z n converges strongly to the fixed point of T .

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 453452, 14 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475819

Digital Object Identifier
doi:10.1155/2012/453452

Mathematical Reviews number (MathSciNet)
MR2970004

Zentralblatt MATH identifier
1253.49007

Citation

Wu, Xionghua; Liou, Yeong-Cheng; Wu, Zhitao; Yang, Pei-Xia. Viscosity Methods of Asymptotically Pseudocontractive and Asymptotically Nonexpansive Mappings for Variational Inequalities. Abstr. Appl. Anal. 2012 (2012), Article ID 453452, 14 pages. doi:10.1155/2012/453452. https://projecteuclid.org/euclid.aaa/1364475819


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