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

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

Abstract and Applied Analysis, 2012, vol. 2012, issue 1

Abstract: Let {tn}⊂(0,1) be such that tn → 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 {tn}, we show the existence of a sequence {xn} n satisfying the relation xn = (1 − tn/kn)f(xn) + (tn/kn)Tnxn and prove that {xn} 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 z0 ∈ K, zn+1 = (1 − tn/kn)f(zn) + (αtn/kn)Tnzn + (βtn/kn)zn converges strongly to the fixed point of T.

Date: 2012
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https://doi.org/10.1155/2012/453452

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:453452

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