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

• Main Authors: Xionghua Wu, Yeong-Cheng Liou, Zhitao Wu, Pei-Xia Yang
• Format: Article
• Language: English
• Published: Hindawi Limited 2012-01-01
• Series: Abstract and Applied Analysis
• Online Access: http://dx.doi.org/10.1155/2012/453452

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.