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

Let { ð ‘¡ ð ‘› } ⊂ ( 0 , 1 ) be such that ð ‘¡ ð ‘› → 1 as ð ‘› → ∞ , let ð ›¼ and ð ›½ be two positive numbers such that ð ›¼ + ð ›½ = 1 , and let ð ‘“ be a contraction. If 𠑇 be a continuous asymptotically pseudocontractive self-mapping of a nonempty bounded closed convex subset ð ¾ of a real reflexive Banach space with a uniformly Gateaux differentiable norm, under suitable conditions on the sequence { ð ‘¡ ð ‘› } , we show the existence of a sequence { ð ‘¥ ð ‘› } ð ‘› satisfying the relation ð ‘¥ ð ‘› = ( 1 − ð ‘¡ ð ‘› / 𠑘 ð ‘› ) ð ‘“ ( ð ‘¥ ð ‘› ) + ( ð ‘¡ ð ‘› / 𠑘 ð ‘› ) 𠑇 ð ‘› ð ‘¥ ð ‘› and prove that { ð ‘¥ ð ‘› } converges strongly to the fixed point of 𠑇 , which solves some variational inequality provided 𠑇 is uniformly asymptotically regular. As an application, if 𠑇 be an asymptotically nonexpansive self-mapping of a nonempty bounded closed convex subset ð ¾ 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 𠑧 0 ∈ ð ¾ , 𠑧 ð ‘› + 1 = ( 1 − ð ‘¡ ð ‘› / 𠑘 ð ‘› ) ð ‘“ ( 𠑧 ð ‘› ) + ( ð ›¼ ð ‘¡ ð ‘› / 𠑘 ð ‘› ) 𠑇 ð ‘› 𠑧 ð ‘› + ( ð ›½ ð ‘¡ ð ‘› / 𠑘 ð ‘› ) 𠑧 ð ‘› converges strongly to the fixed point of 𠑇 .