Viscosity approximation methods for nonexpansive nonself-mappings

Abstract

Let E a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E * , and K be a closed convex subset of E which is also a sunny nonexpansive retract of E, and T : K → E be nonexpansive mappings satisfying the weakly inward condition and F (T ) = ∅, and f : K → K be a fixed contractive mapping. The implicit iterative sequence {x t } is defined by for t ∈ (0, 1)The explicit iterative sequence {x n } is given bywhere α n ∈ (0, 1) and P is sunny nonexpansive retraction of E onto K. We prove that {x t } strongly converges to a fixed point of T as t → 0, and {x n } strongly converges to a fixed point of T as α n satisfying appropriate conditions. The results presented extend and improve the corresponding results of [H.K. Xu, Viscosity approximation methods for nonexpansive mappings,

Viscosity Approximation Method for Nonexpansive Nonself-Mapping and Variational Inequality

“…Let K be a nonempty convex subset of a Banach space E. Then for x ∈ K, set I K (x) is called inward set [2,7], where…”

“…T : K → E is a nonexpansive nonself-mapping. Inspired by Xu [6], in 2006, Y.Song and R.Chen [2] considered the following algorithm, (i) α n → 0, as n → ∞, and…”

The New Viscosity Approximation Methods for Nonexpansive Nonself-Mappings

“…Helpern [3] was the first to study the strong convergence of the iteration process (1). In 1992, Albert [4] studied the convergence of the Ishikawa iteration process in Banach space, which was extended the results of Mann iteration process [5].…”

“…In 2007, Yisheng Song and Qingchun Li [2] found a new viscosity approximation method for nonexpansive nonself-mappings as follows ( ) ( ) where X is a real reflexive Banach space, and C is a closed subset of X which is also a sunny nonexpansive retract of X. :…”

“…In this paper, we will study two new viscosity approximation methods for nonexpansive nonself-mappings in reflexive Banach space X, which can extend the results of Song-Chen [1] and Song-Li [2] on the twodimensional space.…”