Construction of sunny nonexpansive retractions in Banach spaces

Abstract

Let J be a commutative family of nonexpansive self-mappings of a closed convex subset C of a uniformly smooth Banach space X such that the set of common fixed points is nonempty. It is shown that if a certain regularity condition is satisfied, then the sunny nonexpansive retraction from C to F can be constructed in an iterative way.

On viscosity iterative methods for variational inequalities

“…Closely related to this field is the problem of finding a common fixed point of a given family of operators. In this direction, several iterative methods have been developed for these problems, see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. In recent years, viscosity approximation methods have been developed for finding the approximate solutions of the family of operators.…”

Viscosity Approximation Methods for Zeros of Accretive Operators and Fixed Point Problems in Banach Spaces

“…We know that Fi x(S) is nonempty if C is bounded (see [44]). Now, we present the concept of a uniformly asymptotically regular semigroup (see [34][35][36]). …”

Modified noor iterations for nonexpansive semigroups with generalized contraction in Banach spaces

“…Suzuki [7] was the first to introduce again in a Hilbert space the following implicit iteration process: 4) for the nonexpansive semigroup case. In 2002, Benavides et al [8] in a uniformly smooth Banach space, showed that if S satisfies an asymptotic regularity condition and {a n } fulfills the control conditions lim n→∞ α n = 0, ∞ n=1 α n = ∞, and lim n→∞ α n α n +1 = 0 , then both the implicit iteration process (1.4) and the explicit iteration process (1.5) 5) converge to a same point of F(S). In 2005, Xu [9] studied the strong convergence of the implicit iteration process (1.1) and (1.4) in a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping.…”

“…The strong convergence theorem of {x n } is proved in a reflexive Banach space which admits a weakly sequentially continuous duality mapping. Very recently, motivated by the above results, Chen et al [11] proposed the following two modified Mann iterations for nonexpansive semigroups {T(t) : 0 ≤ t < ∞} and obtained the strong convergence theorems in a reflexive Banach space E which admits a weakly sequentially continuous duality mapping: 8) and 9) where f : C C is a contraction. They proved that the implicit iterative scheme {x n } defined by (1.8) converges to an element q of Fix(S), which solves the following variation inequality problem:…”