A New Iterative Algorithm for Approximating Common Fixed Points for Asymptotically Nonexpansive Mappings

Abstract

Suppose that K is a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction. Let T 1 ,T 2 : K → E be two weakly inward and asymptotically nonexpansive mappings with respect to P with sequences {K n }, {l n } ⊂ [1,∞), lim n→∞ k n = 1, lim n→∞ l n = 1, F(T 1) ∩ F(T 2) = {x ∈ K : T 1 x = T 2 x = x} = ∅, respectively. Suppose that {x n } is a sequence in K generated iteratively by x 1 ∈ K, x n+1 = α n x n + β n (PT 1) n x n + γ n (PT 2) n x n , for all n ≥ 1, where {α n }, {β n }, and {γ n } are three real sequences in [ ,1 − ] for some > 0 which satisfy condition α n + β n + γ n = 1. Then, we have the following. (1) If one of T 1 and T 2 is completely continuous or demicompact and ∞ n=1 (k n − 1) < ∞, ∞ n=1 (l n − 1) < ∞, then the strong convergence of {x n } to some q ∈ F(T 1) ∩ F(T 2) is established. (2) If E is a real uniformly convex Banach space satisfying Opial's condition or whose norm is Fréchet differentiable, then the weak convergence of {x n } to some q ∈ F(T 1) ∩ F(T 2) is proved.

Approximating Common Fixed Points of One-step Iterative Scheme with Error for Asymptotically Quasi-nonexpansive Type Nonself-Mappings

“…In this paper, a new one-step iterative scheme with error for approximating common fixed points of asymptotically quasi-nonexpansive type nonself-mappings in Banach space is defined. The results obtained in this paper extend and improve the recent ones, announced by H. Y. Zhou, Y. J. Cho, and S. M. Kang [Zhou et al,(2007), namely, A new iterative algorithm for approximating common fixed points for asymptotically nonexpansive mappings, published to Fixed Point Theory and Applications 2007 : 1-9], and many others. …”

“…Recently, H. Y. Zhou, Y. J. Cho, and S. M. Kang [8] gave a new iterative scheme(5) for approximating common fixed point of two asymptotically nonexpansive nonself-mappings with respect to P and proving some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces.…”

“…The purpose of this paper is to study approximating common fixed point of two asymptotically quasi-nonexpansive type nonself-mappings with respect to P and to a strong convergence theorem, and to define one-step iterative scheme with error which modified of H. Y. Zhou, Y. J. Cho, and S. M. Kang [8] iteration as follows.…”

“…If ν n ≡ 0, then (4) reduced to the iterative scheme, defined by H. Y. Zhou, Y. J. Cho, and S. M. Kang [8], as follows:…”

A hybrid iteration scheme for equilibrium problems and common fixed point problems of generalized quasi-ϕ-asymptotically nonexpansive mappings in Banach spaces

“…Further, the set F(T) is closed and convex. Since 1972, a host of authors have studied the weak and strong convergence problems of the iterative algorithms for such a class of mappings (see, e.g., [4][5][6] and the references therein).…”