Viscosity approximation methods for nonexpansive mappings

Abstract

Viscosity approximation methods for nonexpansive mappings are studied. Consider a nonexpansive self-mapping T of a closed convex subset C of a Banach space X. Suppose that the set Fix(T ) of fixed points of T is nonempty. For a contraction f on C and t ∈ (0, 1), let x t ∈ C be the unique fixed point of the contraction x → tf (x) + (1 − t)T x. Consider also the iteration process {x n }, where x 0 ∈ C is arbitrary and x n+1 = α n f (x n ) + (1 − α n )T x n for n 1, where {α n } ⊂ (0, 1). If X is either Hilbert or uniformly smooth, then it is shown that {x t } and, under certain appropriate conditions on {α n }, {x n } converge strongly to a fixed point of T which solves some variational inequality.  2004 Published by Elsevier Inc.

The modified viscosity implicit rules for uniformly L-Lipschitzian asymptotically pseudocontractive mappings in Banach spaces

“…Since every nonexpansive mapping is asymptotically nonexpansive and an asymptotically nonexpansive mapping is both asymptotically pseudocontractive and uniformly L-Lipschitzian, in Theorems 2.1 and 2.2, if T is a nonexpansive mapping in Hilbert spaces, then it is the main results of Yu et al [15]. Thus, Theorem 2.1 improves and extends the Yu et al's theorem in several respects and improves some other results (see [1,5,7,8,11,12,14]). Remark 2.6.…”

“…Therefore, we extend the main results of Yu et al (see [15]) from Hilbert spaces to Banach spaces when T is an asymptotically nonexpansive respective asymptotically pseudocontractive map. Further, some other results are also improved (see [1,5,7,8,11,12,14]). …”

“…In 2000, moudafi [7] proved the strong convergence theorem for nonexpansive mappings in real Hilbert spaces. In 2004, Xu [11] studied the following viscosity approximation methods of nonexpansive mapping in a uniformly smooth Banach:…”

Approximating Fixed Points of Nonexpansive Type Mappings in Banach Spaces Without Uniform Convexity

“…Let {δ n }, {β n } and {γ n } be three sequences of nonnegative numbers such that β n ≥ 1 and δ n+1 ≤ β n δ n + γ n for all n ∈ N. If ∞ n=1 (β n − 1) < ∞ and ∞ n=1 γ n < ∞, then lim n→∞ δ n exists. Lemma 2.4 ( [30]). Let {α n } and {γ n } be two real sequences satisfying…”

The viscosity approximation forward-backward splitting method for solving quasi inclusion problems in Banach spaces

“…It is well-known [22,31] that under certain conditions, the sequence {x n } converges in norm to a fixed point q of T .…”