Abstract
The main purpose of this article is to introduce the concept of total quasi-ϕ-asymptotically nonexpansive multi-valued mapping and prove the strong convergence theorem in a real uniformly smooth and strictly convex Banach space with Kadec-Klee property. In order to get the theorems, the hybrid algorithms are presented and are used to approximate the fixed point. The results presented in this article improve and extend some recent results announced by some authors.
Highlights
1 Introduction Throughout this article, we always assume that X is a real Banach space with the dual X* and J : X ® 2X is the normalized duality mapping defined by
We use F(T) to denote the set of fixed points of a mapping T, and use R and R+ to denote the set of all real numbers and the set of all nonnegative real numbers, respectively
X is said to be smooth if the limit x + ty − x lim t→0 t exists for all x, y Î U
Summary
(3) A mapping T : C ® C is said to be total quasi-φ-asymptotically nonexpansive if F(T) = 0 and there exist nonnegative real sequences {νn}, {μn} with νn ® 0, μn ® 0 (as n ® ∞) and a strictly increasing continuous function ζ : R+ ® R+ with ζ(0) = 0 such that for all x Î C, p Î F(T) (2) A multi-valued mapping T : C ® N(C) is said to be quasi-φ-asymptotically nonexpansive if F(T) = 0 and there exists a real sequence {kn} ⊂ [1, ∞) with kn ® 1 such that φ(p, wn) ≤ knφ(p, x), ∀n ≥ 1, x ∈ C, wn ∈ Tnx, p ∈ F(T).
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