A fixed point theorem for asymptotically nonexpansive mappings

Abstract

Let K be a subset of a Banach space X. A mapping F : K → K F:K \to K is said to be asymptotically nonexpansive if there exists a sequence { k i } \{ {k_i}\} of real numbers with k i → 1 {k_i} \to 1 as i → ∞ i \to \infty such that ‖ F i x − F i y ‖ ≦ k i ‖ x − y ‖ , x , y ∈ K \left \| {{F^i}x - {F^i}y} \right \| \leqq {k_i}\left \| {x - y} \right \|,x,y \in K . It is proved that if K is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if F : K → K F:K \to K is asymptotically nonexpansive, then F has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. Kirk.

Strong and Weak Convergence of Implicit Iterative Process with Errors for Asymptotically Nonexpansive Mappings

“…The results presented in this paper not only generalized and extend the corresponding results in [1], [2], [4]- [9], [11]- [15], but also give an affirmative to the open question suggested by Xu and Ori [14]. Moreover the results even in the case of u n = v n = 0 or β n = 0, v n = 0, ∀n ≥ 1 are also new.…”

“…T is said to be semi-compact, if for any bounded sequence {x n } in D such that ||x n − T x n || → 0 (n → ∞), then there exists a subsequence {x n i } ⊂ {x n } such that x n i → x * ∈ D. 3. T is said to be asymptotically nonexpansive [4], if there exists a sequence {k n } ⊂ [1, ∞) with lim n→∞ k n = 1 such that…”

“…Recently, concerning the convergence problems of an implicit (or nonimplicit) iterative process to a common fixed point for a finite family of asymptotically nonexpansive mappings (or nonexpansive mappings) in Hilbert spaces or uniformly convex Banach spaces have been considered by several authors (see, for example, Bauschke [1], Chang and Cho [2], Goebel and Kirk [4], Górnicki [5], Halpern [6], Lions [7], Osilike [8], Reich [9], Schu [10], Sun [11], Tan and Xu [12], Wittmann [13], Xu and Ori [14] and Zhou and Chang [15]). …”

Fixed point theorems for a sum of two mappings in locally convex spaces

“…Goebel and Kirk [5] introduced the concept of asymptotically nonexpansive mappings in Banach spaces and proved a theorem on the existence of fixed points for such mappings in uniformly convex Banach spaces.…”

Fixed points of a certain class of mappings in spaces with uniformly normal structure

“…However, there are classes of transformations which lie between the nonexpansive transformation and those with Lipschitz constant k > 1 for which fixed point theorems do exist; in particular, the asymptotically nonexpansive mappings (cf. [6]) form such a class. These are mappings T K K having the property that T has Lipschitz constant/c with --, 1 as n oo.…”