The class of asymptotically nonexpansive maps was introduced by Goebel and Kirk [18] as a generalization of the class of nonexpansive maps. They proved that if K is a nonempty closed convex bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive self-mapping of K , then T has a fixed point. Alber and Guerre-Delabriere have studied in [3–5] weakly contracti...

In this paper, we prove a mean ergodic theorem for nonexpansive mappings in Hadamard (nonpositive curvature metric) spaces, which extends the Baillon nonlinear theorem. The main result shows that sequence given by Karcher means of iterations mapping with nonempty fixed point set converges weakly to mapping. This also remains true 1-parameter continuous semigroup contractions.

Let C be a closed convex subset of a uniformly smooth Banach space E, and T : C → E a nonexpansive nonself-mapping satisfying the weakly inwardness condition such that F(T) = ∅, and f : C → C a fixed contractive mapping. For t ∈ (0,1), the implicit iterative sequence {xt} is defined by xt = P(t f (xt) + (1− t)Txt), the explicit iterative sequence {xn} is given by xn+1 = P(αn f (xn) + (1−αn)Txn)...

Let H be a Hilbert space and let C be a closed convex nonempty subset of H and [Formula: see text] a non-self nonexpansive mapping. A map [Formula: see text] defined by [Formula: see text]. Then, for a fixed [Formula: see text] and for [Formula: see text], Krasnoselskii-Mann algorithm is defined by [Formula: see text] where [Formula: see text]. Recently, Colao and Marino (Fixed Point Theory App...

In this paper, we prove the analog to Browder and Göhde fixed point theorem for G-nonexpansive mappings in complete hyperbolic metric spaces uniformly convex. In the linear case, this result is refined. Indeed, we prove that if X is a Banach space uniformly convex in every direction endowed with a graph G, then every G-nonexpansive mapping T : A → A, where A is a nonempty weakly compact convex ...

Definition . Let T : C → C be a mapping. T is said to be total asymptotically nonexpansive if there exist sequences {μn}, {νn} with μn,νn →  as n → ∞ and a strictly increasing continuous function ψ : R → R with ψ() =  such that ‖Tnx – Tny‖ ≤ ‖x – y‖ +μnψ(‖x – y‖) + νn holds for all x, y ∈ C and all n ∈N. T is said to be total asymptotically quasi-nonexpansive if F(T) = ∅, there exist seque...

The purpose of this paper is first to introduce the concept of total quasi-φ-asymptotically nonexpansive mapping which contains many kinds of mappings as its special cases and then to use a hybrid algorithm to introduce a new iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems and the set of common fixed points for a count...

In this paper, we introduce an iterative sequence by using a hybrid generalized f−projection algorithm for finding a common element of the set of fixed points of a relatively weak nonexpansive mapping and the set of solutions of a generalized variational inequality in a Banach space. Our results extend and improve the recent ones announced by Y. Liu [Strong convergence theorems for variational ...

Let X be a Banach space and K a nonempty subset of X. The set K is called proximinal if for each x ∈ X, there exists an element y ∈ K such that ‖x − y‖ d x,K , where d x,K inf{‖x − z‖ : z ∈ K}. Let CB K , C K , P K , F T denote the family of nonempty closed bounded subsets, nonempty compact subsets, nonempty proximinal bounded subsets of K, and the set of fixed points, respectively. A multivalu...