Minimal invariant sets for nonexpansive mappings share some singular geometrical properties. Here we present some seemingly unknown ones.

The main purpose of this paper is to give weak and strong convergence theorems of a new three-step iterative scheme for asymptotically nonexpansive mappings in Banach spaces and we also give several weak and strong convergence theorems of the three-step iterative scheme with errors for asymptotically nonexpansive mappings in Banach spaces. Mann-type and Ishikawa -type iterations are included by...

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 ...

A generalization of Halpern’s iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern’s iteration, a contractive mapping, and...

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...

In this paper, we introduce a new iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inversestrongly-monotone mapping in a Banach space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common el...

Suppose C is a nonempty bounded closed convex retract of a real uniformly convex Banach space X with uniformly Gâteaux differentiable norm and P as a nonexpansive retraction of X onto C. Let T : C −→ X be an asymptotically nonexpansive nonself-map with sequence {kn}n≥1 ⊂ [1,∞), lim kn = 1, F (T ) = {x ∈ C : Tx = x}, and let u ∈ C. In this paper we study the convergence of the sequences {xn} and...

We propose the class of uniformly convex W -hyperbolic spaces with monotone modulus of uniform convexity (UCW -hyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. UCW -hyperbolic spaces are a natural generalization both of uniformly convex normed spaces and CAT (0)-spaces. Furthermore, we apply proof mining techniques to get effective rates of asympt...

Various random fixed point theorems for different classes of 1-set-contractive random operator are proved. The class of 1-set-contractive random operators includes condensing and nonexpansive random operators. It also includes semicontractive type random operators and locally almost nonexpansive random operators. w Ž . x Thus results due to S. Itoh J. Math. Anal. Appl. 67 1979 , 261]273 , T. C....