the notion of a bead metric space is defined as a nice generalization of the uniformly convex normed space such as $cat(0)$ space, where the curvature is bounded from above by zero. in fact, the bead spaces themselves can be considered in particular as natural extensions of convex sets in uniformly convex spaces and normed bead spaces are identical with uniformly convex spaces. in this paper, w...

and Applied Analysis 3 Let E be a real Banach space and let C be a nonempty closed and convex subset of E, and A : C → E∗ be a mapping. The classical variational inequality problem for a mapping A is to find x∗ ∈ C such that 〈 Ax∗, y − x∗〉 ≥ 0, ∀y ∈ C. 1.7 The set of solutions of 1.4 is denoted by VI A,C . Recall that A is called i monotone if 〈 Ax −Ay, x − y〉 ≥ 0, ∀x, y ∈ C, 1.8 ii an α-invers...

and Applied Analysis 3 Lemma 2.2 cf., 4 . Let D be a nonempty subset of a reflexive, strictly convex, and smooth Banach space E. Let R be a retraction from E onto D. Then R is sunny and generalized nonexpansive if and only if 〈 x − Rx, JRx − Jy ≥ 0, 2.2 for all x ∈ E and y ∈ D. A generalized resolvent Jr of a maximal monotone operator B ⊂ E∗ × E is defined by Jr I rBJ −1 for any real number r >...

In this paper, by using generalized viscosity mappings, we prove two strong convergence theorems for finding fixed points of a nonexpansive mapping which is also unique solution the variational inequality. Our results extend and improve recent ones announced some authors.

The authors have obtained the following results: 1 the definition of uniformly closed countable family of nonlinear mappings, 2 strong convergence theorem by the monotone hybrid algorithm for two countable families of hemirelatively nonexpansive mappings in a Banach space with new method of proof, 3 two examples of uniformly closed countable families of nonlinear mappings and applications, 4 an...

Let K be a convex subset of a uniformly convex Banach space. It is proved that if K is compact, then the fixed points of a continuous generalized nonexpansive self-mapping T on K can be approximated by the iterates of T, with t B (0,1), T,(x) = (1 t)x + tT(x), x e K; T, is asymptotically regular if T has a fixed point. Let (X, d) be a (nonempty) metric space. A function a of X X X into [0, oo) ...

The purpose of this article is to prove strong convergence theorems for weak relatively nonexpansive mapping which is firstly presented in this article. In order to get the strong convergence theorems for weak relatively nonexpansive mapping, the monotone CQ iteration method is presented and is used to approximate the fixed point of weak relatively nonexpansive mapping, therefore this article a...

We introduce viscosity approximations by using the shrinking projection method established by Takahashi, Takeuchi, and Kubota, for finding a common element of the set of solutions of the generalized equilibrium problem and the set of fixed points of a quasi-nonexpansive mapping. Furthermore, we also consider the viscosity shrinking projection method for finding a common element of the set of so...

Recently, Chang, et al introduce the concept of total asymptotically nonexpansive mapping which contain the asymptotically nonexpansive mapping. The purpose of the paper is to analyze a three-step iterative scheme for total asymptotically nonexpansive mapping in uniformly convex hyperbolic spaces. Meanwhile, we obtain a ∆-convergence theorem of the three-step iterative scheme for total asymptot...