Nonexpansive Mappings In Hyperbolic Spaces Research Articles

Fixed point approximation of Picard normal S-iteration process for generalized nonexpansive mappings in hyperbolic spaces

In this paper, we establish strong and $$\Delta$$ -convergence theorems for a relatively new iteration process generated by generalized nonexpansive mappings in uniformly convex hyperbolic spaces. The theorems presented in this paper generalizes corresponding theorems for uniformly convex normed spaces of Kadioglu and Yildirim (Approximating fixed points of nonexpansive mappings by faster iteration process, arXiv:1402.6530v1 [math.FA], 2014) and CAT(0)-spaces of Abbas et al. (J Inequal Appl 2014:212, 2014) and many others in this direction.

Convergence theorems for finite families of total asymptotically nonexpansive mappings in hyperbolic spaces

In this paper, using a multistep iterative scheme, we establish strong and Δ-convergence theorems for finite families of total asymptotically quasi-nonexpansive mappings in uniformly convex hyperbolic spaces. We then establish Δ- and polar convergence theorems for finite families of total asymptotically nonexpansive mappings in CAT(0) spaces. These new theorems are extensions, improvements, and generalizations of some recently announced results by many authors.

Convergence theorems for a family of multivalued nonexpansive mappings in hyperbolic spaces

Abstract In this article we modify an iteration process to prove strong convergence and Δ— convergence theorems for a finite family of nonexpansive multivalued mappings in hyperbolic spaces. The results presented here extend some existing results in the literature.

Two porosity theorems for nonexpansive mappings in hyperbolic spaces

In a previous paper of ours we used the notion of porosity to show that most of the nonexpansive self-mappings of bounded, closed and convex subsets of a Banach space are contractive and possess a unique fixed point which is the uniform limit of all iterates. In this paper we extend this result to nonexpansive self-mappings of closed and convex sets in a Banach space which are not necessarily bounded. As a matter of fact, it turns out that our results are true for all complete hyperbolic metric spaces.

STRONG AND Δ-CONVERGENCE OF A FASTER ITERATION PROCESS IN HYPERBOLIC SPACE

Abstract. In this article, we first give metric version of an iterationscheme of Agarwal et al. [1] and approximate fixed points of two finitefamilies of nonexpansive mappings in hyperbolic spaces through this iter-ation scheme which is independent of but faster than Mann and Ishikawascheme. Also we consider case of three finite families of nonexpansivemappings. But, we need an extra condition to get convergence. Our con-vergence theorems generalize and refine many know results in the currentliterature. 1. IntroductionThroughout the article, Ndenotes the set of positive integers and I denotesthe set of first N natural numbers. Let (X,d) be a metric space and K bea nonempty subset of X. A selfmap T on K is said to be nonexpansive ifd(Tx,Ty) ≤ d(x,y). Denote by F (T) the set of fixed points of T and byF = ∩ Ni=1 (F (T i ) ∩F (S i )) the set of common fixed points of two finite familiesof mappings {T i : i ∈ I} and {S i : i ∈ I}.We know that Mann and Ishikawa iteration processes are defined for givenx

Demiclosed Principle and Δ-Convergence of Fixed Points for Total Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

Demiclosed Principle and Δ-Convergence of Fixed Points for Total Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

Fixed Point Approximation of Generalized Nonexpansive Mappings in Hyperbolic Spaces

We prove strong and Δ-convergence theorems for generalized nonexpansive mappings in uniformly convex hyperbolic spaces using S-iteration process due to Agarwal et al. As uniformly convex hyperbolic spaces contain Banach spaces as well as CAT(0) spaces, our results can be viewed as extension and generalization of several well-known results in Banach spaces as well as CAT(0) spaces.

Formula omitted]-convergence theorems for multi-valued nonexpansive mappings in hyperbolic spaces

The purpose of this paper is to introduce the mixed Agarwal–O’Regan–Sahu type iterative scheme (Agarwal et al., 2007) for finding a common fixed point of the multi-valued nonexpansive mappings in the setting of hyperbolic spaces. Under suitable conditions, some Δ-convergence theorems of the iterative sequence generated by the proposed scheme to approximate a common fixed point of multi-valued nonexpansive mappings are proved. The results presented in the paper extend and improve some recent results announced in the current literature.

Mixed type iterations for multivalued nonexpansive mappings in hyperbolic spaces

Abstract The purpose of this paper is to extend the iteration scheme of multivalued nonexpansive mappings from a Banach space to a hyperbolic space by proving Δ-convergence theorems for two multivalued nonexpansive mappings in terms of mixed type iteration processes to approximate a common fixed point of two multivalued nonexpansive mappings in hyperbolic spaces. The results presented in this paper are new and can be regarded as an extension of corresponding results from Banach spaces to hyperbolic spaces in the literature. MSC:47H10, 54H25.

Some convergence results for modified SP-iteration scheme in hyperbolic spaces

In this paper, we prove some strong and Δ-convergence theorems for a modified SP-iteration scheme for total asymptotically nonexpansive mappings in hyperbolic spaces by employing recent technical results of Khan et al. (Fixed Point Theory Appl. 2012:54, 2012). The results presented here extend and improve some well-known results in the current literature. MSC:47H09, 47H10.

Fixed point approximation of asymptotically nonexpansive mappings in hyperbolic spaces

Convergence theorems are established in a hyperbolic space for the modified Noor iterations with errors of asymptotically nonexpansive mappings. The obtained results extend and improve the several known results in Banach spaces and CAT(0) spaces simultaneously.

Δ−Convergence Theorems for Multivalued Non-expansive Mappings in Hyperbolic Spaces

The purpose of this paper is that iteration scheme of multivalued non-expansive mappings in Banach spaces is extended to hyperbolic spaces and to prove some Δ−convergence theorems of the mixed type iteration process to approximating a common fixed point for two multivalued non-expansive mappings and two non-expansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current literature.

Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces

Abstract The purpose of this paper is to introduce the concept of total asymptotically nonexpansive mappings and to prove some Δ-convergence theorems of the mixed type iteration process to approximating a common fixed point for two asymptotically nonexpansive mappings and two total asymptotically nonexpansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current literature. MSC:47H09, 47H10.

△-convergence for mixed-type total asymptotically nonexpansive mappings in hyperbolic spaces

In this paper, we prove some △-convergence theorems in a hyperbolic space. A mixed Agarwal-O’Regan-Sahu type iterative scheme for approximating a common fixed point of total asymptotically nonexpansive mappings is constructed. Our results extend some results in the literature.MSC:47H09, 49M05.

Convergence Theorems for Total Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

The purpose of this paper is to introduce the concept oftotal asymptotically nonexpansive mappingsand to prove someΔ-convergence theorems of the iteration process for this kind of mappings in the setting ofhyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current literature.

Mann Iterates of Directionally Nonexpansive Mappings in Hyperbolic Spaces

In a previous paper, the first author derived an explicit quantitative version of a theorem due to Borwein, Reich and Shafrir on the asymptotic behaviour of Mann iterations of nonexpansive mappings of convex sets C in normed linear spaces. This quantitative version, which was obtained by a logical analysis of the ineffective proof given by Borwein, Reich and Shafrir, could be used to obtain strong uniform bounds on the asymptotic regularity of such iterations in the case of bounded C and even weaker conditions. In this paper we extend these results to hyperbolic spaces and directionally nonexpansive mappings. In particular, we obtain significantly stronger and more general forms of the main results of a recent paper by W.A. Kirk with explicit bounds. As a special feature of our approach, which is based on logical analysis instead of functional analysis, no functional analytic embeddings are needed to obtain our uniformity results which contain all previously known results of this kind as special cases.

Mann iterates of directionally nonexpansive mappings in hyperbolic spaces

In a previous paper, the first author derived an explicit quantitative version of a theorem due to Borwein, Reich, and Shafrir on the asymptotic behaviour of Mann iterations of nonexpansive mappings of convex sets C in normed linear spaces. This quantitative version, which was obtained by a logical analysis of the ineffective proof given by Borwein, Reich, and Shafrir, could be used to obtain strong uniform bounds on the asymptotic regularity of such iterations in the case of bounded C and even weaker conditions. In this paper, we extend these results to hyperbolic spaces and directionally nonexpansive mappings. In particular, we obtain significantly stronger and more general forms of the main results of a recent paper by W. A. Kirk with explicit bounds. As a special feature of our approach, which is based on logical analysis instead of functional analysis, no functional analytic embeddings are needed to obtain our uniformity results which contain all previously known results of this kind as special cases.