Nonexpansive Nonself-mappings Research Articles

Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces

Abstract In this article, we introduce and study some strong convergence theorems for a mixed-type SP-iteration for three asymptotically nonexpansive self-mappings and three asymptotically nonexpansive nonself-mappings in uniformly convex hyperbolic spaces. In addition to that, we provide an illustrative example. The findings here expand and improve upon some of the relevant conclusions found in the published literature.

A new iterative method for approximating common fixed points of two non-self mappings in a CAT(0) space

In this paper, we introduce a new iterative process for approximating common fixed points of two non-self mappings in the setting of CAT(0) spaces. Then we establish Delta -convergence and strong convergence results for two nonexpansive non-self mappings under appropriate conditions. Moreover, we establish strong convergence theorems for approximating common fixed points of two Lipschitz quasi-nonexpansive non-self mappings under some additional conditions. Our results improve, complement and unify most of the results in the literature.

New Iterative Methods for Finding Common Fixed Points of Two Non-self Mappings in a Real Hilbert Space

Abstract In this paper, we introduce new iterative methods for approximating common fixed points of two non-self mappings in the framework of real Hilbert spaces. We establish weak and strong convergence results for approximating common fixed points of two nonexpansive non-self mappings. In addition, we establish strong convergence results for approximating common fixed points of two quasi-nonexpansive non-self mappings under appropriate conditions. Our results improve and generalize many of the results in the literature. Moreover, our findings will open the way forward for the study of iterative methods for finding common fixed points of two non-self mappings in Banach spaces more general than Hilbert spaces.

Convergence theorems and stability results of a two-step iteration scheme for pointwise asymptotically nonexpansive self mappings and pointwise asymptotically nonexpansive nonself mappings in uniformly convex Banach spaces

Convergence theorems and stability results of a two-step iteration scheme for pointwise asymptotically nonexpansive self mappings and pointwise asymptotically nonexpansive nonself mappings in uniformly convex Banach spaces

Convergence of a Three-step Iteration Scheme to the Common Fixed Points of Mixed-Type Total Asymtotically Nonexpansive Mappings in Uniformly Convex Banach Spaces

We propose a three-step iteration scheme of hybrid mixed-type for three total asymptotically nonexpansive self mappings and three total asymptotically nonexpansive nonself mappings. In addition, we establish some weak convergence theorems of the scheme to the common fixed point of the mappings in uniformly convex Banach spaces. Our results extend and generalize numerous results currently in literature.

Mixed Type Algorithms for Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

In this paper, a new mixed type iteration process for approximating a common fixed point of two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings is constructed. We then establish a strong convergence theorem under mild conditions in a uniformly convex hyperbolic space. The results presented here extend and improve some related results in the literature.

Controllability and Accumulation of Errors Arising in a General Iteration Method

In this paper, we propose and analyze a three-step general iteration method which is a special case of an iteration method proposed in (S. Thianwan and S. Suantai, Convergence criteria of a new three-step iteration with errors for nonexpansive nonself-mappings, Comput. Math. Appl. 52 (2006), 1107-1118). Here we intend to study directly the accumulation, estimation and control of random errors in the newly proposed general iteration method. We give conditions under which the accumulated-error in our iteration method is bounded and controllable in a permissible range.

Strong and $$\varDelta -$$convergence of Ishikawa iterates of mixed type nonexpansive mappings in hyperbolic spaces

We consider a pair of mixed type non-expansive mappings S and T on a nonempty, closed and convex subset K of a uniformly convex hyperbolic space (X, d, W) such that $$S:K\rightarrow K$$ and $$T:K\rightarrow X$$ and we discuss strong and $$\varDelta -$$ convergence theorems based on a Ishikwa type iterative scheme. The results are further extended to two pairs of self and nonself nonexpansive mappings

Approximating best proximity points for Reich type non-self nonexpansive mappings

In this paper, we consider a general class of non-self non expansive mappings and present best proximity point results in Hilbert spaces. More precisely, we employ a Krasnosel’skiĭ–Mann type algorithm to approximate best proximity points of non-self mappings. We also propose some hybrid algorithms and obtain certain strong convergence results.

Parallel extragradient algorithms for a family of pseudomonotone equilibrium problems and fixed point problems of nonself-nonexpansive mappings in Hilbert space

Parallel extragradient algorithms for a family of pseudomonotone equilibrium problems and fixed point problems of nonself-nonexpansive mappings in Hilbert space

Convergence Theorems of Variational Inequality for Asymptotically Nonexpansive Nonself Mapping in CAT(0) Spaces

The aim of this manuscript is to get the strong convergence theorems of the Moudafi’s viscosity approximation methods for an asymptotically nonexpansive nonself mapping in C A T ( 0 ) spaces.

Viscosity extragradient method with Armijo linesearch rule for pseudomonotone equilibrium problem and fixed point problem in Hilbert spaces

In this paper, we introduce a viscosity extragradient method with Armijo linesearch rule to find a common element of solution set of a pseudomonotone equilibrium problem and fixed point set of a nonexpansive nonself-mapping in Hilbert space. The strong convergence of the algorithm is proved. As the application, a common fixed point theorem for two nonexpansive nonself-mappings is proved. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm. Our result improves the ones of others in the literature.

Strong convergence of approximants to best proximity points of nonself nonexpansive mappings

Let us consider a nonself nonexpansive mapping $$T:A\rightarrow B$$, where A and B are nonempty subsets of a real Hilbert space H. It is worth mentioning that the sequence of iteration is meaningless for this type of nonself mappings. We introduced Browder-type net $$\{x_t\}$$ in A for T and established necessary and sufficient conditions for the existence of a best proximity point of T in terms of the net $$\{x_t\}$$.

Existence and Convergence Theorems for Global Minimization of Best Proximity Points in Hilbert Spaces

In order to solve global minimization problems involving best proximity points, we introduce general Mann algorithm for nonself nonexpansive mappings and then prove weak and strong convergence of the proposed algorithm under some suitable conditions in real Hilbert spaces. Furthermore, we also provide numerical experiment to illustrate the convergence behavior of our proposed algorithm.

A new hybrid algorithm for global minimization of best proximity points in Hilbert spaces

The purpose of this paper is to introduce a new hybrid algorithm for finding a global minimization of best proximity points for a new class of mappings, called best proximally nonexpansive (BPNE), which is weaker than nonself nonexpansive mappings and then prove strong convergence of the proposed method under some suitable conditions in real Hilbert spaces. Finally, some numerical experiment is also given for demonstrating our main result.

Convergence theorems for a new iteration scheme for mixed-type asymptotically nonexpansive mappings

In this paper, we introduce a new iteration scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings, and prove some weak and strong convergence theorems of the proposed iteration scheme in uniformly convex Banach spaces. Our results improve and extend the corresponding results given by some authors.

Hybrid algorithm for common best proximity points of some generalized nonself nonexpansive mappings

In this paper, we first introduce a new class of generalized nonself nonexpansive mappings, called C‐Berinde nonexpansive mappings, and study common best proximity point of two mappings in this class. We propose two hybrid algorithms for finding common best proximity points of two best proximally nonexpansive mappings for which one of them is C‐Berinde nonself nonexpansive mapping. Strong convergence analysis of the proposed methods are discussed in a real Hilbert space. Moreover, we also give a numerical example to support our main results.

A new approximation method for common fixed points of a finite family of nonexpansive non-self mappings in Banach spaces

In this paper, we introduce a new iterative scheme to approximate a common fixed point for a finite family of nonexpansive non-self mappings. Strong convergence theorems of the proposed iteration in Banach spaces.

Strong Convergence Theorems for Hybrid Mixed Type Nonlinear Mappings in Banach Spaces

Abstract In this paper, we introduce a new two-step iteration scheme of hybrid mixed type for two asymptotically nonexpansive self mappings and two asymptotically nonexpansive non-self mappings in the intermediate sense and establish some strong convergence theorems for mentioned scheme and mappings in Banach spaces. Our results extend and generalize the corresponding results recently announced by Wei and Guo [16] (Comm. Math. Res. 31(2015), 149-160) and many others.

Fixed points of nonexpansive and quasi-nonexpansive mappings

In the paper Krasnoselskii–Mann method for non-self mappings in the journal of Fixed Point Theory and Applications, Colao and Marino proved strong convergence of Krasnoselskii–Mann algorithm defined by $$x_{n+1}=\alpha _nx_n+(1-\alpha _n)Tx_n$$ for a non-expansive non-self mapping in a Hilbert space and they proposed three open questions. In this paper we have proved theorems that are answers to all the open questions raised in that paper by relaxing the space, involved map and inward condition to be uniformly convex Banach space, quasi-nonexpansive and weakly inward condition respectively. An application of non-linear parabolic partial differential equation is discussed.We also provide numerical example to verify our main result.