Three-step Iterative Scheme Research Articles

Convergence to common fixed points of multi-step iteration process for generalized asymptotically quasi-nonexpansive mappings in convex metric spaces

In this paper, we study strong convergence of multi-step iterations with errors for a finite family of generalized asymptotically quasinonexpansive mappings in the framework of convex metric spaces. The new iteration scheme includes modified Mann and Ishikawa iterations with errors, the three-step iteration scheme of Xu and Noor as special cases in Banach spaces. Our results extend and generalize many known results from the current literature.

Fixed Point Theorems for Nonself Asymptotically Nonexpansive type Mappings in CAT(0) Spaces

In this paper, we propose a three-step iteration scheme for nonself asymptotically nonexpansive type mappings in a complete CAT(0) metric space and establish necessary and sufficient conditions for convergence of this process to a fixed point of nonself asymptotically nonexpansive type mappings. We also establish a strong convergence result. These results generalize and unify many important results in the literature.

Approximating common Fixed points for four asymptotically quasi-nonexpansive mappings in CAT(0) spaces

The aim of this paper to study a three-step iterative algorithm for four asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces (for definition, see page 2). Also we establish some strong convergence theorems for above said scheme and mappings by using semi-compactness and condition (GA) (see, definition 2) which is more general than condition (A) (see, definition 1). Our results improve and extend the corresponding results from the previous work. Keywords:Asymptotically quasi-nonexpansive mapping, three-step iteration scheme,common Fixed point, strong convergence, CAT(0) space. 2010 Mathematics Subject Classification: 54H25, 54E40.

Three step iterative algorithms for solving a class of quasi variational inequalities

In this paper, we discuss the existence of a solution of extended general quasi variational inequalities using projection technique. Using the technique of Noor, we suggest and analyze a new class of three-step iterative schemes for solving extended general quasi variational inequalities. Under some certain conditions on operators, we also discuss the convergence of the proposed iterative scheme. Results proved in this paper continue to hold for previously known and new classes of quasi variational inequalities. Ideas and techniques of this paper may stimulate further research in this field. The implementation and comparison of the proposed new algorithms is an interesting problem for further research.

Convergence of three-step iterations for total asymptotically nonexpansive mappings in uniformly convex Banach spaces

In this paper, we study a three-step iterative scheme for total asymptotically nonexpansive mappings in uniformly convex Banach spaces. As an application, several convergence theorems of this scheme are established for total asymptotically nonexpansive mappings. The results obtained in this paper are an extension and refinement of some previous results.

The Hybrid Projection Methods for Pseudocontractive, Nonexpansive Semigroup, and Monotone Mapping

We modify the three-step iterative schemes to prove the strong convergence theorems by using the hybrid projection methods for finding a common element of the set of solutions of fixed points for a pseudocontractive mapping and a nonexpansive semigroup mapping and the set of solutions of a variational inequality problem for a monotone mapping in a Hilbert space under some appropriate control conditions. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.

Fixed point iterations of three asymptotically pseudocontractive mappings

Abstract In this paper, we establish the strong convergence for a modified three-step iterative scheme with errors associated with three mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest. Remark at the end simplifies many known results.

THREE STEP ITERATION SCHEME FOR M-CONTRACTIVE CONDITION IN BANACH SPACE

Our aim in this paper is to prove a strong convergence theorem by three-step iteration scheme with respect to contractive condition in a Banach space. Our results obtained in this paper is new extension as well as refinement of previous known results. AMS Subject Classification: 47H05, 47H10, 49M05

Fixed Point of a New Three-Step Iteration Algorithm under Contractive-Like Operators over Normed Spaces

We introduce a new three-step iteration scheme and prove that this new iteration scheme is convergent to fixed points of contractive-like operators. Also, by providing an example, we show that our new iteration method is faster than another iteration method due to Suantai (2005). Furthermore, it is shown that this new iteration method is equivalent to some other iteration methods in the sense of convergence. Finally, it is proved that this new iteration method isT-stable.

A three-step iterative scheme for solving nonlinear ϕ-strongly accretive operator equations in Banach spaces

In this paper, we study a three-step iterative scheme with error terms for solving nonlinear ϕ-strongly accretive operator equations in arbitrary real Banach spaces.

Image segmentation based on multiresolution Markov random field with fuzzy constraint in wavelet domain

This study proposes a multiresolution Markov random field model with fuzzy constraint in wavelet domain (MRMRF-F). In this model, a fuzzy field is introduced into the multiresolution Markov random field model to estimate parameters, by which the spatial constraint between neighbouring features can be reflected. There are three subfields on each resolution in the MRMRF-F model: one feature field, one label field and one fuzzy field. Among these fields, a three-step iteration scheme is designed to realise image segmentation. Namely, the label field renews the fuzzy field; then the fuzzy field estimates the parameters of the feature field; and a renewed label field is obtained by maximising the products of both the probability of the feature field and the probability of current label field; image segmentation is finally obtained by the cycle iteration among these fields. Texture and natural images are used to test the performance of the new model, and experimental results illustrate a better and higher accuracy compared with other existing methods about fuzzy methods or Markov random field models.

Approximation of Common Fixed Points of Generalized Non-Lipschitzian Mappings

In this paper, we prove strong convergence of a generalized three-step iterative scheme with errors to approximate common flxed points of three generalized Non-Lipschitzian mappings in uniformly convex Banach spaces. Our results generalize and improve several known results.

Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces

Abstract In this article, we introduce a new three-step iterative scheme for the mappings which are asymptotically nonexpansive in the intermediate sense in Banach spaces. Weak convergence theorem is established for this three-step iterative scheme in a uniformly convex Banach space that satisfies Opial's condition or whose dual space has the Kadec-Klee property. Furthermore, we give an example of the nonself mapping which is asymptotically nonexpansive in the intermediate sense but not asymptotically nonexpansive. The results obtained in this article extend and improve many recent results in this area. AMS classification: 47H10; 47H09; 46B20.

Strong convergence theorems with three-step iteration in star-shaped metric spaces

The author considers a Noor-type three-step iterative scheme including Ishikawa-type scheme as a special case to approximate common fixed points of an infinite family of uniformly quasi-Lipschitzian mappings and an infinite family of nonexpansive mappings in star-shaped metric spaces. His results are cases of star-shaped metric space of results shown in [7].

Strong convergence of a general iteration scheme in [formula omitted] spaces

We introduce and study strong convergence of a general iteration scheme for a finite family of asymptotically quasi-nonexpansive maps in convex metric spaces and C A T ( 0 ) spaces. The new iteration scheme includes modified Mann and Ishikawa iterations, the three-step iteration scheme of Xu and Noor and the scheme of Khan, Domlo and Fukhar-ud-din as special cases in Banach spaces. Our results are refinements and generalizations of several recent results from the current literature.

Some iterative schemes for general mixed variational inequalities

In this paper, we consider a class of variational inequalities which is called the general mixed variational inequality. It is known that the general mixed variational inequalities are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some three-step iterative schemes for finding the common element of the set of fixed points of a nonexpansive mappings and the set of solutions of the mixed variational inequalities. We also study the convergence criteria of three-step iterative method under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.

Convergence Theorems of Three-Step Iterative Scheme for a Finite Family of Uniformly Quasi-Lipschitzian Mappings in Convex Metric Spaces

We consider a new Noor-type iterative procedure with errors for approximating the common fixed point of a finite family of uniformly quasi-Lipschitzian mappings in convex metric spaces. Under appropriate conditions, some convergence theorems are proved for such iterative sequences involving a finite family of uniformly quasi-Lipschitzian mappings. The results presented in this paper extend, improve and unify some main results in previous work.

Common Fixed Points of Multistep Noor Iterations with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces. The new iterative scheme includes the multistep Noor iterations with errors, modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor, and Khan and Takahashi scheme as special cases. Our results generalize and improve the recent ones announced by Khan et al. (2008), H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006), and many others.

STRONG CONVERGENCE FOR THREE CLASSES OF UNIFORMLY EQUI-CONTINUOUS AND ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS

In this paper, we introduce a modified three-step iteration scheme with errors for three classes of uniformly equi-continuous and asymptotically quasi-nonexpansive mappings in the framework of uniformly convex Banach spaces. We then use this scheme to approximate a common fixed point of these mappings. The results obtained in this paper extend and improve the recent ones announced by Khan, Fukhar-ud-di, Zhou, Cho, Noor and some others.

Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive Mappings

We introduce a new three-step iterative scheme with errors. Several convergence theorems of this scheme are established for common fixed points of nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces. Our theorems improve and generalize recent known results in the literature.