Nonexpansive Mappings In Hilbert Spaces Research Articles

Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

Convergence Analysis of Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

Inertial reflected Krasnosel’ski-Mann iteration with applications to monotone inclusion problems

This paper studies an inertial reflected method for the approximation of fixed points (assuming existence) of nonexpansive mappings in Hilbert spaces. The proposed method is a combination of Krasnosel'ski-Mann iteration, inertial extrapolation step and reflected step. The aim is to improve and complement various versions of inertial Krasnosel'ski-Mann iterations and the recently proposed reflected Krasnosel'ski-Mann iteration in the literature for approximating fixed points of nonexpansive mappings in Hilbert spaces. Some applications to problems of finding zeros of the sum of monotone operators are given. Finally, numerical simulations including image restoration problems are performed using standard test examples to show the superiority of our proposed method.

Modified Mann-Type Algorithm for Two Countable Families of Nonexpansive Mappings and Application to Monotone Inclusion and Image Restoration Problems

In this paper, we introduce and study a modified Mann-type algorithm that combines inertial terms for solving common fixed point problems of two countable families of nonexpansive mappings in Hilbert spaces. Under appropriate assumptions on the sequences of parameters, we establish a strong convergence result for the sequence generated by the proposed method in finding a common fixed point of two countable families of nonexpansive mappings. This method can be applied to solve the monotone inclusion problem. Additionally, we employ a modified Mann-type iterative algorithm to address image restoration problems. Furthermore, we present numerical results across different scenarios to demonstrate the superior efficiency of our algorithm compared to existing algorithms.

Inertial iterative method for solving variational inequality problems of pseudo-monotone operators and fixed point problems of nonexpansive mappings in Hilbert spaces

<p style='text-indent:20px;'>In this paper, we propose a new inertial viscosity iterative algorithm for solving the variational inequality problem with a pseudo-monotone operator and the fixed point problem involving a nonexpansive mapping in real Hilbert spaces. The advantage of the proposed algorithm is that it can work without the prior knowledge of the Lipschitz constant of the mapping. The strong convergence of the sequence generated by the proposed algorithm is proved under some suitable assumptions imposed on the parameters. Some numerical experiments are given to support our main results.</p>

The prooposition for an CAK-generalized nonexpansive mapping in Hilbert spaces

In this paper, we introduce a new class of nonexpansive type of mapping namely, CAK-generalized nonexpansive mapping, which is more general than an AK-generalized nonexpansive mapping and $\alpha$-nonexpansive mapping. Then, we obtain the proposition of the approximation method for an CAK-generalized nonexpansive in Hilbert spaces.

On split equality fixed-point problems

In this present work, we study new algorithms to solve the split equality fixed-point problems for total quasi-asymptotically nonexpansive mappings in Hilbert spaces. In addition, we have established the convergence criteria for the proposed algorithms, at the end, we provided numerical results that justified our theoretical results. The results presented in this paper, provided a unified framework for studying this kind of problem involving different classes of mappings.

Generalized Viscosity Approximation Methods of Ishikawa Type for Nonexpansive Mappings in Hilbert Spaces

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 a unique solution of the variational inequality. Our results extend and improve the recent ones announced by some authors.

"Viscosity algorithm for solving split generalized equilibrium problem and fixed-point problem"

"This article considers a split generalized equilibrium problem and fixed point problem for infi- nite family of nonexpansive mapping in Hilbert space. We propose an algorithm for finding a common solution of these problems. Under mild assumptions, we establish a strong convergence theorem for the sequence gen- erated by the proposed algorithm using viscosity technique. We present the implementation of the proposed algorithm by considering some numerical illustrations and comparison of the proposed algorithm with an ex- isting algorithm in the literature."

A Modified Krasnosel’skiǐ–Mann Iterative Algorithm for Approximating Fixed Points of Enriched Nonexpansive Mappings

For approximating the fixed points of enriched nonexpansive mappings in Hilbert spaces, we consider a modified Krasnosel’skiǐ–Mann algorithm for which we prove a strong convergence theorem. We also empirically compare the rate of convergence of the modified Krasnosel’skiǐ–Mann algorithm and of the simple Krasnosel’skiǐ fixed point algorithm. Based on the numerical experiments reported in the paper we conclude that, for the class of enriched nonexpansive mappings, it is more convenient to work with the simple Krasnosel’skiǐ fixed point algorithm than with the modified Krasnosel’skiǐ–Mann algorithm.

Iterated nonexpansive mappings in Hilbert spaces

In [T. Dominguez Benavides and E. Llorens-Fuster, Iterated nonexpansive mappings, J. Fixed Point Theory Appl. 20 (2018), no. 3, Paper No. 104, 18 pp.], the authors raised the question about the existence of a fixed point free continuous INEA mapping T defined on a closed convex and bounded subset (or on a weakly compact convex subset) of a Banach space with normal structure. Our main goal is to give the affirmative answer to this problem in the very special case of a Hilbert space.

The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

The generalized viscosity implicit rules of nonexpansive asymptotically mappings in Hilbert spaces are considered. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. An application of it in the convex minimization problem is considered. The results presented in this paper improve and extend some recent corresponding results in the literature.

Convergence of an inertial-type Krasnosel’skii-Mann-type iterative sequence to fixed points of nonexpansive mappings in Hilbert spaces

Let H be a real Hilbert space and C a nonempty closed convex subset of H. Let T: C→C be a nonexpansive self mapping. We introduce a new inertial-type iterative sequence which converges to a fixed point of T. Our results are new and complement several important existing results in the literature.

Convergence results for a modified inertial-type Krasnosel'skii-Mann algorithm to fixed points of asymptotically nonexpansive mappings in Hilbert spaces

Convergence results for a modified inertial-type Krasnosel'skii-Mann algorithm to fixed points of asymptotically nonexpansive mappings in Hilbert spaces

A modified Krasnoselskii-Mann algorithm for equilibrium and fixed point problems for nonexpansive mappings in Hilbert spaces

A modified Krasnoselskii-Mann algorithm for equilibrium and fixed point problems for nonexpansive mappings in Hilbert spaces

A New Iterative Algorithm for Multivalued Nonexpansive Mappping and Equlibruim Problems with Applications

In this paper, we introduce two iterative schemes by a modified Krasnoselskii-Mann algorithm for finding a common element of the set of solutions of equilibrium problems and the set of fixed points of multivalued nonexpansive mappings in Hilbert space. We prove that the sequence generated by the proposed method converges strongly to a common element of the set of solutions of equilibruim problems and the set of fixed points of multivalued nonexpansive mappings which is also the minimum-norm element of the above two sets. Finally, some applications of our results to optimization problems with constraint and the split feasibility problem are given. No compactness assumption is made. The methods in the paper are novel and different from those in early and recent literature.

Modified CQ-Algorithms for G-Nonexpansive Mappings in Hilbert Spaces Involving Graphs

In this paper, we introduce four new iterative schemes by modifying the CQ-method with Ishikawa and [Formula: see text]-iterations. The strong convergence theorems are given by the CQ-projection method with our modified iterations for obtaining a common fixed point of two [Formula: see text]-nonexpansive mappings in a Hilbert space with a directed graph. Finally, to compare the rate of convergence and support our main theorems, we give some numerical experiments.

An inertial Mann-like algorithm for fixed points of nonexpansive mappings in Hilbert spaces

An inertial Mann-like algorithm for fixed points of nonexpansive mappings in Hilbert spaces

Viscosity method for generalized equilibrium problems with perturbation and nonexpansive mappings in Hilbert spaces

In this paper, we introduce a new iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem with perturbation and the fixed points for a new kind of combination of two nonexpansive mapping in Hilber space, we prove that the sequence converges strongly to a common element of two sets. Our results generalize and improve some results in the litretures.

The modified viscosity implicit rules for variational inequality problems and fixed point problems of nonexpansive mappings in Hilbert spaces

The purpose of this paper is to introduce a modified viscosity implicit rules for finding a common element of the set of solutions of variational inequality problems for two inverse-strongly monotone operators and the set of fixed points of one nonexpansive mapping in Hilbert spaces. Under some suitable assumptions imposed on the parameters, we obtain some strong convergence theorems. We also apply our main results to solve fixed point problems for strict pseudocontractive mappings and equilibrium problems in Hilbert spaces. A numerical example is given for supporting our main results.

Inertial Krasnosel’skiǐ–Mann type hybrid algorithms for solving hierarchical fixed point problems

In this paper, we suggest two inertial Krasnosel’skiǐ–Mann type hybrid algorithms to approximate a solution of a hierarchical fixed point problem for nonexpansive mappings in Hilbert space. We prove strong convergence theorems for these algorithms and the conditions of the convergence are very weak comparing other algorithms for the hierarchical fixed point problems. Further, we derive some consequences from the main results. Finally, we present two academic numerical examples for comparing these two algorithms with the algorithm in Dong et al. (J Fixed Point Theory A 19(4):3097–3118, 2017), which illustrate the advantage of the proposed algorithms. The methods and results presented in this paper generalize and unify previously known corresponding methods and results of this area.