Viscosity Approximation Methods for Generalized Mixed Equilibrium Problems and Fixed Points of a Sequence of Nonexpansive Mappings

In this work, we introduce notions of generalized firmly nonexpansive (G-firmly non expansive) and fundamentally firmly nonexpansive (F-firmly nonexpansive) mappings and utilize to the same to prove Ray's theorem for G-firmly and F-firmly nonexpansive mappings in Hilbert Spaces. Our results extend the result due to F. Kohsaka [ Ray's theorem revisited: a fixed point free firmly nonexpansive mapping in Hilbert spaces, Journal of Inequalities and Applications (2015) 2015:86 ].

Convergence of a general iterative method for nonexpansive mappings in Hilbert spaces

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 i ˘ -Mann iteration, inertial extrapolation step and reflected step. The aim is to improve and complement various versions of inertial Krasnosel'ski i ˘ -Mann iterations and the recently proposed reflected Krasnosel'ski i ˘ -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.

In this paper, we suggest and analyze two iterative algorithms with perturbations for non-expansive mappings in Hilbert spaces. We prove that the proposed iterative algorithms converge strongly to a fixed point of some non-expansive mapping.2000 Mathematics Subject Classification 47H09, 47H10.

Iterative algorithms have been extensively studied over the class of nonexpansive mappings in Hilbert spaces. Recall that nonexpansive mappings belong to quasi-nonexpansive mappings. The aim of this article is expanding the general approximation method proposed by Marino and Xu to quasi-nonexpansive mappings in Hilbert spaces.

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.

We first extend the definition of Wn from an infinite family of nonexpansive mappings to an infinite family of strictly pseudocontractive mappings, and then propose an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of an infinite family of ki‐strictly pseudocontractive mappings in Hilbert spaces. The results obtained in this paper extend and improve the recent ones announced by many others. Furthermore, a numerical example is presented to illustrate the effectiveness of the proposed scheme.

In this paper, we consider split equality generalized mixed equilibrium problem, which is more general than many problems such as split feasibility problem, split equality problem, split equilibrium problem, and so on. We propose a new modified algorithm to obtain strong and weak convergence theorems for split equality generalized mixed equilibrium problem for nonexpansive mappings in Hilbert spaces. Also, we give some applications to other problems. Our results extend some results in the literature.

Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces

In this article, we study an iterative method over the class of quasi-nonexpansive mappings which are more general than nonexpansive mappings in Hilbert spaces. Our strong convergent theorems include several corresponding authors’ results. 2000 MSC: 58E35; 47H09; 65J15.

In this paper, we introduce, for the first time, the viscosity rules for common fixed points of two nonexpansive mappings in Hilbert spaces. The strong convergence of this technique is proved under certain assumptions imposed on the sequence of parameters.

A general iterative algorithm for nonexpansive mappings in Hilbert spaces

The purpose of this paper is to introduce the implicit midpoint rule of asymptotically nonexpansive mappings in Hilbert spaces. The strong convergence of this viscosity method is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit solves an additional variational inequality. Applications to nonlinear variational inclusion problem, nonlinear Volterra integral equations, variational inequality problem and hierarchical minimization problems are included. The results presented in the paper extend and improve some recent results announced in the current literature.

Solutions of system of equilibrium and variational inequality problems on fixed points of infinite family of nonexpansive mappings

Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings