Strong and weak convergence theorems for split equality generalized mixed equilibrium problem

In this paper, we adopt an iterative approach to solve the class of optimization problem for the sum of finite functions over split equality optimization problems for the sum of two functions. This type of problem contains many optimization problems, and bilevel problems, as well as split equality problems, and split feasibility problems as special cases. Here, we are able to establish a strong convergence theorem for an iterative method for solving this problem. As consequences of this convergence theorem, we study the following problems: optimization for the sum of finite functions over the common solution set of optimization problems for the sum of two functions; optimization for the sum of finite functions; optimization for the sum of finite functions with split equality inconsistent feasibility constraints; optimization for the sum of finite functions over the solution set for split equality constrained quadratic signal recovery problem; optimization for the sum of finite functions over the solution set of generalized split equality multiple set feasibility problem, and optimization for the sum of finite functions over the solution set of split equality linear equations problem. We use simultaneous iteration to establish strong convergence theorems for these problems. Our results generalize and improve many existing theorems for these types of problems in the literature and will have applications in nonlinear analysis, optimization problems and signal processing problems.

In this paper, we propose an iterative algorithm for finding a common solution of a system of generalized equilibrium problems, a split equilibrium problem and a hierarchical fixed point problem over the common fixed points set of a finite family of nonexpansive mappings in Hilbert spaces. Furthermore, we prove that the proposed iterative method has strong convergence under some mild conditions imposed on algorithm parameters. The results presented in this paper improve and extend the corresponding results reported by some authors recently.

In this paper, we introduce a new iterative algorithm for solving the split equality generalized mixed equilibrium problems. The weak and strong convergence theorems are proved for demi-contractive mappings in real Hilbert spaces. Several special cases are also discussed. As applications, we employ our results to get the convergence results for the split equality convex differentiable optimization problem, the split equality convex minimization problem, and the split equality mixed equilibrium problem. The results in this paper generalize, extend, and unify some recent results in the literature.

The split feasibility problem is an inverse problem which arises in signal processing and medical image reconstruction. So there is practical value in studying it. While both the split equality problem and the split variational inclusion problem are generalized form of the split feasibility problem which are more meaningful than the split feasibility problem. In this paper, fusing the two problems, we research a split inclusion problem and propose relevant methods for solving it. What counts is that not only the proposed algorithms have strong convergence, but also the limit points of the algorithms are the minimal norm solution of the split inclusion problem.

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

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.

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.

In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature.

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.

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications

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

We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a strong convergence theorem under some suitable conditions.