STRONG CONVERGENCE OF A GENERAL VISCOSITY EXPLICIT RULE FOR THE SUM OF TWO MONOTONE OPERATORS IN HILBERT SPACES

2019 ◽  
Vol 9 (6) ◽  
pp. 2137-2155 ◽  
Author(s):  
Prasit Cholamjiak ◽  
◽  
Suthep Suantai ◽  
Pongsakorn Sunthrayuth ◽  
◽  
...  
Keyword(s):  
Strong Convergence ◽  
Hilbert Spaces ◽  
Monotone Operators ◽  
Explicit Rule

scholarly journals Strong Convergence Theorems of theCQAlgorithm forH-Monotone Operators in Hilbert Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Huimin He ◽  
Sanyang Liu
Keyword(s):  
Mathematical Programming ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Hybrid Method ◽  
Monotone Operators ◽  
Convergence Theorems

The aim of this paper is to show the strong convergence theorems of theCQalgorithm forH-monotone operators in Hilbert spaces by hybrid method in the mathematical programming. The main results extend and improve the corresponding results. Moreover, the assumption conditions of our results are weaker than those of the corresponding results.

scholarly journals Weak and Strong Convergence Theorems for the Inclusion Problem and the Fixed-Point Problem of Nonexpansive Mappings

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 167 ◽  
Author(s):  
Prasit Cholamjiak ◽  
Suparat Kesornprom ◽  
Nattawut Pholasa
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Inclusion Problem ◽  
Point Problem ◽  
Weak And Strong Convergence ◽  
Convergence Theorems

In this work, we study the inclusion problem of the sum of two monotone operators and the fixed-point problem of nonexpansive mappings in Hilbert spaces. We prove the weak and strong convergence theorems under some weakened conditions. Some numerical experiments are also given to support our main theorem.

scholarly journals Generalized split null point of sum of monotone operators in Hilbert spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 359-376
Author(s):  
Akindele A. Mebawondu ◽  
Hammed A. Abass ◽  
Olalwale K. Oyewole ◽  
Kazeem O. Aremu ◽  
Ojen K. Narain
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Monotone Operators ◽  
Convergence Result ◽  
Null Point ◽  
Iterative Technique ◽  
New Type ◽  
Inertial Extrapolation ◽  
Nonlinear Mappings

Abstract In this paper, we introduce a new type of a generalized split monotone variational inclusion (GSMVI) problem in the framework of real Hilbert spaces. By incorporating an inertial extrapolation method and an Halpern iterative technique, we establish a strong convergence result for approximating a solution of GSMVI and fixed point problems of certain nonlinear mappings in the framework of real Hilbert spaces. Many existing results are derived as corollaries to our main result. Furthermore, we present a numerical example to support our main result and propose an open problem for interested researchers in this area. The result obtained in this paper improves and generalizes many existing results in the literature.

scholarly journals Strong Convergence Theorems for Maximal Monotone Operators, Fixed-Point Problems, and Equilibrium Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng ◽  
De-ning Qu
Keyword(s):  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Optimization Problems ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Convergence Theorems ◽  
New Iterative Method

We present a new iterative method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions to an equilibrium problem, and the set of zeros of the sum of maximal monotone operators and prove the strong convergence theorems in the Hilbert spaces. We also apply our results to variational inequality and optimization problems.

scholarly journals Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

2020 ◽  
Vol 10 (1) ◽  
pp. 450-476
Author(s):  
Radu Ioan Boţ ◽  
Sorin-Mihai Grad ◽  
Dennis Meier ◽  
Mathias Staudigl
Keyword(s):  
Dynamical Systems ◽  
Strong Convergence ◽  
Equilibrium Problems ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Minimum Norm ◽  
Monotone Inclusion ◽  
Inclusion Problems ◽  
Monotone Inclusion Problem ◽  
Maximally Monotone Operators

Abstract In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong convergence of the trajectories towards minimum norm solutions to an underlying monotone inclusion problem, and we illustrate numerically qualitative differences between these two complementary regularization strategies. The so-constructed dynamical systems are either of Krasnoselskiĭ-Mann, of forward-backward type or of forward-backward-forward type, and with the help of injected regularization we demonstrate seminal results on the strong convergence of Hilbert space valued evolutions designed to solve monotone inclusion and equilibrium problems.

scholarly journals On a System of Nonlinear Variational Inclusions withHh,η-Monotone Operators

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Zeqing Liu ◽  
Jeong Sheok Ume ◽  
Shin Min Kang
Keyword(s):  
Hilbert Spaces ◽  
Monotone Operators ◽  
Variational Inclusions ◽  
Iterative Approximation

This paper is concerned mainly with the existence and iterative approximation of solutions for a system of nonlinear variational inclusions involving the stronglyHh,η-monotone operators in Hilbert spaces. The results presented in this paper extend, improve, and unify many known results in the literature.

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