scholarly journals Inertial Krasnoselski–Mann Iterative Method for Solving Hierarchical Fixed Point and Split Monotone Variational Inclusion Problems with Its Applications

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2460
Author(s):  
Preeyanuch Chuasuk ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Hilbert Spaces ◽  
Numerical Experiments ◽  
Variational Inclusion ◽  
Strong Convergence Theorem ◽  
Step Size ◽  
Inclusion Problems ◽  
Hierarchical Fixed Point ◽  
New Iterative Method

In this article, we discuss the hierarchical fixed point and split monotone variational inclusion problems and propose a new iterative method with the inertial terms involving a step size to avoid the difficulty of calculating the operator norm in real Hilbert spaces. A strong convergence theorem of the proposed method is established under some suitable control conditions. Furthermore, the proposed method is modified and used to derive a scheme for solving the split problems. Finally, we compare and demonstrate the efficiency and applicability of our schemes for numerical experiments as well as an example in the field of image restoration.

General iterative algorithm for demicontractive-type mapping in real Hilbert spaces

2020 ◽  
Vol 29 (1) ◽  
pp. 91-99
Author(s):  
T. M. M. SOW
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Fixed Point Problem ◽  
Strong Convergence Theorem ◽  
Point Problem ◽  
Type Mapping ◽  
Demicontractive Mappings ◽  
New Iterative Method

In this paper, we investigate the problem of finding a solution to fixed point problem involving demicontractive mappings in the framework of Hilbert spaces. Inspired by general iterative algorithm, a new iterative method for solving the problem is introduced. Strong convergence theorem of the proposed method is established without any compactness assumption. Our theorems are significant improvements on several important recent results.

scholarly journals Strong Convergence of a New Iterative Algorithm for Split Monotone Variational Inclusion Problems

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 123 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Qing Yuan
Keyword(s):  
Iterative Method ◽  
Hilbert Spaces ◽  
Optimization Problem ◽  
Variational Inclusion ◽  
Hierarchical Optimization ◽  
Inclusion Problem ◽  
Inclusion Problems ◽  
Infinite Dimensional ◽  
Variational Inclusion Problem ◽  
General Iterative Method

The main aim of this work is to introduce an implicit general iterative method for approximating a solution of a split variational inclusion problem with a hierarchical optimization problem constraint for a countable family of mappings, which are nonexpansive, in the setting of infinite dimensional Hilbert spaces. Convergence theorem of the sequences generated in our proposed implicit algorithm is obtained under some weak assumptions.

scholarly journals Approximations for Equilibrium Problems and Nonexpansive Semigroups

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng
Keyword(s):  
Iterative Method ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Common Fixed Points ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Nonexpansive Semigroup ◽  
New Iterative Method

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of all common fixed points of a nonexpansive semigroup and prove the strong convergence theorem in Hilbert spaces. Our result extends the recent result of Zegeye and Shahzad (2013). In the last part of the paper, by the way, we point out that there is a slight flaw in the proof of the main result in Shehu's paper (2012) and perfect the proof.

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

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 123
Author(s):  
Vasile Berinde
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Fixed Points ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Numerical Experiments ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Fixed Point Algorithm ◽  
Mann Algorithm

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.

scholarly journals Krasnoselski–Mann-type inertial method for solving split generalized mixed equilibrium and hierarchical fixed point problems

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Preeyanuch Chuasuk ◽  
Anchalee Kaewcharoen
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Fixed Point Problems ◽  
Step Size ◽  
Inertial Method ◽  
Oligopolistic Market ◽  
Generalized Mixed Equilibrium ◽  
Hierarchical Fixed Point ◽  
Mixed Equilibrium ◽  
Hierarchical Fixed Point Problems

AbstractIn this paper, we present Krasnoselski–Mann-type inertial method for solving split generalized mixed equilibrium and hierarchical fixed point problems for k-strictly pseudocontractive nonself-mappings. We establish that the weak convergence of the proposed accelerated iterative method with inertial terms involves a step size which does not require any prior knowledge of the operator norm under several suitable conditions in Hilbert spaces. Finally, the application to a Nash–Cournot oligopolistic market equilibrium model is discussed, and numerical examples are provided to demonstrate the effectiveness of our iterative method.

scholarly journals A Strong Convergence Theorem for Total Asymptotically Pseudocontractive Mappings in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Chuan Ding ◽  
Jing Quan
Keyword(s):  
Fixed Point ◽  
Iterative Method ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Strong Convergence Theorem ◽  
Demiclosedness Principle ◽  
Pseudocontractive Mappings ◽  
Asymptotically Pseudocontractive Mappings ◽  
Projective Operator ◽  
Hybrid Iterative Method

Demiclosedness principle for total asymptotically pseudocontractive mappings in Hilbert spaces is established. The strong convergence to a fixed point of total asymptotically pseudocontraction in Hilbert spaces is obtained based on demiclosedness principle, metric projective operator, and hybrid iterative method. The main results presented in this paper extend and improve the corresponding results of Zhou (2009), Qin, Cho, and Kang (2011) and of many other authors.

scholarly journals The zeros of monotone operators for the variational inclusion problem in Hilbert spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pattanapong Tianchai
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Monotone Operators ◽  
Fixed Point Problem ◽  
Variational Inclusion ◽  
Feasibility Problem ◽  
Strong Convergence Theorem ◽  
Inclusion Problem ◽  
Point Problem ◽  
Variational Inclusion Problem

AbstractIn this paper, we introduce a regularization method for solving the variational inclusion problem of the sum of two monotone operators in real Hilbert spaces. We suggest and analyze this method under some mild appropriate conditions imposed on the parameters, which allow us to obtain a short proof of another strong convergence theorem for this problem. We also apply our main result to the fixed point problem of the nonexpansive variational inequality problem, the common fixed point problem of nonexpansive strict pseudocontractions, the convex minimization problem, and the split feasibility problem. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique.

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