scholarly journals A new system of variational inclusions with (H, η)-monotone operators

2006 ◽  
Vol 74 (2) ◽  
pp. 301-319 ◽  
Author(s):  
Jianwen Peng ◽  
Jianrong Huang
Keyword(s):  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Iterative Algorithms ◽  
Operator Method ◽  
Monotone Operators ◽  
Variational Inclusions ◽  
Uniqueness Of Solutions ◽  
Special Cases ◽  
The Resolvent Operator ◽  
New System

In this paper, We introduce and study a new system of variational inclusions involving(H, η)-monotone operators in Hilbert spaces. By using the resolvent operator method associated with (H, η)-monotone operators, we prove the existence and uniqueness of solutions and the convergence of some new three-step iterative algorithms for this system of variational inclusions and its special cases. The results in this paper extends and improves some results in the literature.

scholarly journals Approximation solvability for a system of implicit nonlinear variational inclusions with Η-monotone operators

2018 ◽  
Vol 51 (1) ◽  
pp. 241-254
Author(s):  
Jong Kyu Kim ◽  
Muhammad Iqbal Bhat
Keyword(s):  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Monotone Operators ◽  
Inner Product ◽  
Operator Technique ◽  
Variational Inclusions ◽  
Inclusion Problems ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator ◽  
New System

AbstractIn this paper, we introduce and study a new system of variational inclusions which is called a system of nonlinear implicit variational inclusion problems with A-monotone and H-monotone operators in semi-inner product spaces. We define the resolvent operator associated with A-monotone and H-monotone operators and prove its Lipschitz continuity. Using resolvent operator technique, we prove the existence and uniqueness of solution for this new system of variational inclusions. Moreover, we suggest an iterative algorithm for approximating the solution of this system and discuss the convergence analysis of the sequences generated by the iterative algorithm under some suitable conditions.

scholarly journals An iterative algorithm for generalized nonlinear variational inclusions with relaxed strongly accretive mappings in Banach spaces

2004 ◽  
Vol 2004 (20) ◽  
pp. 1035-1045 ◽  
Author(s):  
A. H. Siddiqi ◽  
Rais Ahmad
Keyword(s):  
Banach Spaces ◽  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Existence Of Solutions ◽  
Operator Technique ◽  
Variational Inclusions ◽  
Special Cases ◽  
Accretive Mappings ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator

We use Nadler's theorem and the resolvent operator technique form-accretive mappings to suggest an iterative algorithm for solving generalized nonlinear variational inclusions with relaxed strongly accretive mappings in Banach spaces. We prove the existence of solutions for our inclusions without compactness assumption and the convergence of the iterative sequences generated by the algorithm in real Banach spaces. Some special cases are also discussed.

scholarly journals Sensitivity Analysis for a System of Generalized Nonlinear Mixed Quasi Variational Inclusions withH-Monotone Operators

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Han-Wen Cao
Keyword(s):  
Sensitivity Analysis ◽  
Hilbert Spaces ◽  
Monotone Operators ◽  
Variational Inclusions ◽  
New System

The existence of the solution for a new system of generalized nonlinear mixed quasi variational inclusions withH-monotone operators is proved by using implicit resolvent technique, and the sensitivity analysis of solution in Hilbert spaces is given. Our results improve and generalize some results of the recent ones.

scholarly journals Iterative Methods for Computing the Resolvent of Composed Operators in Hilbert Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 131 ◽  
Author(s):  
Yixuan Yang ◽  
Yuchao Tang ◽  
Chuanxi Zhu
Keyword(s):  
Linear Operator ◽  
Resolvent Operator ◽  
Operator Splitting ◽  
Iterative Algorithms ◽  
Monotone Operators ◽  
Linear Operators ◽  
Bounded Linear ◽  
Lower Semi ◽  
Finite Sum ◽  
The Resolvent Operator

The resolvent is a fundamental concept in studying various operator splitting algorithms. In this paper, we investigate the problem of computing the resolvent of compositions of operators with bounded linear operators. First, we discuss several explicit solutions of this resolvent operator by taking into account additional constraints on the linear operator. Second, we propose a fixed point approach for computing this resolvent operator in a general case. Based on the Krasnoselskii–Mann algorithm for finding fixed points of non-expansive operators, we prove the strong convergence of the sequence generated by the proposed algorithm. As a consequence, we obtain an effective iterative algorithm for solving the scaled proximity operator of a convex function composed by a linear operator, which has wide applications in image restoration and image reconstruction problems. Furthermore, we propose and study iterative algorithms for studying the resolvent operator of a finite sum of maximally monotone operators as well as the proximal operator of a finite sum of proper, lower semi-continuous convex functions.

scholarly journals Generalized System for Relaxed Cocoercive Mixed Variational Inequalities and Iterative Algorithms in Hilbert Spaces

2012 ◽  
Vol 20 (3) ◽  
pp. 131-139
Author(s):  
Shuyi Zhang ◽  
Xinqi Guo ◽  
Dan Luan
Keyword(s):  
Variational Inequality ◽  
Hilbert Spaces ◽  
Resolvent Operator ◽  
Iterative Algorithms ◽  
Operator Technique ◽  
Mixed Variational Inequality ◽  
Generalized System ◽  
Resolvent Operator Technique ◽  
Mixed Variational Inequalities ◽  
The Resolvent Operator

Abstract The approximate solvability of a generalized system for relaxed co- coercive mixed variational inequality is studied by using the resolvent operator technique. The results presented in this paper extend and improve the main results of Chang et al.[1], He and Gu [2] and Verma [3, 4].

scholarly journals H(⋅,⋅)-Cocoercive Operator and an Application for Solving Generalized Variational Inclusions

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Rais Ahmad ◽  
Mohd Dilshad ◽  
Mu-Ming Wong ◽  
Jen-Chin Yao
Keyword(s):  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Lipschitz Continuity ◽  
Monotone Operators ◽  
Variational Inclusions ◽  
The Resolvent Operator ◽  
Cocoercive Operator

The purpose of this paper is to introduce a newH(⋅,⋅)-cocoercive operator, which generalizes many existing monotone operators. The resolvent operator associated withH(⋅,⋅)-cocoercive operator is defined, and its Lipschitz continuity is presented. By using techniques of resolvent operator, a new iterative algorithm for solving generalized variational inclusions is constructed. Under some suitable conditions, we prove the convergence of iterative sequences generated by the algorithm. For illustration, some examples are given.

scholarly journals Iterative Algorithms for New General Systems of Set-Valued Variational Inclusions Involving(A,η)-Maximal Relaxed Monotone Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ting-jian Xiong ◽  
Heng-you Lan
Keyword(s):  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
General System ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Operator Technique ◽  
Variational Inclusions ◽  
Variational Inclusion Problem ◽  
General Systems ◽  
Resolvent Operator Technique

We introduce and study a class of new general systems of set-valued variational inclusions involving(A,η)-maximal relaxed monotone operators in Hilbert spaces. By using the general resolvent operator technique associated with(A,η)-maximal relaxed monotone operators, we construct some new iterative algorithms for finding approximation solutions to the general system of set-valued variational inclusion problem and prove the convergence of this algorithm. Our results improve and extend some known results.

scholarly journals Algorithm for Solving a New System of Generalized Variational Inclusions in Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Shamshad Husain ◽  
Sanjeev Gupta
Keyword(s):  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Existence Of Solutions ◽  
Variational Inclusions ◽  
Special Cases ◽  
Systems Of Variational Inequalities ◽  
Cocoercive Operators ◽  
New System

We introduce and study a new system of generalized variational inclusions involving -cocoercive and relaxed -cocoercive operators, which contain the systems of variational inclusions and the systems of variational inequalities, variational inclusions, and variational inequalities as special cases. By using the resolvent technique for the -cocoercive operators, we prove the existence of solutions and the convergence of a new iterative algorithm for this system of variational inclusions in Hilbert spaces. An example is given to justify the main result. Our results can be viewed as a generalization of some known results in the literature.

scholarly journals --Accretive Mappings and a New System of Generalized Variational Inclusions with --Accretive Mappings in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Sayyedeh Zahra Nazemi
Keyword(s):  
Banach Spaces ◽  
Resolvent Operator ◽  
Lipschitz Continuity ◽  
Operator Technique ◽  
Variational Inclusions ◽  
New Class ◽  
Accretive Mappings ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator ◽  
New System

We introduce a new class of generalized accretive mappings, named --accretive mappings, in Banach spaces. We define a resolvent operator associated with --accretive mappings and show its Lipschitz continuity. We also introduce and study a new system of generalized variational inclusions with --accretive mappings in Banach spaces. By using the resolvent operator technique associated with --accretive mappings, we construct a new iterative algorithm for solving this system of generalized variational inclusions in Banach spaces. We also prove the existence of solutions for the generalized variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.

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