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 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 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 A New System of Random Generalized Variational Inclusions with Random Fuzzy Mappings and Random --Accretive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Sayyedeh Zahra Nazemi
Keyword(s):  
Banach Spaces ◽  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Lipschitz Continuity ◽  
Variational Inclusions ◽  
Accretive Mappings ◽  
New System

We introduce a new notion of random --accretive mappings and prove the Lipschitz continuity of the random resolvent operator associated with the random --accretive mappings. We introduce and study a new system of random generalized variational inclusions with random --accretive mappings and random fuzzy mappings in Banach spaces. By using the random resolvent operator, an iterative algorithm for solving such system of random generalized variational inclusions is constructed in Banach spaces. Under some suitable conditions, we prove the convergence of the iterative sequences generated by the algorithm.

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.

Generalized variational-like inclusion involving relaxed monotone operators

2017 ◽  
Vol 8 (2) ◽  
Author(s):  
Syed Shakaib Irfan ◽  
Mohammad F. Khan ◽  
Ali P. Farajzadeh ◽  
Allahkaram Shafie
Keyword(s):  
Hilbert Space ◽  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Recent Literature ◽  
Monotone Operators ◽  
Inclusion Problem ◽  
Proximal Operator ◽  
New Class

Abstract In this paper, we introduce a new class of resolvent operator, the η-proximal operator, and discuss some of its properties. We consider a new generalized variational-like inclusion problem involving relaxed monotone operators in Hilbert space and construct a new iterative algorithm for proving the existence of the solutions of our problem. Our results improve and generalize many corresponding results in the recent literature.

scholarly journals Resolvent operator technique for solving a system of generalized variational-like inclusions in Banach spaces

Filomat ◽  
2012 ◽  
Vol 26 (5) ◽  
pp. 897-908
Author(s):  
Rais Ahmad ◽  
Mohammad Dilshad ◽  
Mohammad Akram
Keyword(s):  
Banach Spaces ◽  
Iterative Algorithm ◽  
Resolvent Operator ◽  
Existence Of Solutions ◽  
Operator Technique ◽  
Uniformly Smooth ◽  
Resolvent Operator Technique ◽  
Uniformly Smooth Banach Spaces ◽  
Matlab Programming ◽  
Cocoercive Operator

In this paper, we apply H(?,?)-?-cocoercive operator introduced in [2] for solving a system of generalized variational-like inclusions in q-uniformly smooth Banach spaces. By using the approach of resolvent operator associated with H(?,?)-?-cocoercive operator, an iterative algorithm for solving a system of generalized variational-like inclusions is constructed. We prove the existence of solutions of system of generalized variational-like inclusions and convergence of iterative sequences generated by the algorithm. An example through Matlab programming is constructed.

scholarly journals System of Nonlinear Set-Valued Variational Inclusions Involving a Finite Family ofH(·,·)-Accretive Operators in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Prapairat Junlouchai ◽  
Somyot Plubtieng
Keyword(s):  
Banach Spaces ◽  
Resolvent Operator ◽  
Iterative Scheme ◽  
Finite Family ◽  
Strong Convergence Theorem ◽  
Operator Technique ◽  
Variational Inclusions ◽  
Accretive Operators ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator

We study a new system of nonlinear set-valued variational inclusions involving a finite family ofH(·,·)-accretive operators in Banach spaces. By using the resolvent operator technique associated with a finite family ofH(·,·)-accretive operators, we prove the existence of the solution for the system of nonlinear set-valued variational inclusions. Moreover, we introduce a new iterative scheme and prove a strong convergence theorem for finding solutions for this system.

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.

Sign in / Sign up

Export Citation Format

Share Document