scholarly journals Generalized -Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Shamshad Husain ◽  
Sanjeev Gupta ◽  
Vishnu Narayan Mishra
Keyword(s):  
Iterative Algorithm ◽  
Inclusion Problem ◽  
Operator Technique ◽  
Convergence Criteria ◽  
Lipschitz Continuous ◽  
New Class ◽  
Resolvent Operator Technique ◽  
Definition Of ◽  
Cocoercive Operator ◽  
Cocoercive Operators

We investigate a new class of cocoercive operators named generalized -cocoercive operators in Hilbert spaces. We prove that generalized -cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with -cocoercive operators to the generalized -cocoercive operators. Some examples are given to justify the definition of generalized -cocoercive operators. Further, we consider a generalized set-valued variational-like inclusion problem involving generalized -cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the 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 Strong Convergence of Iterative Algorithm for a New System of GeneralizedH·,·-η-Cocoercive Operator Inclusions in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Saud M. Alsulami ◽  
Eskandar Naraghirad ◽  
Nawab Hussain
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Resolvent Operator ◽  
Operator Technique ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator ◽  
Cocoercive Operator ◽  
Cocoercive Operators ◽  
Convergence Of Iterative Algorithm ◽  
New System

We introduce and study a new system of generalizedH·,·-η-cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated withH·,·-η-cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.

A Unified Iteration Method with Error for Inclution Problem

2011 ◽  
Vol 403-408 ◽  
pp. 1584-1587
Author(s):  
Han Jun Chen ◽  
Yang Xue ◽  
Hong Zhao
Keyword(s):  
Iteration Method ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Inclusion Problem ◽  
Operator Technique ◽  
Convergence Criteria ◽  
Special Cases ◽  
Resolvent Operator Technique ◽  
The Common ◽  
The Resolvent Operator

In this paper, we suggest and analyze a unified iteration method with error for finding the common element of the set of fixed points of nonexpansive mappings and the set of the solutions of the inclusion problem using the resolvent operator technique. We also study the convergence criteria of the unified iteration method with error under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement the previously known results.

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 Completely generalized multivalued nonlinear quasi-variational inclusions

2002 ◽  
Vol 30 (10) ◽  
pp. 593-604 ◽  
Author(s):  
Zeqing Liu ◽  
Lokenath Debnath ◽  
Shin Min Kang ◽  
Jeong Sheok Ume
Keyword(s):  
Existence Theorems ◽  
Maximal Monotone ◽  
Iterative Algorithms ◽  
Monotone Mappings ◽  
Operator Technique ◽  
Variational Inclusions ◽  
New Class ◽  
Convergence Results ◽  
Resolvent Operator Technique ◽  
The Resolvent Operator

We introduce and study a new class of completely generalized multivalued nonlinear quasi-variational inclusions. Using the resolvent operator technique for maximal monotone mappings, we suggest two kinds of iterative algorithms for solving the completely generalized multivalued nonlinear quasi-variational inclusions. We establish both four existence theorems of solutions for the class of completely generalized multivalued nonlinear quasi-variational inclusions involving strongly monotone, relaxed Lipschitz, and generalized pseudocontractive mappings, and obtain a few convergence results of iterative sequences generated by the algorithms. The results presented in this paper extend, improve, and unify a lot of results due to Adly, Huang, Jou-Yao, Kazmi, Noor, Noor-Al-Said, Noor-Noor, Noor-Noor-Rassias, Shim-Kang-Huang-Cho, Siddiqi-Ansari, Verma, Yao, and Zhang.

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 Stability and convergence analysis for a new class of GNYIP involving XOR-operation in ordered positive Hilbert spaces

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2877-2895
Author(s):  
Iqbal Ahmad ◽  
A Abdullah ◽  
Khaled Khedher ◽  
Syed Irfan
Keyword(s):  
Convergence Analysis ◽  
Iterative Algorithm ◽  
Hilbert Spaces ◽  
Operator Method ◽  
Stability And Convergence ◽  
Inclusion Problem ◽  
Operation Technique ◽  
New Class ◽  
Stability And Convergence Analysis ◽  
Xor Operation

In the setting of real ordered positive Hilbert spaces, a new class of general nonlinear ordered Yosida inclusion problem involving ? operation has been considered and solved by employing a perturbed two step-iterative algorithm. The stability and convergence analysis of solution of new class of Yosida inclusion problem involving ? operation has been substantiated by applying a new resolvent operator and Yosida approximation operator method with XOR-operation technique. The iterative algorithm and results demonstrated in this article have witnessed, a significant improvement for many previously known results of this domain. Further, we give a numerical example in support of our main result by using MATLprogramming.

scholarly journals An Iterative Algorithm for the Nonlinear MC2 Model with Variational Inequality Method

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 514
Author(s):  
Chao Min ◽  
Feifei Fan ◽  
Zhaozhong Yang ◽  
Xiaogang Li
Keyword(s):  
Iterative Algorithm ◽  
Decision Makers ◽  
Iterative Sequence ◽  
Operator Technique ◽  
Multiple Decision Makers ◽  
Multiple Constraint ◽  
Resolvent Operator Technique ◽  
The Stability ◽  
The Resolvent Operator ◽  
Constraint Level

The multiple criteria and multiple constraint level (MC 2 ) model is a useful tool to deal with the decision programming problems, which concern multiple decision makers and uncertain resource constraint levels. In this paper, by regarding the nonlinear MC 2 problems as a class of mixed implicit variational inequalities, we develop an iterative algorithm to solve the nonlinear MC 2 problems through the resolvent operator technique. The convergence of the generated iterative sequence is analyzed and discussed by a calculation example, and the stability of Algorithm 1 is also verified by error propagation. By comparing with two other MC 2 -algorithms, Algorithm 1 performs well in terms of number of iterations and computation complexity.

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.

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