scholarly journals Solvability and iterative algorithms for a system of generalized nonlinear mixed quasivariational inclusions with (Hi,ηi)-monotone operators

2012 ◽  
Vol 2012 (1) ◽  
pp. 235
Author(s):  
Zeqing Liu ◽  
Lili Wang ◽  
Jeong Ume ◽  
Shin Kang
Keyword(s):  
Iterative Algorithms ◽  
Monotone Operators ◽  
Quasivariational Inclusions

scholarly journals Iterative Algorithms for Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Caiping Yang ◽  
Songnian He
Keyword(s):  
Variational Inequality ◽  
Unique Solution ◽  
Real Hilbert Space ◽  
Iterative Algorithms ◽  
Monotone Operators ◽  
Strongly Monotone ◽  
Lipschitz Constants ◽  
Strongly Monotone Operators ◽  
Original Domain ◽  
Self Adaptive

Consider the variational inequalityVI(C,F)of finding a pointx*∈Csatisfying the property〈Fx*,x-x*〉≥0for allx∈C, whereCis a level set of a convex function defined on a real Hilbert spaceHandF:H→His a boundedly Lipschitzian (i.e., Lipschitzian on bounded subsets ofH) and strongly monotone operator. He and Xu proved that this variational inequality has a unique solution and devised iterative algorithms to approximate this solution (see He and Xu, 2009). In this paper, relaxed and self-adaptive iterative algorithms are proposed for computing this unique solution. Since our algorithms avoid calculating the projectionPC(calculatingPCby computing a sequence of projections onto half-spaces containing the original domainC) directly and select the stepsizes through a self-adaptive way (having no need to know any information of bounded Lipschitz constants ofF(i.e., Lipschitz constants on some bounded subsets ofH)), the implementations of our algorithms are very easy. The algorithms in this paper improve and extend the corresponding results of He and Xu.

scholarly journals Strong Convergence Theorems for Maximal Monotone Operators with Nonspreading Mappings in a Hilbert Space

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Hongjie Liu ◽  
Junqing Wang ◽  
Qiansheng Feng
Keyword(s):  
Hilbert Space ◽  
Strong Convergence ◽  
Monotone Mapping ◽  
Maximal Monotone ◽  
Iterative Algorithms ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Common Element ◽  
Convergence Theorems ◽  
Nonspreading Mappings

We prove the strong convergence theorems for finding a common element of the set of fixed points of a nonspreading mappingTand the solution sets of zero of a maximal monotone mapping and anα-inverse strongly monotone mapping in a Hilbert space. Manaka and Takahashi (2011) proved weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space; there we introduced new iterative algorithms and got some strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space.

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 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.

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