An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces

Sun, Jiachang; Zhang, Liwei; Xiao, Xiantao

doi:10.1016/j.na.2007.09.026

Cited by 24 publications

(22 citation statements)

References 11 publications

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“…Remark 3.2 Theorem 3.1 and Theorem 3.2 improve the similar conclusions (see Sun et al(2008), Huang and Fang (2003), Zhang (2007), Feng and Ding (2009) …”

Section: Preliminariessupporting

confidence: 67%

“…In order to study various variational inequalities and variational inclusions, Ding(2000), Huang and Fang (2003), Fang and Huang (2003), Fang et al(2005), , Zhang (2007), Sun et al(2008), Xia and Huang (2007), Feng and Ding (2009) and He et al(2008) have introduced the concepts of η-subdifferential operators, maximal η-monotone operators, H-monotone operators, (H, η)-monotone operators, A-monotone operators, (A, η)-monotone operators, G-η-monotone operators, M-monotone operators in Hilbert spaces, H-monotone operators, A-monotone operators and H-η-monotone operators in Banach spaces and their resolvent operators, respectively. Further, by using the resolvent operator technique, a number of nonlinear variational inclusions and many systems of variational inequalities and variational inclusions have been studied by some authors in recent years (for example Lan (2007), Ding and Feng (2008), Peng and Zhu (2007), Zeng (2007), Ding and Wang (2009)).…”

Section: Introductionmentioning

confidence: 99%

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A Generalized Mixed Variational Inclusion Involving (H(·, ·), ?)-Monotone Operators in Banach Spaces

Xu¹,

Wang²

2010

JMR

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In this paper, we introduce a new class of monotone operators − (H(·, ·), η)-monotone operators, which generalize many existing monotone operators. The resolvent operator associated with an (H(·, ·), η)-monotone operator is defined and its Lipschitz continuity is presented. As an application, we also consider a new generalized mixed variational inclusion involving (H(·, ·), η)-monotone operators and construct a new algorithm for solving the generalized mixed variational inclusion. Under some suitable conditions, we prove the convergence of the iterative sequences generated by the algorithm. These results improve and generalize many corresponding results in recently literatures.

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“…Remark 3.2 Theorem 3.1 and Theorem 3.2 improve the similar conclusions (see Sun et al(2008), Huang and Fang (2003), Zhang (2007), Feng and Ding (2009) …”

Section: Preliminariessupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

A Generalized Mixed Variational Inclusion Involving (H(·, ·), ?)-Monotone Operators in Banach Spaces

Xu¹,

Wang²

2010

JMR

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“…Among these methods, the proximal-point mapping method for solving variational inclusions has been widely used by many authors. For details, we refer to see [2,3,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][26][27][28] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Since then a number of researchers investigated several classes of generalized m-accretive mappings such as H-accretive, (H, g)-accretive, and (A, g)-accretive mappings, see for examples [2,[4][5][6][8][9][10][11][12][13][16][17][18]24,26]. Sun et al [23] introduced a new class of M-monotone mapping in Hilbert spaces. Recently, Zou and Huang [27,28] and Kazmi et al [18] introduced and studied a class of H(Á, Á)-accretive mappings in Banach spaces, an natural extension of M-monotone mapping and studied variational inclusions involving these mappings.…”

Section: Introductionmentioning

confidence: 99%

A system of generalized variational inclusions involving generalized H(·,·)-accretive mapping in real q-uniformly smooth Banach spaces

Kazmi

Khan

Shahzad

2011

Applied Mathematics and Computation

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“…Recently, Ding and Luo [12] , Huang and Fang [13] , Fang and Huang [14] , Fang et al [15] , Verma [16][17] , Verma [18] , Zhang [19] , and Sun et al [20] introduced the concepts…”

Section: Introductionmentioning

confidence: 99%

Generalized H-η-accretive operators in Banach spaces with application to variational inclusions

Luo

Huang

2010

Appl. Math. Mech.-Engl. Ed.

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In this paper, a new notion of a generalized H-η-accretive operator is introduced and studied, which provides a unifying framework for the generalized m-accretive operator and the H-η-monotone operator in Banach spaces. A resolvent operator associated with the generalized H-η-accretive operator is defined, and its Lipschitz continuity is shown. As an application, the solvability for a class of variational inclusions involving the generalized H-η-accretive operators in Banach spaces is considered. By using the technique of the resolvent mapping, an iterative algorithm for solving the variational inclusion in Banach spaces is constructed. Under some suitable conditions, it is proven that the solution for the variational inclusion and the convergence of the iterative sequence generated by the algorithm exist. 502Xue-ping LUO and Nan-jing HUANG of η-subdifferential operators, maximal η-monotone operators, H-monotone operators, (H, η)-monotone operators, A-monotone operators, (A, η)-monotone operators, G-η-monotone operators, and M -monotone operators in Hilbert spaces, respectively. Huang and Fang [21] were the first to introduced the generalized m-accretive operators which extended the notion of maximal η-monotone operators to Banach spaces. Fang and Huang [22][23] , Lan et al. [24] , Lan [25] , and Zou and Huang [26][27] investigated many accretive operators such as H-accretive operators, (H, η)-accretive operators, (A, η)-accretive operators, and H(·, ·)-accretive operators in Banach spaces, which generalized the notions of H-monotone operators, (H, η)-monotone operators, (A, η)-monotone operators, and M -monotone operators in Hilbert spaces. They also defined the associated resolvent operators and used the resolvent operator technique. They developed some iterative algorithms to approximate the solutions of variational inclusions. Further, Xia and Huang [28] , Ding and Feng [29] , Feng and Ding [30] , Lou et al. [31] , Ding and Wang [32] , and Luo and Huang [33] introduced the notions of general H-monotone operators, A-monotone operators, H-η-monotone operators, and B-monotone operators in Banach spaces, which generalized the corresponding notions of the monotone type operators mentioned above. In addition, the H-η-monotone operators are different from the generalized m-accretive operators in Banach spaces.Motivated and inspired by the research in this field, in this paper, we introduce a new notion of generalized H-η-accretive operators which can provide a unifying framework for the generalized m-accretive operators and the H-η-monotone operators in Banach spaces. Moreover, we give a definition of the resolvent operator for the generalized H-η-accretive operator and prove its Lipschitz continuity in Banach spaces. As an application, we consider the solvability for a class of variational inclusions involving the generalized H-η-accretive operators in Banach spaces. By using the technique of resolvent mapping, we construct an iterative algorithm for solving the variational inclusion in Banach spaces. Under some su...

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An algorithm based on resolvent operators for solving variational inequalities in Hilbert spaces

Cited by 24 publications

References 11 publications

A Generalized Mixed Variational Inclusion Involving (H(·, ·), ?)-Monotone Operators in Banach Spaces

A Generalized Mixed Variational Inclusion Involving (H(·, ·), ?)-Monotone Operators in Banach Spaces

A system of generalized variational inclusions involving generalized H(·,·)-accretive mapping in real q-uniformly smooth Banach spaces

Generalized H-η-accretive operators in Banach spaces with application to variational inclusions

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