• 제목/요약/키워드: variational

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A VARIANT OF THE GENERALIZED VECTOR VARIATIONAL INEQUALITY WITH OPERATOR SOLUTIONS

  • Kum, Sang-Ho
    • 대한수학회논문집
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    • 제21권4호
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    • pp.665-673
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    • 2006
  • In a recent paper, Domokos and $Kolumb\'{a}}n$ [2] gave an interesting interpretation of variational inequalities (VI) and vector variational inequalities (VVI) in Banach space settings in terms of variational inequalities with operator solutions (in short, OVVI). Inspired by their work, in a former paper [4], we proposed the scheme of generalized vector variational inequality with operator solutions (in short, GOVVI) which extends (OVVI) into a multivalued case. In this note, we further develop the previous work [4]. A more general pseudomonotone operator is treated. We present a result on the existence of solutions of (GVVI) under the weak pseudomonotonicity introduced in Yu and Yao [8] within the framework of (GOVVI) by exploiting some techniques on (GOVVI) or (GVVI) in [4].

GENERAL VARIATIONAL INCLUSIONS AND GENERAL RESOLVENT EQUATIONS

  • Liu, Zeqing;Ume, Jeong-Sheok;Kang, Shin-Min
    • 대한수학회보
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    • 제41권2호
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    • pp.241-256
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    • 2004
  • In this paper, we introduce and study a new class of variational inclusions, called the general variational inclusion. We prove the equivalence between the general variational inclusions, the general resolvent equations, and the fixed-point problems, using the resolvent operator technique. This equivalence is used to suggest and analyze a few iterative algorithms for solving the general variational inclusions and the general resolvent equations. Under certain conditions, the convergence analyses are also studied. The results presented in this paper generalize, improve and unify a number of recent results.

ON A SYSTEM OF GENERALIZED NONLINEAR VARIATIONAL INEQUALITIES

  • Li, Jingchang;Guo, Zhenyu;Liu, Zeqing;Kang, Shin-Min
    • 대한수학회논문집
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    • 제22권2호
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    • pp.247-258
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    • 2007
  • In this paper a new class of system of generalized nonlinear variational inequalities involving strongly monotone, relaxed co coercive and relaxed generalized monotone mappings in Hilbert spaces is introduced and studied. Based on the projection method, an equivalence between the system of generalized nonlinear variational inequalities and the fixed point problem is established, which is used to suggest some new iterative algorithms for computing approximate solutions of the system of generalized nonlinear variational inequalities. A few sufficient conditions which ensure the existence and uniqueness of solution of the system of generalized nonlinear variational inequalities are given, and the convergence analysis of iterative sequences generated by the algorithms are also discussed.

STRONG CONVERGENCE OF EXTENDED GENERAL VARIATIONAL INEQUALITIES AND NONEXPANSIVE MAPPINGS

  • Chen, Jun-Min;Zhang, Li-Juan;He, Zhen
    • East Asian mathematical journal
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    • 제26권1호
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    • pp.59-67
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    • 2010
  • In this paper, we suggest and analyze some three step iterative scheme for finding the common elements of the set of the solutions of the extended general variational inequalities involving three operators and the set of the fixed points of nonexpansive mappings. We also consider the convergence analysis of suggested iterative schemes under some mild conditions. Since the extended general variational inequalities include general variational inequalities and several other classes of variational inequalities as special cases, results obtained in this paper continue to hold for these problems. Results obtained in this paper may be viewed as a refinement and improvement of the previously known results.

THE PERRON AND VARIATIONAL INTEGRALS

  • Park, Jae Myung;Lee, Deok Ho
    • 충청수학회지
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    • 제10권1호
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    • pp.37-41
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    • 1997
  • In this paper, we give a direct proof that the Perron and variational integrals are equivalent.

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GENERALIZED SYSTEMS OF RELAXED $g-{\gamma}-r-COCOERCIVE$ NONLINEAR VARIATIONAL INEQUALITIES AND PROJECTION METHODS

  • Verma, Ram U.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • 제7권2호
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    • pp.83-94
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    • 2003
  • Let K be a nonempty closed convex subset of a real Hilbert space H. Approximation solvability of a system of nonlinear variational inequality (SNVI) problems, based on the convergence of projection methods, is given as follows: find elements $x^*,\;y^*{\in}H$ such that $g(x^*),\;g(y^*){\in}K$ and $$<\;{\rho}T(y^*)+g(x^*)-g(y^*),\;g(x)-g(x^*)\;{\geq}\;0\;{\forall}\;g(x){\in}K\;and\;for\;{\rho}>0$$ $$<\;{\eta}T(x^*)+g(y^*)-g(x^*),\;g(x)-g(y^*)\;{\geq}\;0\;{\forall}g(x){\in}K\;and\;for\;{\eta}>0,$$ where T: $H\;{\rightarrow}\;H$ is a relaxed $g-{\gamma}-r-cocoercive$ and $g-{\mu}-Lipschitz$ continuous nonlinear mapping on H and g: $H{\rightarrow}\;H$ is any mapping on H. In recent years general variational inequalities and their algorithmic have assumed a central role in the theory of variational methods. This two-step system for nonlinear variational inequalities offers a great promise and more new challenges to the existing theory of general variational inequalities in terms of applications to problems arising from other closely related fields, such as complementarity problems, control and optimizations, and mathematical programming.

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A NEW PREDICTOR-CORRECTOR METHOD FOR NONCOERCIVE MIXED VARIATIONAL INEQUALITIES

  • Noor, Muhammad Aslam
    • Journal of applied mathematics & informatics
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    • 제7권2호
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    • pp.483-491
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    • 2000
  • In this paper, we use the auxiliary principle technique to suggest and analyze a class of predictor-corrector methods for solving noncoercive mixed variational inequalities. The convergence of the proposed method requires only the partially relaxed strongly monotonicity. which is even weaker than the co-coercivity. As special cases, we obtain a number of new and known results for classical variational inequalities.

A REMARK ON THE REGULARIZED GAP FUNCTION FOR IQVI

  • Kum, Sangho
    • 충청수학회지
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    • 제28권1호
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    • pp.145-150
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    • 2015
  • Aussel et al. [1] introduced the notion of inverse quasi-variational inequalities (IQVI) by combining quasi-variational inequalities and inverse variational inequalities. Discussions are made in a finite dimensional Euclidean space. In this note, we develop an infinite dimensional version of IQVI by investigating some basic properties of the regularized gap function of IQVI in a Banach space.

EXISTENCE OF SOLUTIONS FOR GENERALIZED NONLINEAR VARIATIONAL-LIKE INEQUALITY PROBLEMS IN BANACH SPACES

  • Jeong, Jae-Ug
    • Journal of applied mathematics & informatics
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    • 제29권5_6호
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    • pp.1453-1462
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    • 2011
  • In this paper, we study a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By using the KKM technique and the concept of the Hausdorff metric, we obtain some existence results for generalized nonlinear variational-like inequalities with generalized monotone multi-valued mappings in Banach spaces. These results improve and generalize many known results in recent literature.

A PARALLEL HYBRID METHOD FOR EQUILIBRIUM PROBLEMS, VARIATIONAL INEQUALITIES AND NONEXPANSIVE MAPPINGS IN HILBERT SPACE

  • Hieu, Dang Van
    • 대한수학회지
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    • 제52권2호
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    • pp.373-388
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    • 2015
  • In this paper, a novel parallel hybrid iterative method is proposed for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of variational inequalities for inverse strongly monotone mappings and the set of fixed points of a finite family of nonexpansive mappings in Hilbert space. Strong convergence theorem is proved for the sequence generated by the scheme. Finally, a parallel iterative algorithm for two finite families of variational inequalities and nonexpansive mappings is established.