• Title, Summary, Keyword: variational

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ON A SYSTEM OF GENERALIZED NONLINEAR VARIATIONAL INEQUALITIES

  • Li, Jingchang;Guo, Zhenyu;Liu, Zeqing;Kang, Shin-Min
    • Communications of the Korean Mathematical Society
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    • v.22 no.2
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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.

THE PERRON AND VARIATIONAL INTEGRALS

  • Park, Jae Myung;Lee, Deok Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.10 no.1
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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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    • v.7 no.2
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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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On vector variational inequality

  • Lee, Gue-Myung;Kim, Do-Sang;Lee, Byung-Soo;Cho, Sung-Jin
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.4
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    • pp.553-564
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    • 1996
  • Since Giannessi [5] introduced the vector variational inequality in a finite dimensional Euclidean space with further application, Chang et al. [17], Chen et al. [1-4] and Lee et al. [10-16] have considered several kinds of vector variational inequalities in abstract spaces and have obtained existence theorems for their inequalities.

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A NOTE ON THE GENERALIZED VARIATIONAL INEQUALITY WITH OPERATOR SOLUTIONS

  • Kum, Sangho
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.3
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    • pp.319-324
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    • 2009
  • In a series of papers [3, 4, 5], the author developed the generalized vector variational inequality with operator solutions (in short, GOVVI) by exploiting variational inequalities with operator solutions (in short, OVVI) due to Domokos and $Kolumb\acute{a}n$ [2]. In this note, we give an extension of the previous work [4] in the setting of Hausdorff locally convex spaces. To be more specific, we present an existence of solutions of (GVVI) under the weak pseudomonotonicity introduced in Yu and Yao [7] within the framework of (GOVVI).

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A REMARK ON THE REGULARIZED GAP FUNCTION FOR IQVI

  • Kum, Sangho
    • Journal of the Chungcheong Mathematical Society
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    • v.28 no.1
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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.

ON SOLVABILITY AND ALGORITHM OF GENERAL STRONGLY NONLINEAR VARIATIONAL-LIKE INEQUALITIES

  • Liu Zeqing;Sun, Juhe;Shim, Soo-Hak;Kang, Shin-Min
    • Bulletin of the Korean Mathematical Society
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    • v.43 no.2
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    • pp.319-331
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    • 2006
  • In this paper, a new class of general strongly nonlinear variational-like inequalities was introduced and studied. The existence and uniqueness of solutions and a new iterative algorithm for the general strongly nonlinear variational-like inequality are established and suggested, respectively. The convergence criteria of the iterative sequence generated by the iterative algorithm are also given.

Random completley generalized nonlinear variational inclusions with non-compact valued random mappings

  • Huang, Nan-Jing;Xiang Long;Cho, Yeol-Je
    • Bulletin of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.603-615
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    • 1997
  • In this paper, we introduce and study a new class of random completely generalized nonlinear variational inclusions with non-compact valued random mappings and construct some new iterative algorithms. We prove the existence of random solutions for this class of random variational inclusions and the convergence of random iterative sequences generated by the algorithms.

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A GENERAL ITERATIVE METHOD BASED ON THE HYBRID STEEPEST DESCENT SCHEME FOR VARIATIONAL INCLUSIONS, EQUILIBRIUM PROBLEMS

  • Tian, Ming;Lan, Yun Di
    • Journal of applied mathematics & informatics
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    • v.29 no.3_4
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    • pp.603-619
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    • 2011
  • To the best of our knowledge, it would probably be the first time in the literature that we clarify the relationship between Yamada's method and viscosity iteration correctly. We design iterative methods based on the hybrid steepest descent algorithms for solving variational inclusions, equilibrium problems. Our results unify, extend and improve the corresponding results given by many others.

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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    • v.29 no.5_6
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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.