• Title, Summary, Keyword: variational

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MINTY'S LEMMA FOR STRONG IMPLICIT VECTOR VARIATIONAL INEQUALITY SYSTEMS

  • Kim, Seung-Hyun;Lee, Byung-Soo
    • The Pure and Applied Mathematics
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    • v.15 no.4
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    • pp.423-432
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    • 2008
  • In this paper, we consider a new Minty's Lemma for strong implicit vector variational inequality systems and obtain some existence results for systems of strong implicit vector variational inequalities which generalize some results in [1].

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MULTIVALUED MIXED QUASI-VARIATIONAL-LIKE INEQUALITIES

  • Lee Byung-Soo
    • The Pure and Applied Mathematics
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    • v.13 no.3
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    • pp.197-206
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    • 2006
  • This paper introduces a class of multivalued mixed quasi-variational-like ineqcalities and shows the existence of solutions to the class of quasi-variational-like inequalities in reflexive Banach spaces.

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PERIODIC SOLUTIONS FOR THE NONLINEAR HAMILTONIAN SYSTEMS

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.17 no.3
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    • pp.331-340
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    • 2009
  • We show the existence of nonconstant periodic solution for the nonlinear Hamiltonian systems with some nonlinearity. We approach the variational method. We use the critical point theory and the variational linking theory for strongly indefinite functional.

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APPROXIMATION OF SOLUTIONS OF A GENERALIZED VARIATIONAL INEQUALITY PROBLEM BASED ON ITERATIVE METHODS

  • Cho, Sun-Young
    • Communications of the Korean Mathematical Society
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    • v.25 no.2
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    • pp.207-214
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    • 2010
  • In this paper, a generalized variational inequality problem is considered. An iterative method is studied for approximating a solution of the generalized variational inequality problem. Strong convergence theorem are established in a real Hilbert space.

MULTIPLICITY RESULTS AND THE M-PAIRS OF TORUS-SPHERE VARIATIONAL LINKS OF THE STRONGLY INDEFINITE FUNCTIONAL

  • Jung, Tack-Sun;Choi, Q-Heung
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.12 no.4
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    • pp.239-247
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    • 2008
  • Let $I{\in}C^{1,1}$ be a strongly indefinite functional defined on a Hilbert space H. We investigate the number of the critical points of I when I satisfies two pairs of Torus-Sphere variational linking inequalities and when I satisfies m ($m{\geq}2$) pairs of Torus-Sphere variational linking inequalities. We show that I has at least four critical points when I satisfies two pairs of Torus-Sphere variational linking inequality with $(P.S.)^*_c$ condition. Moreover we show that I has at least 2m critical points when I satisfies m ($m{\geq}2$) pairs of Torus-Sphere variational linking inequalities with $(P.S.)^*_c$ condition. We prove these results by Theorem 2.2 (Theorem 1.1 in [1]) and the critical point theory on the manifold with boundary.

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GENERAL VARIATIONAL INCLUSIONS AND GENERAL RESOLVENT EQUATIONS

  • Liu, Zeqing;Ume, Jeong-Sheok;Kang, Shin-Min
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.2
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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.

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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    • v.26 no.1
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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.

APPROXIMATION OF SOLUTIONS FOR GENERALIZED WIENER-HOPF EQUATIONS AND GENERALIZED VARIATIONAL INEQUALITIES

  • Gu, Guanghui;Su, Yongfu
    • Journal of applied mathematics & informatics
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    • v.28 no.1_2
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    • pp.465-472
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    • 2010
  • The purpose of this article is to introduce a new generalized class of the Wiener-Hopf equations and a new generalized class of the variational inequalities. Using the projection technique, we show that the generalized Wiener-Hopf equations are equivalent to the generalized variational inequalities. We use this alternative equivalence to suggest and analyze an iterative scheme for finding the solution of the generalized Wiener-Hopf equations and the solution of the generalized variational inequalities. The results presented in this paper may be viewed as significant and improvement of the previously known results. In special, our results improve and extend the resent results of M.A. Noor and Z.Y.Huang[M.A. Noor and Z.Y.Huang, Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings, Appl. Math. Comput.(2007), doi:10.1016/j.amc.2007.02.117].

A VARIANT OF THE GENERALIZED VECTOR VARIATIONAL INEQUALITY WITH OPERATOR SOLUTIONS

  • Kum, Sang-Ho
    • Communications of the Korean Mathematical Society
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    • v.21 no.4
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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].