• Title, Summary, Keyword: Variational Inequalities

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GENERALIZED VECTOR-VALUED VARIATIONAL INEQUALITIES AND FUZZY EXTENSIONS

  • Lee, Byung-Soo;Lee, Gue-Myung;Kim, Do-Sang
    • Journal of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.609-624
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    • 1996
  • Recently, Giannessi [9] firstly introduced the vector-valued variational inequalities in a real Euclidean space. Later Chen et al. [5] intensively discussed vector-valued variational inequalities and vector-valued quasi variationl inequalities in Banach spaces. They [4-8] proved some existence theorems for the solutions of vector-valued variational inequalities and vector-valued quasi-variational inequalities. Lee et al. [14] established the existence theorem for the solutions of vector-valued variational inequalities for multifunctions in reflexive Banach spaces.

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SCALARIZATION METHODS FOR MINTY-TYPE VECTOR VARIATIONAL INEQUALITIES

  • Lee, Byung-Soo
    • East Asian mathematical journal
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    • v.26 no.3
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    • pp.415-421
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    • 2010
  • Many kinds of Minty's lemmas show that Minty-type variational inequality problems are very closely related to Stampacchia-type variational inequality problems. Particularly, Minty-type vector variational inequality problems are deeply connected with vector optimization problems. Liu et al. [10] considered vector variational inequalities for setvalued mappings by using scalarization approaches considered by Konnov [8]. Lee et al. [9] considered two kinds of Stampacchia-type vector variational inequalities by using four kinds of Stampacchia-type scalar variational inequalities and obtain the relations of the solution sets between the six variational inequalities, which are more generalized results than those considered in [10]. In this paper, the author considers the Minty-type case corresponding to the Stampacchia-type case considered in [9].

PERTURBED PROXIMAL POINT ALGORITHMS FOR GENERALIZED MIXED VARIATIONAL INEQUALITIES

  • Jeong, Jae-Ug
    • East Asian mathematical journal
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    • v.18 no.1
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    • pp.95-109
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    • 2002
  • In this paper, we study a class of variational inequalities, which is called the generalized set-valued mixed variational inequality. By using the properties of the resolvent operator associated with a maximal monotone mapping in Hilbert spaces, we have established an existence theorem of solutions for generalized set-valued mixed variational inequalities, suggesting a new iterative algorithm and a perturbed proximal point algorithm for finding approximate solutions which strongly converge to the exact solution of the generalized set-valued mixed variational inequalities.

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AN ITERATIVE METHOD FOR NONLINEAR MIXED IMPLICIT VARIATIONAL INEQUALITIES

  • JEONG, JAE UG
    • Honam Mathematical Journal
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    • v.26 no.4
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    • pp.391-399
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    • 2004
  • In this paper, we develop an iterative algorithm for solving a class of nonlinear mixed implicit variational inequalities in Hilbert spaces. The resolvent operator technique is used to establish the equivalence between variational inequalities and fixed point problems. This equivalence is used to study the existence of a solution of nonlinear mixed implicit variational inequalities and to suggest an iterative algorithm for solving variational inequalities. In our results, we do not assume that the mapping is strongly monotone.

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AUXILIARY PRINCEPLE AND ERROR ESTIMATES FOR VARIATIONAL INEQUALITIES

  • NOOR, MUHAMMED ASLAM
    • Honam Mathematical Journal
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    • v.15 no.1
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    • pp.105-120
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    • 1993
  • The auxiliary principle technique is used to prove the uniqueness and the existence of solutions for a class of nonlinear variational inequalities and suggest an innovative iterative algorithm for computing the approximate solution of variational inequalities. Error estimates for the finite element approximation of the solution of variational inequalities are derived, which refine the previous known results. An example is given to illustrate the applications of the results obtained. Several special cases are considered and studied.

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ON THE GENERALIZED SET-VALUED MIXED VARIATIONAL INEQUALITIES

  • Zhao, Yali;Liu, Zeqing;Kang, Shin-Min
    • Communications of the Korean Mathematical Society
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    • v.18 no.3
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    • pp.459-468
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    • 2003
  • In this paper, we introduce and study a new class of the generalized set-valued mixed variational inequalities. Using the resolvent operator technique, we construct a new iterative algorithm for solving this class of the generalized set-valued mixed variational inequalities. We prove the existence of solutions for the generalized set-valued mixed variational inequalities and the convergence of the iterative sequences generated by the algorithm.

CONVERGENCE OF AN ITERATIVE ALGORITHM FOR SYSTEMS OF GENERALIZED VARIATIONAL INEQUALITIES

  • Jeong, Jae Ug
    • Korean Journal of Mathematics
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    • v.21 no.3
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    • pp.213-222
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    • 2013
  • In this paper, we introduce and consider a new system of generalized variational inequalities involving five different operators. Using the sunny nonexpansive retraction technique we suggest and analyze some new explicit iterative methods for this system of variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Our results can be viewed as a refinement and improvement of the previously known results for variational inequalities.

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].

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.

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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