• Title/Summary/Keyword: generalized vector variational inequality

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GENERALIZED VECTOR MINTY'S LEMMA

  • Lee, Byung-Soo
    • The Pure and Applied Mathematics
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    • v.19 no.3
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    • pp.281-288
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    • 2012
  • In this paper, the author defines a new generalized ${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mapping and considers the equivalence of Stampacchia-type vector variational-like inequality problems and Minty-type vector variational-like inequality problems for generalized (${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mappings in Banach spaces, called the generalized vector Minty's lemma.

A REMARK ON MULTI-VALUED GENERALIZED SYSTEM

  • Kum, Sangho
    • Journal of the Chungcheong Mathematical Society
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    • v.24 no.2
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    • pp.163-169
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    • 2011
  • Recently, Kazmi and Khan [7] introduced a kind of equilibrium problem called generalized system (GS) with a single-valued bi-operator F. In this note, we aim at an extension of (GS) due to Kazmi and Khan [7] into a multi-valued circumstance. We consider a fairly general problem called the multi-valued quasi-generalized system (in short, MQGS). Based on the existence of 1-person game by Ding, Kim and Tan [5], we give a generalization of (GS) in the name of (MQGS) within the framework of Hausdorff topological vector spaces. As an application, we derive an existence result of the generalized vector quasi-variational inequality problem. This result leads to a multi-valued vector quasi-variational inequality extension of the strong vector variational inequality (SVVI) due to Fang and Huang [6] in a general Hausdorff topological vector space.

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 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].

On vector Quasivariational-like inequality

  • Lee, Gue-Myung;Kim, Do-Sang;Lee, Byung-Soo
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.45-55
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    • 1996
  • Recently, Giannessi [1] introduced a vector variational inequalityy for vector-valued functions in an Euclidean space. Since then, Chen et al. [2-6], Lee et al. [7], and Yang [8] have intensively studied vector variational inequalities for vector-valued functions in abstract spaces.

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AN EXTENSION OF GENERALIZED VECTOR QUASI-VARIATIONAL INEQUALITY

  • Kum Sang-Ho;Kim Won-Kyu
    • Communications of the Korean Mathematical Society
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    • v.21 no.2
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    • pp.273-285
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    • 2006
  • In this paper, we shall give an affirmative answer to the question raised by Kim and Tan [1] dealing with generalized vector quasi-variational inequalities which generalize many existence results on (VVI) and (GVQVI) in the literature. Using the maximal element theorem, we derive two theorems on the existence of weak solutions of (GVQVI), one theorem on the existence of strong solution of (GVQVI), and one theorem on strong solution in the 1-dimensional case.

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].