• 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.

• 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].

• 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.

• 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).

• The Pure and Applied Mathematics
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• v.15 no.2
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• pp.203-207
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• 2008

In [1], Mishra and Wang established relationships between vector variational-like inequality problems and non-smooth vector optimization problems under non-smooth invexity in finite-dimensional spaces. In this paper, we generalize recent results of Mishra and Wang to infinite-dimensional case.

• Honam Mathematical Journal
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• v.31 no.2
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• pp.247-258
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• 2009

This paper introduces new mixed vector FQ-implicit variational inequality problems and corresponding mixed vector FQ-implicit complementarity problems for set-valued mappings, and studies the equivalence between them under certain assumptions in Banach spaces. It also derives some new existence theorems of solutions for them with examples under suitable assumptions without monotonicity. This paper generalizes and extends many results in [8, 10, 19-22].

• 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.

• The Pure and Applied Mathematics
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• v.10 no.4
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• pp.225-232
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• 2003

In this paper, we consider the existence of the solutions to the generalized vector variational-type inequalities for set-valued mappings on Hausdorff topological vector spaces using Fan's geometrical lemma.

• Nonlinear Functional Analysis and Applications
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• v.26 no.4
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• pp.781-792
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• 2021

In this paper, we study well-posed problems and variational inequalities in locally convex Hausdorff topological vector spaces. The necessary and sufficient conditions are obtained for the existence of solutions of variational inequality problems and quasi variational inequalities even when the underlying set K is not convex. In certain cases, solutions obtained are not unique. Moreover, counter examples are also presented for the authenticity of the main results.

• East Asian mathematical journal
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• v.26 no.3
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• pp.371-378
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• 2010

In this paper, we study generalized F-implicit multivalued variational inequality problems on a real normed vector space setting. As an application, we study generalized F-implicit multivalued complementarity problems.