• East Asian mathematical journal
• /
• v.26 no.3
• /
• pp.415-421
• /
• 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].

• The Pure and Applied Mathematics
• /
• v.13 no.3 s.33
• /
• pp.197-206
• /
• 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.

• Communications of the Korean Mathematical Society
• /
• v.21 no.4
• /
• pp.689-700
• /
• 2006

In this paper, we introduce a new class of generalized nonlinear multivalued mixed quasi-variational-like inequalities and prove the existence and uniqueness of solutions for the class of generalized nonlinear multivalued mixed quasi-variational-like inequalities in reflexive Banach spaces using Fan-KKM Theorem.

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

• The Pure and Applied Mathematics
• /
• v.6 no.1
• /
• pp.27-32
• /
• 1999

In this paper, we show that the existence of the solutions to the variational-type inequalities for set-valued mappings on normed linear spaces using Fan's section theorem.

• Communications of Mathematical Education
• /
• v.8
• /
• pp.273-278
• /
• 1999

In this paper, we consider the existence of solutions to the variational-type inequalities for single-valued mappings and set-valued mappings on reflexive Banach spaces using Fan's section theorem.

• Journal of the Korean Institute of Intelligent Systems
• /
• v.10 no.2
• /
• pp.133-137
• /
• 2000

In this paper we introduce vector variational-type inequalities for fuzzy mappings on Hausdorff topological vector spaces and obtain an existence theorem of solutions to the inequalities.

• East Asian mathematical journal
• /
• v.26 no.1
• /
• pp.49-58
• /
• 2010

In this paper, we consider and study a new class of generalized vector quasi-variational type inequalities and obtain some existence theorems for both under compact and noncompact assumptions in topological vector spaces without using monotonicity. For the noncompact case, we use the concept of escaping sequences.

• Bulletin of the Korean Mathematical Society
• /
• v.28 no.2
• /
• pp.163-174
• /
• 1991

This is a continuation of the author's previous work [17]. In this paper, we consider mainly variational inequalities for single-valued functions. We first obtain a generalization of the variational type inequality of Juberg and Karamardian [10] and apply it to obtain strengthened versions of the Hartman-Stampacchia inequality and the Brouwer fixed point theorem. Next, we obtain fairly general versions of Browder's variational inequality [5] and its subsequent generalizations due to Brezis et al [4], Takahashk [23], Shih and Tan [19], Simons [20], and others. Finally, in this paper, we obtain a variational inequality for non-real locally convex t.v.s. which generalizes a result of Shih and Tan [19].

• Communications of the Korean Mathematical Society
• /
• v.32 no.4
• /
• pp.933-957
• /
• 2017

In this paper, we prove some existence results of solutions for a new class of generalized bi-quasi-variational-like inequalities (GBQVLI) for (${\eta}-h$)-quasi-pseudo-monotone type I and strongly (${\eta}-h$)-quasi-pseudo-monotone type I operators defined on non-compact sets in locally convex Hausdorff topological vector spaces. To obtain our results on GBQVLI for (${\eta}-h$)-quasi-pseudo-monotone type I and strongly (${\eta}-h$)-quasi-pseudo-monotone type I operators, we use Chowdhury and Tan's generalized version of Ky Fan's minimax inequality as the main tool.