• Bulletin of the Korean Mathematical Society
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• v.54 no.3
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• pp.715-729
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• 2017

In this paper, we investigate the existence and multiplicity of solutions for systems driven by two non-local integrodifferential operators with homogeneous Dirichlet boundary conditions. The main tools are the Saddle point theorem, Ekeland's variational principle and the Mountain pass theorem.

• Nonlinear Functional Analysis and Applications
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• v.27 no.2
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• pp.223-232
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• 2022

The purpose of the paper is to define f-projection operator to develop the f-projection method. The existence of a variational inequality problem is studied using fixed point theorem which establishes the existence of f-projection method. The concept of ρ-projective operator and σ-involutory operator are defined with suitable examples. The relation in between ρ-projective operator and σ-involutory operator are shown. The concept of σ-involutory variational inequality problem is defined and its existence theorem is also established.

• Communications of the Korean Mathematical Society
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• v.14 no.2
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• pp.319-328
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• 1999

In this paper we consider a kind of Minty`s lemma for multifunctions in Banach spaces, and apply it to obtain existence theorems for two kinds of variational-like inequalities using the KKM-Fan theorem.

• The Pure and Applied Mathematics
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• v.6 no.1
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• pp.27-32
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• 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
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• v.8
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• pp.273-278
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• 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.

• Bulletin of the Korean Mathematical Society
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• v.36 no.2
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• pp.371-377
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• 1999

The purpose of this paper is to provide two examples which prove that Cubiotti's theorem and Yao's one on the generalized quasi-variational inequality problem are independent of each other. In addition, we give another example which tells us that certain conditions are essential in Cubiotti's theorem and Yao's one.

• 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.

• Journal of the Korean Mathematical Society
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• v.53 no.5
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• pp.1115-1131
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• 2016

In this paper we develop useful relations which estimate the difference between the solutions of nonlinear impulsive differential systems with different initial values. Then we obtain the converse h-stability theorem of Massera's type for the nonlinear impulsive systems by employing the $t_{\infty}$-similarity of the associated impulsive variational systems and relations.

• Journal of the Chungcheong Mathematical Society
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• v.1 no.1
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• pp.3-6
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• 1988

In a recent paper, Parida and Sen obtained a variational-like inequality. In this note, we obtain another variational-like inequality using Fan's minimax inequality [1] and Michael's selection theorem [2]. Also we generalize the Parida-Sen theorem in Banach spaces.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.139-149
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• 1989

pp.Hartman and G. Stampacchia [6] proved the following theorem in 1966: If f:X.rarw. $R^{n}$ is a continuous map on a compact convex subset X of $R^{n}$ , then there exists $x_{0}$ ..mem.X such that $x_{0}$ , $x_{0}$ -x>.geq.0 for all x.mem.X. This remarkable result has been investigated and generalized by F.E. Browder [1], [2], W. Takahashi [9], S. Park [8] and others. For example, Browder extended this theorem to a map f defined on a compact convex subser X of a topological vector space E into the dual space $E^{*}$; see [2, Theorem 2]. And Takahashi extended Browder's theorem to closed convex sets in topological vector space; see [9, Theorem 3]. In Section 2, we obtain some variational inequalities, especially, generalizations of Browder's and Takahashi's theorems. The generalization of Browder's is an earlier result of the first author [8]. In Section 3, using Theorem 1, we improve and extend some known fixed pint theorems. Theorems 4 and 8 improve Takahashi's results [9, Theorems 5 and 9], respectively. Theorem 4 extends the first author's fixed point theorem [8, Theorem 8] (Theorem 5 in this paper) which is a generalization of Browder [1, Theroem 1]. Theorem 8 extends Theorem 9 which is a generalization of Browder [2, Theorem 3]. Finally, in Section 4, we obtain variational inequalities for multivalued maps by using Theorem 1. We improve Takahashi's results [9, Theorems 21 and 22] which are generalization of Browder [2, Theorem 6] and the Kakutani fixed point theorem [7], respectively.ani fixed point theorem [7], respectively.