• Honam Mathematical Journal
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• v.40 no.3
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• pp.509-519
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• 2018

In this paper, we introduce a new class of reciprocal convex set which is called as relative reciprocal convex set. We establish a necessary and sufficient condition for the minimum of the differentiable relative reciprocal convex function. This condition can be viewed as a new class of variational inequality which is called relative reciprocal variational inequality. Using the auxiliary principle technique we discuss the existence criteria for the solution of relative reciprocal variational inequality. Some special cases which are naturally included in our main results are also discussed.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1453-1462
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• 2011

In this paper, we study a new class of generalized nonlinear variational-like inequalities in reflexive Banach spaces. By using the KKM technique and the concept of the Hausdorff metric, we obtain some existence results for generalized nonlinear variational-like inequalities with generalized monotone multi-valued mappings in Banach spaces. These results improve and generalize many known results in recent literature.

• 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.

• Korean Journal of Mathematics
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• v.16 no.4
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• pp.527-535
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• 2008

Let $I{\in}C^{1,1}$ be a strongly indefinite functional defined on a Hilbert space H. We investigate the number of the critical points of I when I satisfies one pair of Torus-Sphere variational linking inequality. We show that I has at least two critical points when I satisfies one pair of Torus-Sphere variational linking inequality with $(P.S.)^*_c$ condition. We prove this result by use of the limit relative category and critical point theory on the manifold with boundary.

• Journal of the Chungcheong Mathematical Society
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• v.4 no.1
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• pp.39-44
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• 1991

In this note, we shall prove a new variational inequality in non-compact sets and as an application, we prove a generalization of the Schauder-Tychonoff fixed point theorem.

• Journal of the Chungcheong Mathematical Society
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• v.13 no.1
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• pp.13-20
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• 2000

The purpose of this paper is to give a new existence theorem of a generalized weight Nash equilibrium for generalized multiobjective games by using the quasi-variational inequality due to Yuan.

• Korean Journal of Mathematics
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• v.23 no.3
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• pp.457-478
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• 2015

In this paper we propose an iteration hybrid method for approximating a point in the intersection of the solution-sets of pseudomonotone equilibrium and variational inequality problems and the fixed points of a semigroup-nonexpensive mappings in Hilbert spaces. The method is a combination of projection, extragradient-Armijo algorithms and Manns method. We obtain a strong convergence for the sequences generated by the proposed method.

• Journal of the Korean Mathematical Society
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• v.49 no.5
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• pp.881-891
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• 2012

In this paper we investigate the approximate controllability for the following nonlinear functional differential control problem: $$x^{\prime}(t)+Ax(t)+{\partial}{\phi}(x(t)){\ni}f(t,x(t))+h(t)$$ which is governed by the variational inequality problem with nonlinear terms.

• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.191-207
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• 2013

In this paper, we introduce the shrinking projection method for hemi-relatively nonexpansive mappings to find a common solution of variational inequality problems and equilibrium problems in uniformly convex and uniformly smooth Banach spaces and prove some strong convergence theorems to the common solution by using the proposed method.

• 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].