• Nonlinear Functional Analysis and Applications
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• 제27권2호
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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.

• 대한수학회보
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• 제47권4호
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• pp.675-685
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• 2010

In this paper, we consider a new class of regularized (nonconvex) generalized mixed variational inequality problems in real Hilbert space. We give the concepts of partially relaxed strongly mixed monotone and partially relaxed strongly $\theta$-pseudomonotone mappings, which are extension of the concepts given by Xia and Ding [19], Noor [13] and Kazmi et al. [9]. Further we use the auxiliary principle technique to suggest a two-step iterative algorithm for solving regularized (nonconvex) generalized mixed variational inequality problem. We prove that the convergence of the iterative algorithm requires only the continuity, partially relaxed strongly mixed monotonicity and partially relaxed strongly $\theta$-pseudomonotonicity. The theorems presented in this paper represent improvement and generalization of the previously known results for solving equilibrium problems and variational inequality problems involving the nonconvex (convex) sets, see for example Noor [13], Pang et al. [14], and Xia and Ding [19].

• 대한수학회논문집
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• 제18권4호
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• pp.737-743
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• 2003

The purpose of this paper is to consider some existence results for vector nonlinear inequalities without any monotonicity assumption. As consequences of our main result, we give some existence results for vector equilibrium problem, vector variational-like inequality problem and vector variational inequality problems as special cases.

• East Asian mathematical journal
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• 제26권3호
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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].

• Asian Journal for Public Opinion Research
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• 제6권1호
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• pp.1-17
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• 2018

As societal interest in inequality increases in Korea, both public and academic discussion on inequality is also on the rise. In order to more effectively discuss the problems of rising inequality, however, it is essential to study the consequences and implications of inequality. This study examines one of the consequences of inequality, particularly on individuals - the relationship between an individual's perception of inequality and his/her evaluation of societal health, such as social trust and social mobility. According to a statistical analysis of the Korean Academic Multimode Open Survey for Social Sciences (KAMOS), those who perceive the level of income and wealth inequality in Korea as more unequal tend to have a lower level of trust toward Korean society and Korean people, as well as a lower expectation for both intra- and intergenerational social mobility. This study, which shows that rising inequality could have a negative impact at the individual level, not only extends the scope of the consequence-of-inequality studies from the society-oriented toward the individual-oriented, but it also has significant implications for the field, suggesting a new direction for future studies.

• 대한수학회지
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• 제32권3호
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• pp.493-507
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• 1995

The Hartman-Stampacchia variational inequality was first suggested and studied by Hartman and Stampacchia [8] in finite dimensional spaces during the time establishing the base of variational inequality theory in 1960s [4]. Then it was generalized by Lions et al. [6], [9], [10], Browder [3] and others to the case of infinite dimensional inequality [3], [9], [10], and the results concerning this variational inequality have been applied to many important problems, i.e., mechanics, control theory, game theory, differential equations, optimizations, mathematical economics [1], [2], [6], [9], [10]. Recently, the Browder-Hartman-Stampaccnia variational inequality was extended to the case of set-valued monotone mappings in reflexive Banach sapces by Shih-Tan [11] and Chang [5], and under different conditions, they proved some existence theorems of solutions of this variational inequality.

• Korean Journal of Mathematics
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• 제23권3호
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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.

• 대한수학회논문집
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• 제28권1호
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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.

• 호남수학학술지
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• 제31권2호
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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].

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제15권2호
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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.