• Korean Journal of Mathematics
• /
• 제25권4호
• /
• pp.469-481
• /
• 2017

In this paper, we established a general nonconvex split variational inequality problem, this is, an extension of general convex split variational inequality problems in two different Hilbert spaces. By using the concepts of prox-regularity, we proved the convergence of the iterative schemes for the general nonconvex split variational inequality problems. Further, we also discussed the iterative method for the general convex split variational inequality problems.

• 대한수학회보
• /
• 제59권4호
• /
• pp.1019-1044
• /
• 2022

In this paper, we propose new algorithms for solving equilibrium and multivalued variational inequality problems in a real Hilbert space. The first algorithm for equilibrium problems uses only one approximate projection at each iteration to generate an iteration sequence converging strongly to a solution of the problem underlining the bifunction is pseudomonotone. On the basis of the proposed algorithm for the equilibrium problems, we introduce a new algorithm for solving multivalued variational inequality problems. Some fundamental experiments are given to illustrate our algorithms as well as to compare them with other algorithms.

• 호남수학학술지
• /
• 제29권3호
• /
• pp.455-464
• /
• 2007

We consider existence theorems for a new class of mixed vector F-implicit complementarity problems and the corresponding mixed vector F-implicit variational inequality problems.

• 대한수학회논문집
• /
• 제22권3호
• /
• pp.401-410
• /
• 2007

Some existence theorems of solutions of a new class of generalized vector F-implicit complementarity problems with the corresponding generalized vector F-implicit variational inequality problems were established.

• Korean Journal of Mathematics
• /
• 제26권2호
• /
• pp.299-306
• /
• 2018

In this paper, we prove the existence results of the solutions for vector variational inequality problems by using the ${\parallel}{\cdot}{\parallel}$-sequentially continuous mapping.

• East Asian mathematical journal
• /
• 제26권3호
• /
• 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].

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제19권3호
• /
• pp.281-288
• /
• 2012

In this paper, the author defines a new generalized ${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mapping and considers the equivalence of Stampacchia-type vector variational-like inequality problems and Minty-type vector variational-like inequality problems for generalized (${\eta}$, ${\delta}$, ${\alpha}$)-pseudomonotone mappings in Banach spaces, called the generalized vector Minty's lemma.

• Nonlinear Functional Analysis and Applications
• /
• 제27권2호
• /
• pp.223-232
• /
• 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.

• East Asian mathematical journal
• /
• 제26권3호
• /
• pp.371-378
• /
• 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.

• 대한수학회지
• /
• 제32권3호
• /
• pp.493-507
• /
• 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.