• 대한수학회지
• /
• 제45권1호
• /
• pp.1-27
• /
• 2008

We introduce a new concept of convex spaces and a multimap class K having certain KKM property. From a basic KKM type theorem for a K-map defined on an convex space without any topology, we deduce ten equivalent formulations of the theorem. As applications of the equivalents, in the frame of convex topological spaces, we obtain Fan-Browder type fixed point theorems, almost fixed point theorems for multimaps, mutual relations between the map classes K and B, variational inequalities, the von Neumann type minimax theorems, and the Nash equilibrium theorems.

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제11권3호
• /
• pp.231-242
• /
• 2004

In this paper, we introduce weakly relaxed $\alpha$-pseudomonotonicity and weakly relaxed $\alpha$-semi-pseudomonotonicity of set-valued maps. Using the KKM technique, we obtain existence of solutions to the variational-like inequalities with weakly relaxed $\alpha$-pseudomor.otone set-valued maps in reflexive Banach spaces. We also present the solvability of the variational-like inequalities with weakly relaxed $\alpha$-semi-pseudomonotone set-valued maps in arbitrary Banach spaces using Kakutani-Fan-Glicksberg fixed point theorem.

• 대한수학회보
• /
• 제38권3호
• /
• pp.565-573
• /
• 2001

In this paper, we first rove the strict quasi-concavity of maximizing function, and next prove a new maximum theorem using Fan’s generalization of the classical KKM theorem. Also an existence theorem of social equilibrium can be proved when an additional assumption on the constraint correspondence is assumed. Finally, we give illustrative two examples of constrained optimization problems.

• 대한수학회논문집
• /
• 제21권4호
• /
• 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.

• 호남수학학술지
• /
• 제31권2호
• /
• pp.247-258
• /
• 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].

• Journal of applied mathematics & informatics
• /
• 제40권3_4호
• /
• pp.577-585
• /
• 2022

In this work, we introduced and study vector equilibrium problems for trifunction in measurable space (for short, VEPMS). The existence of solutions of (VEPMS) are obtained by employing Aumann theorem and Fan KKM lemma. As an application, we prove an existence result for vector variational inequality problem for measurable space. Our results in this paper are new which can be considered as significant extension of previously known results in the literature.

• 대한수학회지
• /
• 제45권3호
• /
• pp.841-858
• /
• 2008

In this paper, we introduce two classes of generalized variational-like inequalities with compositely monotone multifunctions in Banach spaces. Using the KKM-Fan lemma and the Nadler's result, we prove the existence of solutions for generalized variational-like inequalities with compositely relaxed ${\eta}-{\alpha}$ monotone multifunctions in reflexive Banach spaces. On the other hand we also derive the solvability of generalized variational-like inequalities with compositely relaxed ${\eta}-{\alpha}$ semimonotone multi functions in arbitrary Banach spaces by virtue of the Kakutani-Fan-Glicksberg fixed-point theorem. The results presented in this paper extend and improve some earlier and recent results in the literature.

• 대한수학회보
• /
• 제30권1호
• /
• pp.83-89
• /
• 1993

Applications of the classical Knaster-Kuratowski-Mazurkiewicz theorem [KKM] and the fixed point theory of multifunctions defined on convex subsets of topological vector spaces have been greatly improved by adopting the concept of convex spaces due to Lassonde[L]. Recently, this concept has been extended to pseudo-convex spaces, contractible spaces, or spaces having certain families of contractible subsets by Horvath[H1-4]. In the present paper we give a far-reaching generalization of the best approximation theorem of Ky Fan[F1, 2] to pseudo-metric spaces and improved versions of the well-known fixed point theorems due to Brouwer [B] and Schauder [S] for spaces having certain families of contractible subsets. Our basic tool is a generalized Fan-Browder type fixed point theorem in our previous works [P3, 4].

• Journal of applied mathematics & informatics
• /
• 제27권3_4호
• /
• pp.819-827
• /
• 2009

In this paper, we introduce new pseudomonotonicity and proper quasimonotonicity with respect to a given function, and show some existence results for strong implicit vector variational inequalities by considering new generalized Minty's lemma. Our results generalize and extend some results in [1].

• 한국수학교육학회지시리즈B:순수및응용수학
• /
• 제11권2호
• /
• pp.155-166
• /
• 2004

In this paper, we introduce new monotone concepts for set-valued mappings, called generalized C(x)-L-pseudomonotonicity and weakly C(x)-L-pseudomonotonicity. And we obtain generalized Minty-type lemma and the existence of solutions to vector variational inequality problems for weakly C(x)-L-pseudomonotone set-valued mappings, which generalizes and extends some results of Konnov & Yao [11], Yu & Yao [20], and others Chen & Yang [6], Lai & Yao [12], Lee, Kim, Lee & Cho [16] and Lin, Yang & Yao [18].