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GENERALIZATIONS OF SOME KKM TYPE RESULTS ON HYPERBOLIC SPACES
Sehie Park 경남대학교 기초과학연구소 2018 Nonlinear Functional Analysis and Applications Vol.23 No.4
In some previous works, it is known that the KKM type results on Hadamard manifolds can be extended to hyperbolic spaces originated from Kirk in 1982 and Reich- Shafrir in 1990. Such results are the KKM theorem, the Fan-Browder fixed point theorem, Nash equilibrium theorem, variational inequalities, etc. based on our theory of abstract convex spaces. In the present article, we show that our method can be applied some recent works on Hadamard manifolds. Historical remarks are added on the study of the KKM type results on Hadamard manifolds and hyperbolic spaces.
EXTENSIONS OF ORDERED FIXED POINT THEOREMS
Sehie Park 경남대학교 기초과학연구소 2023 Nonlinear Functional Analysis and Applications Vol.28 No.3
Our long-standing Metatheoremin Ordered Fixed Point Theory is applied to some well-known ordertheoretic fixed point theorems. In the first half of this article,we introduce extended versions of the Zermelo fixed point theorem,Zorn's lemma, and the Caristi fixed point theorem based on theBr{\o}ndsted-Jachymski principle and our 2023 Metatheorem. We showsome of their applications to other fixed point theorems or theoremson the existence of maximal elements in partially ordered sets. Inthe second half, we collect and improve order theoretic fixed pointtheorems in the collection of Howard-Rubin in 1991 and others. Infact, we improve or extend several ordering principles or fixedpoint theorems due to Br\'ezis-Browder, Br{\o}ndsted,Knaster-Tarski, Tarski-Kantorovitch, Turinici, Granas-Horvath,Jachymski, and others.
BEST APPROXIMATIONS FOR MULTIMAPS ON ABSTRACT CONVEX SPACES
Sehie Park 경남대학교 기초과학연구소 2021 Nonlinear Functional Analysis and Applications Vol.26 No.1
In this article we derive some best approximation theorems for multimaps in abstract convex metric spaces. We are based on generalized KKM maps due to Kassay-Kolumbán, Chang-Zhang, and studied by Park, Kim-Park, Park-Lee, and Lee. Our main results are extensions of a recent work of Mitrović-Hussain-Sen-Radenović on G-convex metric spaces to partial KKM metric spaces. We also recall known works related to single-valued maps, and introduce new partial KKM metric spaces which can be applied our new results.
Fixed points and alternative principles
( Sehie Park ),( Hoon Joo Kim ) 호남수학회 2012 호남수학학술지 Vol.34 No.3
In a recent paper, M. Balaj [B] established an alternative principle. The principle was applied to a matching theorem of Ky Fan type, an analytic alternative, a minimax inequality, and existence of solutions of a vector equilibrium theorem. Based on the first author`s xed point theorems, in the present paper, we obtain generalizations of the main result of Balaj [B] and their applications.
On the von Neumann Type Minimax Theorens
Sehie Park 대한민국 학술원 2011 학술원논문집 : 자연과학편 Vol.50 No.2
In this paper, we review generalizations of the won Neumann-Sion minimax theorem mainly due to present author. We give them based on the fixed point theory or the KKM theory on subsets of topological vector spaces, convex space, H-spaces, G-convex space, abstract convex spaces, or other spaces. 이 글에서는 추상블록공간에서의 폰 노이만 형 최대최소정리의 여러 형태 중에서 주로 본인이 발견한 것들을 다룬다. 실제로 위상벡터 공간의 부분집합, 블록공간, H-공간, G-블록공간, 추상블록공간, 그 밖의 공간들에서의 부동점이론과 KKM 이론으로부터 이같은 정리들을 이끌어낸 역사를 체계적으로 다룬다.