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Vietnam Journal of Mathematics 40:2&3(2012) 391-405

 

Topological Minimax Theorems: Old and New

Hoang Tuy

Institute of Mathematics, 18 Hoang Quoc Viet, 10307, Hanoi, Vietnam

Dedicated to Professor Phan Quoc Khanh on the occasion of his 65th birthday

Received October 25, 2011

Revised February 20, 2012

Abstract. We review and extend the main topological minimax theorems based on connectedness that have been developed over the years since the pioneering paper of Wu (1959). It is shown in particular that the topological minimax theorems of Geraghty and Lin (1984) are essentially a rediscovery of much earlier results of Tuy (1974), while the latter can be derived from a minimax theorem recently developed for functions involving a real variable. Several new topological minimax theorems are presented, including a general theorem containing both König’s (1992) and Tuy’s (1974) results and a minimax theorem for increasing-decreasing functions analogous to Sion’s classical theorem.

2000 Mathematics Subject Classification. 49J35, 54D05.

Keywords. Topological minimax theorems, connectedness assumptions, review and extensions of K¨onig’s theorem, minimax theorem for monotonic functions.

 

 

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