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Vietnam Journal of Mathematics 40:2&3(2012)
391-405
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Topological Minimax Theorems: Old and New
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Hoang Tuy
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Institute
of Mathematics, 18 Hoang Quoc Viet, 10307, Hanoi, Vietnam
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Dedicated to Professor
Phan Quoc Khanh on the occasion of his 65th birthday
Received October 25,
2011
Revised February 20,
2012
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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.
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2000 Mathematics
Subject Classification. 49J35, 54D05.
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Keywords.
Topological minimax theorems, connectedness assumptions, review and
extensions of K¨onig’s theorem, minimax theorem for monotonic functions.
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Established
by Vietnam Academy of Science and Technology &
Vietnam Mathematical Society
Published by
Springer since January 2013
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