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Minimax monotonicity. (English) Zbl 0922.47047

Lecture Notes in Mathematics. 1693. Berlin: Springer. xi, 172 p. (1998).
Applications of the von Neumann minimax theorem to functional analysis, in particular, monotone operators are considered. The theorem proves equality of minimax and maximin for convex-concave functionals. This theorem combined with the Hahn-Banach theorem (that a sublinear functional may be majorized by a linear one), and the Banach-Alaoglu theorem (about weak compactness of a ball in dual space) provides the author with the minimax technique to obtain new results and to get simple proofs for well-known theorems. For example, Rockafellar’s characterization of those monotone operators on a reflexive space that are maximal, the local boundedness of any nontrivial monotone multifunction in any absorbing point of its domain, the convexity of the domain of a maximal monotone multifunction, and so on. Some of the known results are strengthened or generalized. The author proves that any point surrounded by the domain of a maximal monotone multifunction is an interior point of that domain. Using the minimax technique, the open mapping theorem is generalized. This helps to give a necessary and sufficient condition for the sum of maximal monotone multifunctions on a reflexive Banach space to be maximal monotone. Some nonreflexive cases are also considered. Note the generalizations of Rockafellar’s result on maximal monotonicity of subdifferentials. For unbounded (discontinuous) positive linear operators from a Banach space to its dual, a criterion is given to be maximal monotone and other related properties are discussed. The most results obtained are based on the notion of “big convexification” of a multifunction graph, the set of probability measures with finite support contained in the graph. Three functionals on the set are introduced such that the relationships between their values give the monotonicity and maximal monotonicity conditions in the book.

MSC:

47H05 Monotone operators and generalizations
47H04 Set-valued operators
49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
46B10 Duality and reflexivity in normed linear and Banach spaces
49J35 Existence of solutions for minimax problems
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
46A20 Duality theory for topological vector spaces
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