Minimax and Monotonicity

Stephen Simons (auth.)3540647554, 9783540647553

Focussing on the theory (both classical and recent) of monotone multifunctions on a (possibly nonreflexive) Banach space, this book looks at the big convexification of a multifunction; convex functions associated with a multifunction; minimax theorems as a tool in functional analysis and convex analysis. It includes new results on the existence of continuous linear functionals; the conjugates, biconjugates and subdifferentials of convex lower semicontinuous functions, Fenchel duality; (possibly unbounded) positive linear operators from a Banach space into its dual; the sum of maximal monotone operators, and a list of open problems. The reader is expected to know basic functional analysis and calculus of variations, including the Bahn-Banach theorem, Banach-Alaoglu theorem, Ekeland’s variational principle.

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Table of contents :
Introduction….Pages 1-11
Functional analytic preliminaries….Pages 13-28
Multifunctions….Pages 29-41
A digression into convex analysis….Pages 43-51
General monotone multifunctions….Pages 53-73
The sum problem for reflexive spaces….Pages 75-95
Special maximal monotone multifunctions….Pages 97-109
Subdifferentials….Pages 111-139
Discontinuous positive linear operators….Pages 141-151
The sum problem for general banach spaces….Pages 153-161
Open problems….Pages 163-164