From Hahn-Banach to Monotonicity

Stephen Simons (auth.)1402069189, 9781402069185, 9781402069192

In this new edition of LNM 1693 the essential idea is to reduce questions on monotone multifunctions to questions on convex functions. However, rather than using a “big convexification” of the graph of the multifunction and the “minimax technique”for proving the existence of linear functionals satisfying certain conditions, the Fitzpatrick function is used. The journey begins with a generalization of the Hahn-Banach theorem uniting classical functional analysis, minimax theory, Lagrange multiplier theory and convex analysis and culminates in a survey of current results on monotone multifunctions on a Banach space.

The first two chapters are aimed at students interested in the development of the basic theorems of functional analysis, which leads painlessly to the theory of minimax theorems, convex Lagrange multiplier theory and convex analysis. The remaining five chapters are useful for those who wish to learn about the current research on monotone multifunctions on (possibly non reflexive) Banach space.

Table of contents :
Front Matter….Pages I-XIV
Introduction….Pages 1-13
The Hahn-Banach-Lagrange theorem and some consequences….Pages 15-39
Fenchel duality….Pages 41-69
Multifunctions, SSD spaces, monotonicity and Fitzpatrick functions….Pages 71-105
Monotone multifunctions on general Banach spaces….Pages 107-115
Monotone multifunctions on reflexive Banach spaces….Pages 117-138
Special maximally monotone multifunctions….Pages 139-195
The sum problem for general Banach spaces….Pages 197-201
Open problems….Pages 203-204
Glossary of classes of multifunctions….Pages 205-206
A selection of results….Pages 207-231
Back Matter….Pages 233-248