From Hahn-Banach to MonotonicityIn 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. |
Contents
Introduction | 1 |
2 | 7 |
Open problems 203 | 8 |
The Htheorem | 14 |
Multifunctions monotonicity and maximality | 17 |
Fitzpatrick functions and fitzpatrifications | 23 |
Monotone multifunctions on reflexive Banach spaces | 29 |
Special maximally monotone multifunctions | 35 |
Maximally monotone multifunctions with convex graph | 180 |
Subtler properties of subdifferentials | 188 |
The sum problem for general Banach spaces | 197 |
Glossary of classes of multifunctions | 205 |
Voiseis theorem | 231 |
References | 233 |
58 | 235 |
| 241 | |