Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-17T12:15:40.589Z Has data issue: false hasContentIssue false
  • English
  • Français

Stronger maximal monotonicity properties of linear operators

Published online by Cambridge University Press:  17 April 2009

H.H. Bauschke  and
S. Simons
H.H. Bauschke
Affiliation:
Department of Mathematics and StatisticsOkanagan University College3333 College WayKelowna, BC V1V 1V7Canada e-mail: bauschke@cecm.sfu.ca
S. Simons
Affiliation:
Department of MathematicsUniversity of CaliforniaSanta Barbara CA 93106–3080United States of America e-mail: simons@math.ucsb.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The subdifferential mapping associated with a proper, convex lower semicontinuous function on a real Banach space is always a special kind of maximal monotone operator. Specifically, it is always “strongly maximal monotone” and of “type (ANA)”. In an attempt to find maximal monotone operators that do not satisfy these properties, we investigate (possibly discontinuous) maximal monotone linear operators from a subspace of a (possibly nonreflexive) real Banach space into its dual. Such a linear mapping is always “strongly maximal monotone”, but we are only able to prove that is of “type (ANA)” when it is continuous or surjective — the situation in general is unclear. In fact, every surjective linear maximal monotone operator is of “type (NA)”, a more restrictive condition than “type (ANA)”, while the zero operator, which is both continuous and linear and also a subdifferential, is never of “type (NA)” if the underlying space is not reflexive. We examine some examples based on the properties of derivatives.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 60 , Issue 1 , August 1999 , pp. 163 - 174
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Fan, K., ‘Minimax theorems’, Proc. Nat. Acad. Sci. U.S.A. 39 (1953), 42–47.CrossRefGoogle ScholarPubMed
[2]Bauschke, H.H., Projection algorithms and monotone operators, PhD thesis (Simon Eraser University, Burnaby BC, Canada, August 1996). This is preprint 96:080 at http://www.cecm.sfu.ca/preprints/1996pp.html.Google Scholar
[3]König, H., ‘Über das Von Neumannsche Minimax-theorem’, Arch. Math. 19 (1968), 482–487.CrossRefGoogle Scholar
[4]Phelps, R.R., Convex functions, monotone pperators and differentiability, Lecture Notes in Mathematics 1364 (Springer-Verlag, Berlin, Heidelberg, New York, 1993).Google Scholar
[5]Phelps, R.R. and Simons, S., ‘Unbounded linear monotone operators on nonreflexive Banach spaces’, J. Convex Anal. 5 (1998), 303–328.Google Scholar
[6]Rockafellar, R.T., ‘On the maximal monotonicity of subdifferential mappings’, Pacific J. Math. 33 (1970), 209–216.CrossRefGoogle Scholar
[7]Saab, E. and Saab, P., ‘On stability problems of some properties in Banach spaces’, in Function spaces, (Jarosz, K., Editor), Lecture Notes in Pure and Applied Mathematics 136 (Marcel Dekker, New York, 1992), pp. 367–394.Google Scholar
[8]Simons, S., ‘Critères de faible compacité en termes du théorème de minimax’, (Seminaire Choquet 1970/1971, no.24, 5 pages).Google Scholar
[9]Simons, S., ‘Subtangents with controlled slope’, Nonlin. Anal. 22 (1994), 1373–1389.CrossRefGoogle Scholar
[10]Simons, S., ‘Subdifferentials of convex functions’, in Recent Developments in Optimization Theory and Nonlinear Analysis, (Censor, Y. and Reich, S., Editors), Contemporary Mathematics 204 (American Mathematical Society, Providence, R.I., 1997), pp. 217–246.CrossRefGoogle Scholar
[11]Simons, S., Minimax and monotonicity, Lecture Notes in Mathematics 1693 (Springer-Verlag, Berlin, Heidelbrg, New York, 1998).CrossRefGoogle Scholar
[12]Simons, S., ‘Techniques for maximal monotonicity’, in Calculus of Variations and Related Topics, (Ioffe, A., Reich, S. and Shafrir, I., Editors), Pitman Research Notes in Mathematics Series (Addison Wesley Longman, Harlow, Essex).Google Scholar
[13]Wilansky, A., Modern methods in topological vector spaces (McGraw-Hill, New York, 1978).Google Scholar