Maximality is Nothing But Continuity

Jean-Pierre Crouzeix; Eladio Ocaña Anaya

doi:10.68381/jca17035

Abstract

We show the connexions between the maximality of a monotone set-valued mapping and some continuity conditions on the domain of the mapping and on the mapping itself. We provide an original and simple proof of the famous characterization of maximal monotone mapping of Minty. Some words are said on monotone variational and generalized equilibrium problems.

Author information

Contact details are reproduced from the original publication and may be historical.

Jean-Pierre Crouzeix

LIMOS, Campus Scientifique des Cézeaux, Université Blaise Pascal, 63170 Aubière, France

jp.crouzeix@isima.fr

Eladio Ocaña Anaya

IMCA, Universidad Nacional de Ingeniería, Calle Los Biólogos 245, Urb. San Cesar, La Molina – Lima 12, Perú

eocana@imca.edu.pe

Keywords

Monotonicity
maximal monotonicity
variational inequality problems
generalized equilibrium problems

References

J.-P. Crouzeix, E. Ocaña Anaya, W. Sosa Sandoval: A construction of a maximal monotone extension of a monotone map, ESAIM, Proc. 20 (2007) 93–104.https://doi.org/10.1051/proc:072009 Google Scholar
C. J. Himmelberg: Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972) 205–207.https://doi.org/10.1016/0022-247X(72)90128-X Google Scholar
G. J. Minty: Monotone (nonlinear) operators in Hilbert Space, Duke Math. J. 29 (1962) 341–346.https://doi.org/10.1215/S0012-7094-62-02933-2 Google Scholar
R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173 Google Scholar
R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar
S. Simons: From Hahn-Banach to Monotonicity, 2nd Ed., Lecture Notes in Mathematics 1693, Springer, Berlin (2008).Google Scholar

Suggested citation

Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.

[1] J.-P. Crouzeix, E. Ocaña Anaya, W. Sosa Sandoval: A construction of a maximal monotone extension of a monotone map, ESAIM, Proc. 20 (2007) 93–104.https://doi.org/10.1051/proc:072009 Google Scholar

[2] C. J. Himmelberg: Fixed points of compact multifunctions, J. Math. Anal. Appl. 38 (1972) 205–207.https://doi.org/10.1016/0022-247X(72)90128-X Google Scholar

[3] G. J. Minty: Monotone (nonlinear) operators in Hilbert Space, Duke Math. J. 29 (1962) 341–346.https://doi.org/10.1215/S0012-7094-62-02933-2 Google Scholar

[4] R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173 Google Scholar

[5] R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3 Google Scholar

[6] S. Simons: From Hahn-Banach to Monotonicity, 2nd Ed., Lecture Notes in Mathematics 1693, Springer, Berlin (2008).Google Scholar