A Maximal Monotone Operator of Type (D) for which Maximal Monotone Extension to the Bidual is Not of Type (D)

Orestes Bueno; Benar Fux Svaiter

doi:10.68381/jca19017

Abstract

We define a family of linear type (D) operators for which the inverse of their maximal monotone extensions to the bidual are not of type (D) and provide an example of an operator in this family.

Author information

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

Orestes Bueno

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil

obueno@impa.br

Benar Fux Svaiter

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, RJ 22460-320, Brazil

benar@impa.br

Keywords

Maximal monotone operator
type (D)
Banach space
extension
bidual

References

J.-P. Gossez: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs, J. Math. Anal. Appl. 34 (1971) 371–395.https://doi.org/10.1016/0022-247X(71)90119-3 Google Scholar
J.-P. Gossez: On the range of a coercive maximal monotone operator in a nonreflexive Banach space, Proc. Amer. Math. Soc. 35 (1972) 88–92.https://doi.org/10.1090/S0002-9939-1972-0298492-7 Google Scholar
J.-P. Gossez: On a convexity property of the range of a maximal monotone operator, Proc. Amer. Math. Soc. 55(2) (1976) 359–360.https://doi.org/10.1090/S0002-9939-1976-0397485-2 Google Scholar
J.-P. Gossez: On the extensions to the bidual of a maximal monotone operator, Proc. Amer. Math. Soc. 62(1) (1977) 67–71.https://doi.org/10.1090/S0002-9939-1977-0428121-5 Google Scholar
R. R. Phelps, S. Simons: Unbounded linear monotone operators on nonreflexive Banach spaces, J. Convex Analysis 5(2) (1998) 303–328.https://doi.org/10.68381/jca05-21 Google Scholar
B. F. Svaiter, M. Marques Alves: On Gossez type (D) maximal monotone operators, J. Convex Analysis 17(3&4) (2010) 1077–1088.https://doi.org/10.68381/jca17066 Google Scholar

Suggested citation

O. Bueno, B. F. Svaiter. “A Maximal Monotone Operator of Type (D) for which Maximal Monotone Extension to the Bidual is Not of Type (D).” Journal of Convex Analysis 19 (2012), No. 1, 295–300. https://doi.org/10.68381/jca19017

Published by Heldermann Verlag, 2012. Rights now held by Banach Press LLC.

[1] J.-P. Gossez: Opérateurs monotones non linéaires dans les espaces de Banach non réflexifs, J. Math. Anal. Appl. 34 (1971) 371–395.https://doi.org/10.1016/0022-247X(71)90119-3 Google Scholar

[2] J.-P. Gossez: On the range of a coercive maximal monotone operator in a nonreflexive Banach space, Proc. Amer. Math. Soc. 35 (1972) 88–92.https://doi.org/10.1090/S0002-9939-1972-0298492-7 Google Scholar

[3] J.-P. Gossez: On a convexity property of the range of a maximal monotone operator, Proc. Amer. Math. Soc. 55(2) (1976) 359–360.https://doi.org/10.1090/S0002-9939-1976-0397485-2 Google Scholar

[4] J.-P. Gossez: On the extensions to the bidual of a maximal monotone operator, Proc. Amer. Math. Soc. 62(1) (1977) 67–71.https://doi.org/10.1090/S0002-9939-1977-0428121-5 Google Scholar

[5] R. R. Phelps, S. Simons: Unbounded linear monotone operators on nonreflexive Banach spaces, J. Convex Analysis 5(2) (1998) 303–328.https://doi.org/10.68381/jca05-21 Google Scholar

[6] B. F. Svaiter, M. Marques Alves: On Gossez type (D) maximal monotone operators, J. Convex Analysis 17(3&4) (2010) 1077–1088.https://doi.org/10.68381/jca17066 Google Scholar