Abstract
In this note, using a technique of Verona and Verona, we show that a result announced in “All maximal monotone operators in a Banach space are of type FPV” by A. Eberhard and R. Wenczel, Set-Valued Var. Anal. 22, 597–615, (2014), implies the truth of the Rockafellar conjecture. We then show that there is a gap in the logic of the Eberhard–Wenczel result, which we tried unsuccessfully to close. We also discuss briefly the connection with maximally monotone multifunctions of type (FPV).
Similar content being viewed by others
References
Eberhard, A., Wenczel, R.: All maximal monotone operators in a Banach space are of type FPV. Set-Valued Var. Anal. 22, 597–615 (2014)
Fitzpatrick, S.P., Phelps, R.R.: Bounded approximants to monotone operators on Banach spaces. Ann. Inst. Henri Poincaré, Analyse Non Linéaire 9, 573–595 (1992)
Fitzpatrick, S.P., Phelps, R.R.: Some properties of maximal monotone operators on nonreflexive Banach spaces. Set–Valued Anal. 3, 51–69 (1995)
Rockafellar, R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Amer. Math. Soc. 149, 75–88 (1970)
Simons, S.: From Hahn–Banach to Monotonicity. Lecture Notes in Mathematics, 2nd edn, vol. 1693. Springer (2008)
Verona, A., Verona, M.E.: Remarks on subgradients and ε-subgradients. Set–Valued Anal. 1, 261–272 (1993)
Verona, A., Verona, M.E.: Regular maximal monotone operators. Set-Valued Anal. 6, 303–312 (1998)
Verona, A., Verona, M.E.: Rockafellar’s sum theorem. arXiv:1506.04207
Author information
Authors and Affiliations
Corresponding author
Additional information
The Editor in Chief thanks the authors of the note for pointing out the problem in the publication by Eberhard and Wenczel.
This comment refers to the article available at: http://dx.doi.org/10.1007/s11228-014-0275-6.
Rights and permissions
About this article
Cite this article
Boţ, R.I., Bueno, O. & Simons, S. The Rockafellar Conjecture and type (FPV). Set-Valued Var. Anal 24, 381–385 (2016). https://doi.org/10.1007/s11228-016-0384-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-016-0384-5