Abstract

The notion of premonotone operator refers to a class of operators more general than monotone ones, which still enjoy a surjectivity property akin to Minty's Theorem for monotone operators. In this paper we prove that operators which are monotone outside a bounded set, are indeed premonotone. As a consequence, we show that premonotocity is preserved under rather arbitrary alterations of the operator values in a bounded subset of its domain. We also develop the basic elements of a polarity theory related to premonotonicity and we deepen the study of maximal premonotone operators. Finally, we prove the one-dimensional version of a previously stated conjecture, namely that any maximal premonotone operator contains a maximal monotone one.

Author information

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

Mohammad Hossein Alizadeh

Dept. of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran

m.alizadeh@iasbs.ac.ir

Alfredo N. Iusem

Escola de Matemática Aplicada, Fundação Getúlio Vargas, Rio de Janeiro, Brazil

alfredo.iusem@fgv.br

Wilfredo Sosa Sandoval

Graduate Program of Economics, Univ. Católica de Brasília, Brasília, Brazil

sosa@ucb.br

Keywords

  • Premonotone operators
  • polarity

Mathematics Subject Classification

90C47, 49J35

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Suggested citation

M. H. Alizadeh, A. N. Iusem, W. Sosa Sandoval. “Some Recent Results on Premonotone Operators.” Journal of Convex Analysis 31 (2024), No. 3, 853–866. https://doi.org/10.68381/jca31042

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Published by Heldermann Verlag, 2024. Rights now held by Banach Press LLC.