Journal of Convex Analysis 31 (2024), No. 3, 853--866
Copyright Heldermann Verlag 2024
Some Recent Results on Premonotone Operators
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, Fundacao Getulio Vargas, Rio de Janeiro, Brazil
alfredo.iusem@fgv.br
Wilfredo Sosa Sandoval
Graduate Program of Economics, Univ. Católica de Brasilia, Brasilia, Brazil
sosa@ucb.br
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.
Keywords: Premonotone operators, polarity.
MSC: 90C47, 49J35.
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