Volume 8, Issue 4 (Vol. 8,No. 4, 2022)                   mmr 2022, 8(4): 151-163 | Back to browse issues page

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Alizadeh M. Generalized monotone operators and polarity approach to generalized monotone sets. mmr 2022; 8 (4) :151-163
URL: http://mmr.khu.ac.ir/article-1-3107-en.html
Abstract:   (599 Views)
Introduction
Many Suppose that is a Banach Space with topological dual space We will denote by  the duality pairing between X and  . For , we denote by the boundary points of Ω and by  the interior of Ω. Also we will denote by  the real nonnegative numbers. Let be a set-valued map from  to . The domain and graph of  are, respectively, defined by

We recall that a set valued operator is monotone if   for all  and   For two multivalued operators  and we write  ifis an extension of , i.e., . A monotone operator is called maximal monotone if it has no monotone extension other than itself.
In 1988, The Fitzpatrick function of a monotone operator was introduced by Fitzpatrick. The Fitzpatrick function makes a bridge between the results of convex functions and results on maximal monotone operators. For a monotone operator , its Fitzpatrick function is defined by

It is a convex and norm to weak lower semicontinuous and function.
Let be an extended real-valued function. Its effective domain is defined by  The function is called proper if . Let  be a proper function. The subdifferential (in the sense of Convex Analysis) of  at  is defined by

Given a proper function  and a map , then  is called -convex if

For all and for all
Given an operator  and a map . Then is called -monotone if for all  and  we have

Also is called maximal -monotone if it has no -monotone extension other than itself. We recall for a proper function  the -subdifferential of  at  is defined by

and  if .
Main results
The definition we use for the Fitzpatrick function is the same as for monotone operators.
Assume that is a proper -convex function its conjugate is defined by

First we have the following refinement of the Fenchel-Moreau inequality:

where is the indicator function and .
Also we have the following refinement, when is a proper, -convex and lower semicontinuous function and  is a maximal -monotone operator:

Moreover, we approach generalized monotonicity from the point of view of the classical concept of polarity. Besides, we introduce and study the notion of generalized monotone polar of a set A. Moreover, we find some equivalent relations between polarity and maximal generalized monotonicity.