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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

URL: http://mmr.khu.ac.ir/article-1-3107-en.html

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 .

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.

Type of Study: S |
Subject:
Mat

Received: 2020/01/1 | Revised: 2023/06/18 | Accepted: 2021/03/7 | Published: 2022/12/31 | ePublished: 2022/12/31

Received: 2020/01/1 | Revised: 2023/06/18 | Accepted: 2021/03/7 | Published: 2022/12/31 | ePublished: 2022/12/31

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