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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
IASBS , m.alizadeh@iasbs.ac.ir
Abstract: (1409 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 operatoris 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 alland for all 
Given an operator and a map 
. Then is called -monotone if for all and 
we have
Alsois 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 thatis 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, whenis 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.
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 byWe 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 byIt 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 byGiven 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 haveAlso
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 byand
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.
Keywords: Monotone and sigma-monotone operators, Fitzpatrik function, Fenchel inequality, Monotone polar
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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