Showing 1 results for Monotone Polar
Dr. Mohammad Alizadeh,
Volume 8, Issue 4 (12-2022)
Abstract
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.
