عملگرهای یکنوای تعمیم یافته و رویکرد قطبیِ مجموعه های یکنوای تعمیم‌یافته

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.