Generalized monotone operators and polarity approach to generalized monotone sets

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 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 ifFor 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.