Maximal monotone operators and maximal monotone bifunctions

In recent years, the introduction of the so-called Fitzpatrick function led to important advances of the theory of maximal monotone operators in Banach spaces. On a parallel path, it has been shown that some of the results on monotone operators can be obtained more simply and much more generally in the broader framework of monotone bifunctions. The present project aims to combine the two approaches by finding the connections between them, in order to make new advances into the theory of maximal monotone operators. It consists of two parts: In the first part, we will explore the relation between the class of representative functions of a maximal monotone operator T defined on a Banach space X, and the class of Fitzpatrick transforms of all bifunctions representing the operator T. More precisely, it is known that for every maximal monotone operator T one can define the class H(T) of all convex, lower semicontinuous functions on the product XX*, which represent T in the following sense: they are greater or equal to the product , with equality holding exactly on the graph of T. On the other hand, there exists a whole class of bifunctions (i.e., functions defined on XX) such that their subdifferential with respect to the second variable at the point (x,x) is equal to T(x). For every such bifunction F one can define its Fitzpatrick transform and one can show that it is a representative function of T. A question that is central to this part of the project is: Can we recover all representative functions in this way? The question is very important for the following reason: The so-called Fitzpatrick function of the operator T is just one of its representative functions, the best known of them. If one shows that every representative function is the Fitzpatrick transform of a bifunction, then we can see each representative function of the operator in a new perspective, as a kind of Fitzpatrick function corresponding not to the operator, but to something more general than the operator. Around this central question there are others that we will also try to answer: For instance, how one can construct bifunctions corresponding to a given operator, and having some desirable properties? In the second part, we will endeavor to answer some questions regarding the maximal monotonicity property of bifunctions. For example, usually the maximal monotonicity of a bifunction is defined as the maximal monotonicity of the corresponding operator. However, bifunctions are just functions with values in the extended real numbers field, so the question arises: What is the relation between this maximal monotonicity, and the maximality with respect to the usual order relation in IR ? Another question: Since monotone bifunctions are closely related to the equilibrium problem, can we have a criterion for maximal monotonicity through the existence of solutions to such a problem?