Maicon Marques Alves

Gossez type (D) operators are defined in non-reflexive Banach spaces and share with the subdifferential a topological related property, characterized by bounded nets. In this work we present new properties and characterizations of these operators. The class (NI) was defined after Gossez defined the class (D) and seemed to generalize the class (D). One of our main results is the proof that these classes, type (D) and (NI), are identical.

We study maximal monotonicity of the sum of maximal monotone operators in general Banach spaces. Under certain qualification condition (QC) on the domain of Fitzpatrick convex representations of the operators the sum is maximal monotone, even in nonreflexive Banach spaces.

Conjugation, or Legendre transformation, is a basic tool in convex analysis, rational mechanics, economics and optimization. It maps a function on a linear topological space into another one, defined in the dual of the linear space by coupling these spaces by means of the duality product. Generalized conjugation extends classical conjugation to any pair of domains, using an arbitrary coupling function between these spaces. This generalization of conjugation is now being widely used in optimal transportation problems, variational analysis and also optimization. If the coupled spaces are equal, generalized conjugations define order reversing maps of a family of functions into itself. In this case, it is natural to ask for the existence of fixed points of the conjugation, that is, functions which are equal to their (generalized) conjugated. Here we prove that any generalized symmetric conjugation has fixed points. The basic tool of the proof is a variational principle involving the order reversing feature of the conjugation. As an application of this abstract result, we will extend to real linear topological spaces a fixed-point theorem for Fitzpatrick's functions, previously proved in Banach spaces.

We are concerned with surjectivity of perturbations of maximal monotone operators in non-reflexive Banach spaces. While in a reflexive setting, a classical surjectivity result due to Rockafellar gives a necessary and sufficient condition to maximal monotonicity, in a non-reflexive space we characterize maximality using an "enlarged" version of the duality mapping, introduced previously by Gossez.

Gossez type (D) operators are defined in non-reflexive Banach spaces and share with the subdifferential operator a topological related property, characterized by bounded nets. In this work we present new properties and characterizations of these operators. The class (NI) was defined after Gossez defined the class (D) and seemed to generalize the class (D). One of our main results is the proof that these classes, type (D) and (NI), are identical.

We study some conditions which guarantee that a convex function represents a maximal monotone operator in non-reflexive Banach spaces.

We study a sufficient condition under which a maximal monotone operator T\colon X\rightrightarrows X^* T  ⁣ : X ⇉ X ∗ admits a unique maximal monotone extension to the bidual \widetilde T\colon X^{**}\rightrightarrows X^* T ~  ⁣ : X ∗ ∗ ⇉ X ∗ . We will prove that for non-linear operators this condition is equivalent to uniqueness of the extension. The central tool in our approach is the \mathcal{S} S -function defined and studied previously by R. S. Burachik and B. F. Svaiter ["Maximal monotone operators, convex functions and a special family of enlargements", Set-Valued Analysis 10 (2002) 297–316]. For a generic operator, this function is the supremum of all convex lower semicontinuous functions which are majorized by the duality product in the graph of the operator. We also prove in this work that if the graph of a maximal monotone operator is convex, then this graph is an affine linear subspace.

In a recent paper ["Brøndsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces", J. Convex Analysis 15(4) (2008) 693–706] the authors studied a class of maximal monotone operators, characterized by the existence of a function in Fitzpatrick's family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a strict version of the Brøndsted-Rockafellar property. In this work we will prove that if a single Fitzpatrick function of a maximal monotone operator has a conjugate above the duality product, then all Fitzpatrick function of the operator have a conjugate above the duality product. As a consequence, the family of maximal monotone operators with this property is just the class NI, previously defined and studied by Simons. We will also prove that an auxiliary condition used by the authors to prove the strict Brøndsted-Rockafellar property is equivalent to the assumption of the conjugate of the Fitzpatrick function to majorize the duality product.