This paper is about the maximally monotone and quasidense subsets of the product of a real Banach space and its dual. We discuss six subclasses of the maximal monotone sets that are equivalent to the quasidense ones. We define the Gossez extension to the dual of a maximally monotone set, and give nine equivalent characterizations of an element of this set in the quasidense case. We discuss maximally monotone sets of “type (NI)” (one of the six classes referred to above) and we show that the “tail operator” is not of type (NI), but it is the Gossez extension of a maximally monotone set that is of type (NI). We generalize Rockafellar's surjectivity theorem for maximally monotone subsets of reflexive Banach spaces to maximally monotone subsets of type (NI) of general Banach spaces. We discuss a generalization of the Brezis-Browder theorem on monotone linear subspaces of reflexive spaces to the nonreflexive situation. We also discuss briefly maximally monotone subsets of “type (D)” and “type (WD)” (two more of the six classes referred to above).

We introduce the concept of the touching of two multifunctions on a real Hilbert space, and deduce that certain multifunctions on the space have a unique fixed point. These result are applied to the theory of genaralized cycles and generalized gap vectors for the composition of the projections onto a finite number of closed convex space in a real Hilbert space.

In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space.

In this paper, we consider the "quasidensity" of a subset of the product of a Banach space and its dual, and give a connection between quasidense sets and sets of "type (NI)". We discuss "coincidence sets" of certain convex functions and prove two sum theorems for coincidence sets. We obtain new results on the Fitzpatrick extension of a closed quasidense monotone multifunction. The analysis in this paper is self-contained, and independent of previous work on "Banach SN spaces".

In three previous papers, we discussed quasidense multifunctions from a Banach space into its dual, or, equivalently, quasidense subsets of the product of a Banach space and its dual. In this paper, we survey (without proofs) some of the main results about quasidensity, and give some simple limiting examples in Hilbert spaces, reflexive Banach spaces, and nonreflexive Banach spaces.

In this note, we give a generalization of Gossez's example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez's original papers. We also discuss some new properties of Gossez's skew linear operator and its adjoint. While most of this paper uses elementary functional analysis, we correlate our results with those obtained by using the Stone-Cech compactification of the integers.

In this note, we give a generalization of Gossez’s example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez’s original papers. We also discuss some new properties of Gossez’s skew linear operator and its adjoint. 2010 Mathematics Subject Classification: Primary 47H05; Secondary 47N10, 46A20, 46A22.

In this paper, we give some extensions of König's extension of the Mazur-Orlicz theorem. These extensions include generalizations of a surprising recent result of Sun Chuanfeng, and generalizations to the product of more than two spaces of the "Hahn-Banach-Lagrange" theorem.

In this paper, we discuss quasidense multifunctions from a Banach space into its dual, and use the two sum theorems proved in a previous paper to give various characterizations of quasidensity, including two fuzzy ones. We investigate the Fitzpatrick extension of such a multifunction. We prove that a closed monotone quasidense multifunction is maximally monotone locally (that is to say, of “type (FPV)”), and strongly maximal. We also prove that a maximally monotone multifunction is quasidense if, and only if, it is locally maximally monotone (that is to say, of “type (FP)”).

In this note, we give an extension of Gossezu0027s example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossezu0027s original papers.

We discuss "Banach SN spaces", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce "L-positive" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce the concepts of "$r_L$-density" and its specialization "quasidensity": the closed quasidense monotone multifunctions from a Banach space into its dual form a (generally) strict subset of the maximally monotone ones, though all surjective maximally monotone and all maximally monotone multifunctions on a reflexive space are quasidense. We give a sum theorem and a parallel sum theorem for closed monotone quasidense multifunctions under very general constraint conditions. That is to say, quasidensity obeys a very nice calculus rule. We give a short proof that the subdifferential of a proper convex lower semicontinuous function on a Banach space is quasidense, and deduce generalizations of the Brezis-Browder theorem on linear relations to non reflexive Banach spaces. We also prove that any closed monotone quasidense multifunction has a number of other very desirable properties. This version differs from the previous version in the removal of Sections 13-16, part of Section 17, and Section 18.

In this note, using a technique of Verona and Verona, we show that a result announced in “All maximal monotone operators in a Banach space are of type FPV” by A. Eberhard and R. Wenczel, Set-Valued Var. Anal. 22, 597–615, (2014), implies the truth of the Rockafellar conjecture. We then show that there is a gap in the logic of the Eberhard–Wenczel result, which we tried unsuccessfully to close. We also discuss briefly the connection with maximally monotone multifunctions of type (FPV).

In this paper, we first investigate an abstract subdifferential for which (using Ekeland’s variational principle) we can prove an analog of the Brøndsted–Rockafellar property. We introduce the “ r L –density” of a subset of the product of a Banach space with its dual. A closed r L –dense monotone set is maximally monotone, but we will also consider the case of nonmonotone closed r L –dense sets. As a special case of our results, we can prove Rockafellar’s result that the subdifferential of a proper convex lower semicontinuous function is maximally monotone.

This paper is about the technique of shadow variables that was used in the theory of monotone operators. In this paper, we use it to show that certain results that were originally proved for lower semicontinuous convex functions are in fact true for marginable convex functions.

This paper is about certain linear subspaces of Banach SN spaces (that is to say Banach spaces which have a symmetric nonexpansive linear map into their dual spaces). We apply our results to monotone linear subspaces of the product of a Banach space and its dual. In this paper, we establish several new results and also give improved proofs of some known ones in both the general and the special contexts.

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q -positive, where q is the quadratic form induced by the original bilinear form. The notion of q -positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q -positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q -positive extensions of a q -positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q -positivity. The paper finishes with two new examples.

In this paper, we define a Banach SNL space to be a Banach space with a certain kind of linear map from it into its dual, and we develop the theory of linear L-positive subsets of Banach SNL spaces with Banach SNL dual spaces. We use this theory to give simplified proofs of some recent results of Bauschke, Borwein, Wang and Yao, and also of the classical Brezis-Browder theorem.