Stephen Simons

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…

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…

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…

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…

The creation of machine learning algorithms for intelligent agents capable of continuous, lifelong learning is a critical objective for algorithms being deployed on real-life systems in dynamic environments. Here we present an algorithm…

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…

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…

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…

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. We investigate the Fitzpatrick extension of…

In this paper, we give some extensions of K\"onig'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…

This paper is about the technique of {\em 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 this paper, we first investigate an abstract subdifferential for which (using Ekeland's variational principle) we can prove an analog of the Br{\o}ndsted-Rockafellar property. We introduce the "$r_L$-density" of a subset of the product…

We discuss the asymmetric sandwich theorem, a generalization of the Hahn-Banach theorem. As applications, we derive various results on the existence of linear functionals that include bivariate, trivariate and quadrivariate generalizations…

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…

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…

In this paper, we give conditions under which we can compute the conjugate of a convex function on the product of two Frechet spaces defined in terms of another convex function on the product of two (possibly different) Frechet spaces. We…

In this paper, we develop some of the theory of SSD spaces and SSDB spaces, and deduce some results on maximally monotone multifunctions on a reflexive Banach space.

In this paper, we unify the theory of SSD spaces and the theory of strongly representable sets, and we apply our results to the theory of the various classes of maximally monotone sets. In particular, we prove that type (ED), dense type,…

In this paper, we show how the Brezis-Browder theorem for maximally monotone multifunctions with a linear graph on a reflexive Banach space, and a consequence of it due to Yao, can be generalized to SSDB spaces.

In this paper, we unify the theory of SSD spaces, part of the theory of strongly representable multifunctions, and the theory of the equivalence of various classes of maximally monotone multifunctions.