The relevance of convex analysis for the study of monotonicity

Autoconjugate functions and representations of monotone operators

“…The preceding proof is similar to an argument due to Martinez-Legaz and Svaiter; see also [20,Proposition 3] in which its origins are described. The assumptions on c are satisfied in each of the following examples.…”

“…available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700033748 [6] In the following proposition which casts the preceding statement in a more general 2 z framework, Z is any set and D : R -)• R is a duality satisfying the conditions: The first part of the following proof is a simplified form due to C. Zalinescu of a proof given in a preliminary version of the paper [20]. The second part fills a gap disclosed by B.F. Svaiter while reading a draft of [20].…”

“…In particular, Martinez-Legaz and Thera [13] have characterised the image of the Fitzpatrick representation and described the inverse correspondence; Burachik and Svaiter [2] have given a criterion ensuring that a convex function o n X x X* represents a maximal monotone operator and have introduced a whole class of such functions [3]. Penot [20] has used a special kind of representations to deduce results about operations on maximal monotone operators from classical results of convex analysis. His approach is connected with the following problem: given a monotone operator M : X =t X*, it is possible to get a closed convex function q on X x X* such that q*(x*,x) = q(x,x*) for any (x,x*) € X x X* and f M ^ q ^ p M , where f M is the Fitzpatrick representation of M and p M = f M .…”

Autoconjugate representers for linear monotone operators

“…Before listing some of the key properties of the Fitzpatrick function, we introduce a convenient notation utilized by Penot [29]: If F : X × X * → ]−∞, +∞], set F ⊺ : X * × X : (x * , x) → F (x, x * ),…”

On the convergence of maximal monotone operators

“…Other motivations stem from the analogies between many results concerning the class of maximal monotone operators with corresponding results about closed proper convex functions. Up to now, the representations introduced in [6], [7], [8], [13], [14], [20] have enabled one to devise simple proofs of known results, but they have not been used to establish new results. In the present note we give a general convergence result for sums of maximal monotone operators.…”