Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on ?² is skew. We show its domain is a proper subset of the domain of its adjoint S^∗, and −S^∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L²[0,1]. We compare the domain of T with the domain of its adjoint T^∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.     

On a Sufficient Condition for Equality of Two Maximal Monotone Operators

We establish minimal conditions under which two maximal monotone operators coincide. Our first result is inspired by an analogous result for subdifferentials of convex functions. In particular, we prove that two maximal monotone operators T,S which share the same convex-like domain D coincide whenever $T(x)\cap S(x)\not=\emptyset $ for every x?∈?D. We extend our result to the setting of enlargements of maximal monotone operators. More precisely, we prove that two operators coincide as long as the enlargements have nonempty intersection at each point of their common domain, assumed to be open. We then use this to obtain new facts for convex functions: we show that the difference of two proper lower semicontinuous and convex functions whose subdifferentials have a common open domain is constant if and only if their ε-subdifferentials intersect at every point of that domain.     

Local boundedness of monotone bifunctions

We consider bifunctions ${F : C\times C\rightarrow \mathbb{R}}$ where C is an arbitrary subset of a Banach space. We show that under weak assumptions, monotone bifunctions are locally bounded in the interior of their domain. As an immediate corollary, we obtain the corresponding property for monotone operators. Also, we show that in contrast to maximal monotone operators, monotone bifunctions (maximal or not maximal) can also be locally bounded at the boundary of their domain; in fact, this is always the case whenever C is a locally polyhedral subset of ${\mathbb{R}^{n}}$ and F(x, ·) is quasiconvex and lower semicontinuous.     

Multiplicative perturbation of nonlinear m-accretive operators

Criteria are obtained for when an accretive product (i.e., composition) BA of nonlinear m-accretive operators A and B in a Banach space X will be itself m-accretive; and, in particular, when a monotone product of two maximal monotone operators in a Hilbert space will be maximal monotone. This extends the theory of multiplicative perturbation of infinitesimal generators of contraction semigroups to the nonlinear case. Also obtained as a biproduct are existence theorems for certain Hammerstein integral equations.     

Transformations of discrete closure systems

Discrete systems such as sets, monoids, groups are familiar categories. The internal structure of the latter two is defined by an algebraic operator. In this paper we concentrate on discrete systems that are characterized by unary operators; these include choice operators σ, encountered in economics and social theory, and closure operators φ, encountered in discrete geometry and data mining. Because, for many arbitrary operators α, it is easy to induce a closure structure on the base set, closure operators play a central role in discrete systems. Our primary interest is in functions f that map power sets 2 ^U into power sets 2 ^U′, which are called transformations. Functions over continuous domains are usually characterized in terms of open sets. When the domains are discrete, closed sets seem more appropriate. In particular, we consider monotone transformations which are “continuous”, or “closed”. These can be used to establish criteria for asserting that “the closure of a transformed image under f is equal to the transformed image of the closure”. Finally, we show that the categories MCont and MClo of closure systems with morphisms given by the monotone continuous transformations and monotone closed transformations respectively have concrete direct products. And the supercategory Clo of MClo whose morphisms are just the closed transformations is shown to be cartesian closed.     

Remarks on p-cyclically monotone operators

ABSTRACT

In this paper, we deal with three aspects of p-cyclically monotone operators. First, we introduce a notion of monotone polar adapted for p-cyclically monotone operators and study these kinds of operators with a unique maximal extension (called pre-maximal), and with a convex graph. We then deal with linear operators and provide characterizations of p-cyclical monotonicity and maximal p-cyclical monotonicity. Finally, we show that the Brézis-Browder theorem preserves p-cyclical monotonicity in reflexive Banach spaces.     

Preserving maximal monotonicity with applications in sum and composition rules

In this paper, we study maximal monotonicity preserving mappings on the Banach space X × X ^*. Indeed, for a maximal monotone set ${M \subset X\times X^*}$ and for a multifunction ${T: X \times X^* \multimap Y \times Y^*}$ , under some sufficient conditions on M and T we show that T(M) is maximal monotone. As two consequences of this result we get sum and composition rules for maximal monotone operators.     

The Rockafellar Conjecture and type (FPV)

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).     

An alternative approach to well-posedness of a class of differential inclusions in Hilbert spaces

We show the well-posedness of initial value problems for differential inclusions of a certain type using abstract perturbation results for maximal monotone operators in Hilbert spaces. For this purpose the time derivative is established in an exponentially weighted _L2$L_2$ space. The problem of well-posedness then reduces to show that the sum of two maximal monotone operators in time and space is again maximal monotone. The theory is exemplified by three inclusions describing phenomena in mathematical physics involving hysteresis.     

Perturbations of maximal monotone random operators

Let X be a Banach space, X¹ its dual, and Ω a measurable space. We study the solvability of nonlinear random equations involving operators of the form L + T, where L is a maximal monotone random operator from Ω × X into X¹ and T : Ω × X → X¹ a random operator of monotone type.     

An extension of the Córdoba–Fefferman theorem on the equivalence between the boundedness of certain classes of maximal and multiplier operators

The Córdoba–Fefferman theorem involving the equivalence between boundedness properties of certain classes of maximal and multiplier operators is extended utilizing the recent work of Bateman on directional maximal operators as well as the work of Hagelstein and Stokolos on geometric maximal operators associated to homothecy invariant bases of convex sets satisfying Tauberian conditions. To cite this article: P. Hagelstein, A. Stokolos, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On the convergence of maximal monotone operators

We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch-Brezis type.

     

A monotone iterative technique for stationary and time dependent problems in Banach spaces

An abstract monotone iterative method is developed for operators between partially ordered Banach spaces for the nonlinear problem Lu=Nu and the nonlinear time dependent problem u^′=(L+N)u. Under appropriate assumptions on L and N we obtain maximal and minimal solutions as limits of monotone sequences of solutions of linear problems. The results are illustrated by means of concrete examples.     

Sur l'interpolation des opérateurs maximaux monotones

On the Interpolation of Maximal Monotone Operators. We study here one way to extend to the maximal monotone case the results of linear interpolation, exposed bybalakrishnan in [2]. We obtain a necessary and sufficient condition of convergence for sequences(A ⁿ ) _n of maximal monotone operators on a real Hilbert spaceH.     

Regularity and the Br?ndsted–Rockafellar Properties of Maximal Monotone Operators

The purpose of this paper is to establish connections between the class of maximal monotone operators of Br?ndsted–Rockafellar type and that of regular maximal monotone operators. ^†Partially supported by a WISE grant.     

Maximality of the Sum of a Maximally Monotone Linear Relation and a Maximally Monotone Operator

The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar’s constraint qualification holds. In this paper, we prove the maximal monotonicity of A?+?B provided that A, B are maximally monotone and A is a linear relation, as soon as Rockafellar’s constraint qualification holds: ${\operatorname{dom}}\,A\cap{\operatorname{int}}\,{\operatorname{dom}}\,B\neq\varnothing$ . Moreover, A?+?B is of type (FPV).     

Notes on the difference of two monotone operators

In this note, we present some results on maximality of the difference of two monotone operators. It is shown that, for two multifunctions S and T from X to X ^*, maximal monotonicity of S and monotonicity of both T and S?T imply maximal monotonicity of S?T.     

Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces

In a Hilbert space setting we introduce dynamical systems, which are linked to Newton and Levenberg–Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M=A+B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy–Lipschitz theorem, and involve separately B and the resolvents of A. In the convex subdifferential case, by using Lyapunov asymptotic analysis, we prove a descent minimizing property and weak convergence to equilibria of the trajectories. Time discretization of these dynamics gives algorithms combining Newton’s method and forward-backward methods for solving structured monotone inclusions.     

Fixed points for mixed monotone multivalued operators in Banach spaces with applications

In this paper, the existence and iterative approximation of fixed points for a class of systems of mixed monotone multivalued operator are discussed. We present some new fixed point theorems of mixed monotone operators and increasing operators which need not be continuous or satisfy a compactness condition. We also give some applications to differential inclusions with discontinuous right hand side in Banach spaces and to Hammerstein integral inclusions on RN.     

An explicit example of a maximal 3-cyclically monotone operator with bizarre properties

Subdifferential operators of proper convex lower semicontinuous functions and, more generally, maximal monotone operators are ubiquitous in optimization and nonsmooth analysis. In between these two classes of operators are the maximal n$n$-cyclically monotone operators. These operators were carefully studied by Asplund, who obtained a complete characterization within the class of positive semidefinite (not necessarily symmetric) matrices, and by Voisei, who presented extension theorems à la Minty.