Maximal Monotone Multifunctions Research Articles

New Variational Principles for Solving Extended Dirichlet-Neumann Problems

We extend in this paper the classical variational methods devoted to solve the Dirichlet-Neumann problems. We assume that the intensive and extensive parameters are related by a maximal monotone multifunction. The Fitzpatrick’s method allows us to elaborate new variational principles.

Sum Formula for Maximal Abstract Monotonicity and Abstract Rockafellar’s Surjectivity Theorem

In this paper, we present an example in which the sum of two maximal abstract monotone operators is maximal. Also, we shall show that the necessary condition for Rockafellar's surjectivity which was obtained in ((19), Theorem 4.3) can be sufficient. Abstract convexity has found many applications in the study of problems of mathematical analysis and optimization. Abstract convexity has mainly been used for the study of point-to-point functions. Examples of its use in the analysis of multifunctions can be found in (3, 13, 14, 24). Several approaches to the theory of monotone multifunctions have established links between maximal monotone multifunctions and convex functions (see

Modelling of implicit standard materials. Application to linear coaxial non-associated constitutive laws

We analyze the relation between Géry de Saxcé's bipotentials representing non-associated constitutive laws and Fitzpatrick's functions representing maximal monotone multifunctions. Revisiting the model of elastic materials initiated by Robert Hooke, we illustrate that Fitzpatrick's representation of monotone operators coming from convex analysis provides a constructive method to discover the best bipotential for modelling an Implicit Standard Material.

The Convexity of the Closure of the Domain and the Range of a Maximal Monotone Multifunction of Type NI

It is shown that the closure of the domain and the closure of the range of a maximal monotone set of type NI are convex sets. This is a positive answer to the question concerning the convexity of the closure of the range of a maximal monotone multifunction, posed in Simons (Minimax and Monotonicity, Lecture Notes in Mathematics 1693. Springer-Verlag, Berlin, 1998, X Open Problems, Problem 27.7).

Dualized and scaled Fitzpatrick functions

In this paper, we obtain an explicit formula for the interior of the domain of a maximal monotone multifunction in terms of its Fitzpatrick function.

A new version of the Hahn-Banach theorem

We discuss a new version of the Hahn-Banach theorem, with applications to linear and nonlinear functional analysis, convex analysis, and the theory of monotone multifunctions. We show how our result can be used to prove a “localized” version of the Fenchel-Moreau formula - even when the classical Fenchel-Moreau formula is valid, the proof of it given here avoids the problem of the “vertical hyperplane”. We give a short proof of Rockafellar’s fundamental result on dual problems and Lagrangians - obtaining a necessary and sufficient condition instead of the more usual sufficient condition. We show how our result leads to a proof of the (well-known) result that if a monotone multifunction on a normed space has bounded range then it has full domain. We also show how our result leads to generalizations of an existence theorem with no a priori scalar bound that has proved very useful in the investigation of monotone multifunctions, and show how the estimates obtained can be applied to Rockafellar’s surjectivity theorem for maximal monotone multifunctions in reflexive Banach spaces. Finally, we show how our result leads easily to a result on convex functions that can be used to establish a minimax theorem.

Five Kinds of Maximal Monotonicity

We discuss Gossez's ‘type (D)’ maximal monotone multifunctions and the newer ‘type (ED)’ subfamily (for which an analog of the Brondsted-Rockafellar property holds). We then discuss the ‘locally maximal monotone’ (= type (FP)) and ‘maximal monotone locally’ (= type (FPV)) multifunctions of Fitzpatrick-Phelps and Fitzpatrick-Phelps-Verona-Verona. Finally, we discuss the strongly maximal monotone multifunctions. We prove that every maximal monotone multifunction of type (D) is locally maximal monotone, and every maximal monotone multifunction of type (ED) is both maximal monotone locally and strongly maximal monotone.

Solutions to Affine Generalized Equations Using Proximal Mappings

The normal map has proven to be a powerful tool for solving generalized equations of the form: find z ∈ C, with 0 ∈ F(z) + N c (z), where C is a convex set and N c (z) is the normal cone to C at z. In this paper, we use the T-map, a generalization of the normal map, to solve equations of the more general form: find z ∈ dom(T), with 0 ∈ F(z) + T(z), where T is a maximal monotone multifunction. We present a path-following algorithm that determines zeros of coherently oriented piecewise-affine functions, and we use this algorithm, together with the T-map, to solve the generalized equation for affine, coherently oriented functions F, and polyhedral multifunctions T. The path-following algorithm we develop here extends the piecewise-linear homotopy framework of Eaves to the case where a representation of a subdivided manifold is unknown.

Maximal Monotone Multifunctions of Brøndsted–Rockafellar Type

We consider whether the “inequality-splitting” property established in the Brondsted–Rockafellar theorem for the subdifferential of a proper convex lower semicontinuous function on a Banach space has an analog for arbitrary maximal monotone multifunctions. We introduce the maximal monotone multifunctions of type (ED), for which an “inequality-splitting” property does hold. These multifunctions form a subclass of Gossez"s maximal monotone multifunctions of type (D); however, in every case where it has been proved that a multifunction is maximal monotone of type (D) then it is also of type (ED). Specifically, the following maximal monotone multifunctions are of type (ED): • ultramaximal monotone multifunctions, which occur in the study of certain nonlinear elliptic functional equations; • single-valued linear operators that are maximal monotone of type (D); • subdifferentials of proper convex lower semicontinuous functions; • “subdifferentials” of certain saddle-functions. We discuss the negative alignment set of a maximal monotone multifunction of type (ED) with respect to a point not in its graph – a mysterious continuous curve without end-points lying in the interior of the first quadrant of the plane. We deduce new inequality-splitting properties of subdifferentials, almost giving a substantial generalization of the original Brondsted–Rockafellar theorem. We develop some mathematical infrastructure, some specific to multifunctions, some with possible applications to other areas of nonlinear analysis: • the formula for the biconjugate of the pointwise maximum of a finite set of convex functions – in a situation where the “obvious” formula for the conjugate fails; • a new topology on the bidual of a Banach space – in some respects, quite well behaved, but in other respects, quite pathological; • an existence theorem for bounded linear functionals – unusual in that it does not assume the existence of any a priori bound; • the 'big convexification" of a multifunction.

Applications of the method of partial inverses to convex programming: Decomposition

A primal–dual decomposition method is presented to solve the separable convex programming problem. Convergence to a solution and Lagrange multiplier vector occurs from an arbitrary starting point. The method is equivalent to the proximal point algorithm applied to a certain maximal monotone multifunction. In the nonseparable case, it specializes to a known method, the proximal method of multipliers. Conditions are provided which guarantee linear convergence.

Upper semicontinuity of set-valued functions

Various types of upper semcontinuity properties for set-valued functions have been used in the past to obtain closure and lower closure theorems in optimal control theory as well as selection theorems and fixed-point theorems in topology. This paper unifies these various concepts by using semiclosure operators, extended topologies, and lattice theoretic operations and obtains general closure theorems. In addition, analytic criteria are given for this generalized upper semicontinuity. In particular, set-valued functions which are maximal in terms of certain properties (e.g., maximal monotone multifunctions) are shown to be necessarily upper semicontinuous.