Applications sous-linéaires à valeurs dans un espace de fonctions continues

Summary We investigate the possibility of representing a convex mapping f with values in some space C(T) of continuous functions on a compact space T, as a supremum of continuous affine mappings, when the domain D of f is some convex subset of a Banach space X. Various limiting examples are presented.     

Codifferential Calculus

In this paper, some exact calculus rules are obtained for calculating the coderivatives of the composition of two multivalued maps. Similar rules are displayed for sums. A crucial role is played by an intermediate set-valued map called the resolvent. We first establish inclusions for contingent, Fréchet and limiting coderivatives. Combining them, we get equality rules. The qualification conditions we present are natural and less exacting than classical conditions.     

Multipliers and Generalized Derivatives of Performance Functions

It is known that multipliers in mathematical programming have to do with the interpretation of generalized derivatives for the performance function of some perturbed problem. In economics, this fact is known under the term of shadow price. Here, we point out a precise relationship with the subdifferential of the performance function in the contingent sense and in the Fréchet sense, which are the simplest notions of nonsmooth analysis.     

A representation of maximal monotone operators by closed convex functions and its impact on calculus rules

We introduce a new representation for maximal monotone operators. We relate it to previous representations given by Krauss, Fitzpatrick and Mart??nez-Legaz and Théra. We show its usefulness for the study of compositions and sums of maximal monotone operators. To cite this article: J.-P. Penot, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

A fixed-point theorem for asymptotically contractive mappings

We present fixed point theorems for a nonexpansive mapping from a closed convex subset of a uniformly convex Banach space into itself under some asymptotic contraction assumptions. They generalize results valid for bounded convex sets or asymptotically compact sets.

     

Tangent Sets of Order One and Two to the Positive Cones of Some Functional Spaces

We compute the tangent cone and the tangent set of order two to the positive cone of some important functional spaces such as the space C(K) of continuous functions on a compact space and the space L_p(Q) of p-integrable functions on a measured space. The results are applied to the calculus of derivatives of order one and two of marginal functions. Accepted 19 April 1996     

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.

     

Persistence and stability of solutions of Hamilton-Jacobi equations

Given a convergent sequence of Hamiltonians (Hn) and a convergent sequence of initial data (gn) for the first-order evolutionary Hamilton-Jacobi equation, we look for conditions ensuring that the sequences (un) and (vn) of Lax solutions and Hopf solutions respectively converge. The convergences we deal with are variational convergences. We take advantage of several recent results giving criteria for the continuity of usual operations.     

Open mappings theorems and linearization stability

An important computation rule for tangent cones is examined. Two results are given which assume only Hadamard differentiability (and a variant of it) instead of strict Fréchet differentiability. This allows the consideration of concrete examples such as superposition operators and can be applied to the problem of linearizing a nonlinear equation or inequality.