Sard theorems for Lipschitz functions and applications in optimization

International audienceWe establish a " preparatory Sard theorem " for smooth functions with a partial affine structure. By means of this result, we improve a previous result of Rifford [14, 16] concerning the generalized (Clarke) critical values of Lipschitz functions defined as minima of smooth functions. We also establish a nonsmooth Sard theorem for the class of Lipschitz functions from R d to R p that can be expressed as finite selections of C k functions (more generally, continuous selections over a compact countable set). This recovers readily the classical Sard theorem and extends a previous result of Barbet-Daniilidis-Dambrine [1] to the case p > 1. Applications in semi-infinite and Pareto optimization are given

Variational inequalities and Fixed Point theorems in the Euclidean space for non-continuous operators

The existence of solutions of general variational inequalities is obtained for some maps defined on a nonempty closed convex subset of the Euclidean space. The convex subset is possibly unbounded, the operator is possibly neither continuous nor coercive, the convex function is possibly non-lower semi-continuous. Two generalizations of the fixed points theorem of Brouwer are deduced for some operators which are non-continuous and defined on some unbounded closed convex subsets

Convex extensions of convex functions by sequential processes

The aim of this paper is to extend a real-valued convex function f into a realvalued convex function bf, defined on a convex subset of the closure of the domain of f. When f is sequentially lower semi-continuous we study whether bf is sequentially lower semi-continuous. The extended function bf is constructed by a sequential process

Unilateral and bilateral characterizations of increasing maps

We define and study various properties of lateral increasing for mappings de- fined between two ordered spaces. After introducing some particular classes of ordered spaces, we formulate unilateral characterizations of the property of increasing. The main theorem characterizes the increasing of a map defined on a complete totally ordered space with bilateral conditions which generalize the classical notions of right or left increasing (cf. [2]

Sequences of contractions and convergence of fixed points

Stability of fixed points of contraction mappings has been studied by Bonsall (cf. [2]) and Nadler (cf. [4]). These authors consider a sequence (Tn) of maps defined on a metric space (X,d) into itself and study the convergence of the sequence of fixed points for uniform or pointwise convergence of (Tn), under contraction assumptions of the maps. We will first consider k-contractions Tn which are only defined on a subset Xn of the metric space. We note that, in general, we cannot apply their results by using an extension theorem of contractions (cf. [1]). In this general setting, pointwise convergence cannot be defined (except when all Xn are a same subset). We then introduce a new notion of convergence and we obtain a convergence result for the fixed points which generalizes Bonsall¿s theorem. Secondly, after introducing another notion of convergence which generalizes uniform convergence, we obtain a stability result when only the limit map is a contraction. Some other results of stability of fixed points, which generalize Nadler¿s theorems, can be found in [3]

La place des SSIAD dans l'offre de soins du Cotentin vue par les médecins généralistes est-elle en adéquation avec la perception qu'en ont les infirmiers coordonnateurs ?

CAEN-BU Médecine pharmacie (141182102) / SudocSudocFranceF

L'animal en politique

2-7475-5042-7Historiens, liguistes et politologues éclairent ici des feux croisés de leurs problématiques et de leurs méthodologies cet objet : le rapport de l'animal à la politique