We propose a level proximal subdifferential for proper lower semicontinuous function. Level is uniform refinement of the well-known subdifferential, and has pleasant feature that its resolvent always coincides with mapping It turns out representation in terms Mordukhovich limiting only valid hypoconvex functions. also provide properties numerous examples to illustrate our results.

AUTHORS Hugo GROULT / UNIVERSITY OF LA ROCHELLE, UMR 7266, LIENSS, AVENUE MICHEL CREPEAU, LA ROCHELLE Jean-Marie PIOT / UNIVERSITY OF LA ROCHELLE, UMR 7266, LIENSS, AVENUE MICHEL CREPEAU, LA ROCHELLE Nicolas POUPARD / UNIVERSITY OF LA ROCHELLE, UMR 7266, LIENSS, AVENUE MICHEL CREPEAU, LA ROCHELLE Ingrid FRUITIER-ARNAUDIN / UNIVERSITY OF LA ROCHELLE, UMR 7266, LIENSS, AVENUE MICHEL CREPEAU, LA R...

We study properties of functions convex with respect to a given family X of vector fields, a notion that appears natural in Carnot-Carathéodory metric spaces. We define a suitable subdifferential and show that a continuous function is X -convex if and only if such subdifferential is nonempty at every point. For vector fields of Carnot type we deduce from this property that a generalized Fenchel...

In this note we give a subdifferential mean value inequality for every continuous Gâteaux subdifferentiable function f in a Banach space which only requires a bound for one but not necessarily all of the subgradients of f at every point of its domain. We also give a subdifferential approximate Rolle’s theorem stating that if a subdifferentiable function oscillates between −ε and ε on the bounda...

In this paper we study the optimal control of system driven by hemivariational inequality of second order. First, we establish the existence of solutions to hemivariational inequality which contains nonlinear pseudomonotone evolution operator. Introducing a control variable in the multivalued term of the generalized subdifferential, we prove the closedness (in suitable topologies) of the graph ...

Using a quantitative version of the subdifferential characterization of directionally Lipschitz functions, we study the integrability of subdifferentials of such functions over arbitrary Banach space.

Michel Labouesse grew up near Paris. He studied maths and physics for a degree in engineering, but subsequently embraced a career in biology. He learned genetics during his PhD with Piotr Slonimski in Gif/Yvette, where he analysed how the nuclear genome controls expression of mitochondrial genes in yeast. He then went on to learn developmental biology of the nematode Caenorhabditis elegans as a...

In this work, we improve the approach of [15] to nonlinear error bounds for lower semicontinuous functions on complete metric spaces, an approach consisting in reducing the nonlinear case to the linear one through a change of metric. This improvement is basically a technical one, and allows dealing with local error bounds in an appropriate way. We present some consequences of the general result...