# On the range of the derivative of a smooth function and applications.

RACSAM (2006)

- Volume: 100, Issue: 1-2, page 63-74
- ISSN: 1578-7303

## Access Full Article

top## Abstract

top## How to cite

topDeville, Robert. "On the range of the derivative of a smooth function and applications.." RACSAM 100.1-2 (2006): 63-74. <http://eudml.org/doc/41643>.

@article{Deville2006,

abstract = {We survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of L(X,Y) for the existence of a Fréchet-differentiable mapping F from X into Y so that F'(X) = A. Whenever F is only assumed Gâteaux-differentiable, new phenomena appear: we discuss the existence of a mapping F from a Banach space X into a Banach space Y, which is bounded, Lipschitz-continuous, and so that for all x, y ∈ X, if x ≠ y, then ||F'(x) - F'(y)||L(X,Y) > 1. Applications are given to existence and uniqueness of solutions of Hamilton-Jacobi equations.},

author = {Deville, Robert},

journal = {RACSAM},

language = {eng},

number = {1-2},

pages = {63-74},

title = {On the range of the derivative of a smooth function and applications.},

url = {http://eudml.org/doc/41643},

volume = {100},

year = {2006},

}

TY - JOUR

AU - Deville, Robert

TI - On the range of the derivative of a smooth function and applications.

JO - RACSAM

PY - 2006

VL - 100

IS - 1-2

SP - 63

EP - 74

AB - We survey recent results on the structure of the range of the derivative of a smooth real valued function f defined on a real Banach space X and of a smooth mapping F between two real Banach spaces X and Y. We recall some necessary conditions and some sufficient conditions on a subset A of L(X,Y) for the existence of a Fréchet-differentiable mapping F from X into Y so that F'(X) = A. Whenever F is only assumed Gâteaux-differentiable, new phenomena appear: we discuss the existence of a mapping F from a Banach space X into a Banach space Y, which is bounded, Lipschitz-continuous, and so that for all x, y ∈ X, if x ≠ y, then ||F'(x) - F'(y)||L(X,Y) > 1. Applications are given to existence and uniqueness of solutions of Hamilton-Jacobi equations.

LA - eng

UR - http://eudml.org/doc/41643

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.