# Parametrization of Carathéodory multifunctions

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## Abstract

Parametrization of Carathéodory multifunctions RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA ANTÓNIOORNELAS Parametrization of Carathéodorymultifunctions Rendiconti del Seminario Matematico della Università di Padova, tome 83 (1990), p. 33-44. <http://www.numdam.org/item?id=RSMUP_1990__83__33_0> © Rendiconti del Seminario Matematico della Università di Padova, 1990, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/legal. php). Toute utilisation commerciale ou impression systématique est consti- tutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Parametrization of Carathéodory Multifunctions. ANTÓNIO ORNELAS (*) 1. Introduction. Let F: be a multifunction which is Lipschitz with con- stant I and has values bounded by m. We show that co can be represented as f (x, U), with U the unit closed ball in Rn and f Lipschitz with constant 6n(21 + m). Existing representations were: either with U the unit closed ball in Rn but f just continuous in (x, u) (Ekeland-Valadier [3]); or with f Lipschitz in (x, u) but U in some infinite dimensional space (LeDonne-Marchi [6]). More generally, let F: be a multifunction with .F’( ~ , x) measurable and .F(t, ~ ) uniformly continuous. We show that co F(t, x) can be represented as f (t, x, U), where U is either the unit closed ball in Rn (in case the values F(t, x) are compact) or U = Rn (in case the values F(t, x) are unbounded). As to f, we obtain f(., x, u) meas- urable and f (t, ~ , ~ ) uniformly continuous (with modulus of con- tinuity equal to that of F(t, ·) multiplied by a constant). A consequence of this is that differential inclus

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