A note on existence theorem of Peano

Oleg Zubelevich

Address: Department of Theoretical Mechanics, Mechanics and Mathematics Faculty, M. V. Lomonosov Moscow State University, 119899 Moscow, Vorobevy gory, MGU, Russia

Current address: 2-nd Krestovskii Pereulok 12-179, 129110, Moscow, Russia

E-mail: ozubel@yandex.ru

Abstract: An ODE with non-Lipschitz right hand side has been considered. A family of solutions with $L^p$-dependence of the initial data has been obtained. A special set of initial data has been constructed. In this set the family is continuous. The measure of this set has been estimated.

AMSclassification: primary 34A12.

Keywords: Peano existence theorem, non-Lipschitz nonlinearity, non-uniqueness, IVP, ODE, Cauchy problem.