Remarks on the existence theories of ordinary differential equations

Abstract

In this thesis, we review the state-of-art in the existence theories of ordinary differential equations. More precisely, we discuss the Cauchy-Lipschitz theory, Carathéodory theory, Picard-Lindelöf theory, Peano theory and Osgood conditions for the classical solution. For discontinuous forcing, we introduce Filippov theory and generalized theory by Luigi Ambrosio, Camillo De Lellis and Gianluca Crippa.