Topological dynamics of retarded functional differential equations

Abstract

We prove that a local flow can be constructed for a general class of nonautonomous retarded functional differential equations (RFDE). This is an extension to a result of Artstein (J. Differential Equations 23 (1977) 216) and fits in the classical theory of R. Miller and G. Sell. The main tool in this paper are generalized ordinary differential equations according to Kurzweil (Czech. Math. J. 7 (82) (1957) 418). In obtaining our results, we must prove the space of RFDEs can be embedded in a space of generalized ordinary differential equations. In opposition to the technical hypotheses of Oliva and Vorel (Bol. Soc. Mat. Mexicana 11 (1996) 40), this auxiliary result, as we present, is advantageous in the sense that our assumptions have an explanatory character. Applications based on topological dynamics techniques follow naturally from our results. As an illustration of this fact we show how to achieve in this setting a theorem on continuous dependence on initial data of solutions of RFDEs.