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Generalized ODE approach to impulsive retarded functional differential equations. (English) Zbl 1212.34251
Summary: It is known that retarded functional differential equations (RFDEs) can be regarded as generalized ordinary differential equations (GODEs) [see M. Federson, P. Táboas, J. Differ. Equations 195, No. 2, 313–331 (2003; Zbl 1054.34102); C. Imaz, Z. Vorel, Bol. Soc. Mat. Mex., II. Ser. 11, 47–59 (1966; Zbl 0178.44203) and F. Oliva, Z. Vorel, Bol. Soc. Mat. Mex., II. Ser. 11, 40–46 (1966; Zbl 0178.44204)]. In this paper, we prove the equivalence between RFDEs with pre-assigned moments of impulse effects and a certain class of GODEs introduced in [the second author, Series in Real Analysis, 5, Singapore: World Scientific (1992; Zbl 0781.34003)] using some ideas of the above three papers. We state results on the existence, uniqueness and continuous dependence of the solutions for this class of GODEs, and we use them to obtain fine results concerning the corresponding impulsive RFDEs.

34K45 Functional-differential equations with impulses
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations