Sikorska-Nowak, Aneta Retarded functional differential equations in Banach spaces and Henstock-Kurzweil integrals. (English) Zbl 1011.34066 Demonstr. Math. 35, No. 1, 49-60 (2002). The author deals with a functional-differential equation of the form \[ x'(t)= f(t,x_t), \quad\text{a.e. }t\in [0,a], \] with the initial condition \(x_0 = \varphi\) on \([-r,0]\), where the function \(x_t\) defined by \(x_t(\theta)= x(t+\theta)\) provides the finite delay and \(\varphi\) belongs to a suitable subspace of a space of Henstock-Kurzweil integrable functions; moreover, the function \(f:[0,a]\times C([-r,0];E)\to E\) is Carathéodory and \(E\) is a Banach space.The main result is the existence of local solutions for the functional Cauchy problem. To obtain this result, a kind of HK-integrability on \(f\) and an estimate on the Hausdorff measure of noncompactness of the range of \(f\) by means of a Kamke function are assumed.As a consequence, the existence of local solutions for the differential inclusion \(x'(t)\in F(t,x_t)\), a.e. \(t\in[0,a]\), with initial data \(x_0=\varphi\), is derived. Reviewer: Paola Rubbioni (Perugia) Cited in 6 Documents MSC: 34K30 Functional-differential equations in abstract spaces 34G20 Nonlinear differential equations in abstract spaces 34A60 Ordinary differential inclusions Keywords:Cauchy problem; measure of noncompactness; Kamke function; Henstock-Kurzweil integral PDFBibTeX XMLCite \textit{A. Sikorska-Nowak}, Demonstr. Math. 35, No. 1, 49--60 (2002; Zbl 1011.34066) Full Text: DOI