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Retarded functional differential equations in Banach spaces and Henstock-Kurzweil integrals. (English) Zbl 1011.34066

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.

MSC:

34K30 Functional-differential equations in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
34A60 Ordinary differential inclusions
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