Existence theory for integrodifferential equations and Henstock-Kurzweil integral in Banach spaces. (English) Zbl 1148.26010

The author considers the initial value problem for a nonlinear Volterra type integro-differential equation \[ x'(t)=f\left(t, x(t), \int\limits_0^t k(t, s, x(s)) ds \right), \] where \(f\), \(k\) and \(x\) are functions with values in a Banach space, and the integral is taken in the sense of HL [see S. S. Cao, Southeast Asian Bull. Math. 16, No. 1, 35–40 (1992; Zbl 0749.28007)]. In the first part of the paper some well-known definitions and results from the literature are presented and discussed, in particular, the Kuratowski and Hausdorff measures of noncompactness as well as Henstock-Kurzweil integral in Banach space. Using these notions and a fixed point theorem due to H. Mönch [Nonlinear Anal., Theory Methods Appl. 4, 985–999 (1980; Zbl 0462.34041)], the author proves local solvability of the above initial value problem, provided the functions \(f\) and \(k\) belong to the \(L^1\)-Carathéodory class, and satisfy some additional conditions expressed in terms of the measure of noncompactness.


26A39 Denjoy and Perron integrals, other special integrals
45J05 Integro-ordinary differential equations
45D05 Volterra integral equations
26E20 Calculus of functions taking values in infinite-dimensional spaces
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