On retarded functional differential equations and Henstock-Kurzweil integrals. (English) Zbl 0873.34054

The functional differential equation of the form \[ x'(t)=f(t,x_t), \] where \(f\colon {\mathbb{R}}\times C[-r,0]\to C[-r,0]\), \(x_t\in {\mathbb{R}}\), \(x_t(\vartheta)=x(t+\vartheta)\), is considered in the case when \(f(t,x_t)\) may not be Lebesgue integrable. The Cauchy problem for this equation is reduced to the integral equation \[ x(t)=\phi(\sigma)+\int_{\sigma}^{t}f(s,x_s) ds,\quad t\geq\sigma, \] where \(\int\) is regarded as Henstock-Kurzweil integral (see R. Henstock [The general theory of integration, Clarendon Press, Oxford (1991; Zbl 0745.26006)]; J. Kurzweil [Nichtabsolut konvergente Integrale, Leipzig (1980; Zbl 0441.28001)]). Existence and continuous dependence (with respect to a parameter) theorems for the Cauchy problem in described classes of solutions are proved.


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
26A42 Integrals of Riemann, Stieltjes and Lebesgue type