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Ordinary differential equations the solution of which are ACG\(_ *\)- functions. (English) Zbl 0756.34003

This paper is devoted to the fundamental Cauchy problem for ordinary differential systems \(x'=f(t,x)\) when the right-hand member does not satisfy the Carathéodory conditions. Indeed, the equivalent integral equation \[ x(t)=x(r)+\int^ t_ rf(s,x(s))ds \] is considered where the integral is considered in the sense of Perron. Some conditions for the existence of a local solution for this problem were obtained in the book “Lectures on the Theory of Integration” of R. Henstock (1988; Zbl 0668.28001). It is proved in the present paper that the conditions of Henstock hold if and only if \(f(t,x)=h(t,x)+g(t)\), where \(g\) is Perron integrable and \(h\) satisfies the classical Carathéodory conditions.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
26A39 Denjoy and Perron integrals, other special integrals
26A46 Absolutely continuous real functions in one variable

Citations:

Zbl 0668.28001
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