Averaging for ordinary differential equations perturbed by a small parameter.

*(English)*Zbl 1389.34126In averaging theory, it is well known that the solutions to a nonautonomous ordinary differential equation of the form
\[
x'(t)=f(t/\varepsilon,x(t))
\]
are well approximated by solutions of the autonomous equation
\[
y'(t)=f^0(y(t)),
\]
where the right-hand side \(f^0\) is given by
\[
f^0(x)=\lim_{T\to\infty}\frac{1}{T}\int_0^T f(\tau,x)\,\mathrm{d}\tau
\]
(provided that the limit exists). More precisely, for each \(L>0\) and \(\delta>0\), there exists an \(\varepsilon_0>0\) such that for each \(\varepsilon\in(0,\varepsilon_0]\) and for each solution \(x_\varepsilon\) of the original equation, there exists a solution \(y\) of the averaged equation (with the same initial condition at \(t=0\)) such that \(| x_\varepsilon(t)-y(t)| <\delta\) for all \(t\in[0,L]\).

The authors of the present paper show that the usual conditions on the right-hand side \(f\) can be weakened. Their key assumptions are the continuity of \(f\), uniform continuity of \(f\) in the second variable with respect to the first variable, and the inequality \(| f(t,x)| \leq m(t)\), where \(m\) is a Lebesgue integrable function whose indefinite integral is Lipschitz continuous. Note that \(f\) is assumed to be neither uniformly bounded nor Lipschitz-continuous; hence, the averaged equation with a given initial condition does not necessarily have a unique solution.

The authors of the present paper show that the usual conditions on the right-hand side \(f\) can be weakened. Their key assumptions are the continuity of \(f\), uniform continuity of \(f\) in the second variable with respect to the first variable, and the inequality \(| f(t,x)| \leq m(t)\), where \(m\) is a Lebesgue integrable function whose indefinite integral is Lipschitz continuous. Note that \(f\) is assumed to be neither uniformly bounded nor Lipschitz-continuous; hence, the averaged equation with a given initial condition does not necessarily have a unique solution.

Reviewer: Antonín Slavík (Praha)