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Averaging for ordinary differential equations perturbed by a small parameter. (English) Zbl 1389.34126
In 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.
##### MSC:
 34C29 Averaging method for ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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