## Limit processes in ordinary differential equations.(English)Zbl 0616.34004

Let $$g_ k: {\mathbb{R}}^ n\times [0,T]\to {\mathbb{R}}^ n$$ be continuous, let $$g_ k=\partial G_ k/\partial t$$, where $$G_ k$$ is a function with a continuous derivative $$DG_ k$$ with respect to x, Hölderian in t with an exponent $$0<\delta \leq 1$$, and both $$G_ k$$, $$DG_ k$$ tend to zero uniformly with $$k\to \infty$$. Further, let the functions $$\Gamma_ k=DG_ k\cdot g_ k$$ satisfy $$\| \int^{t}_{s}\Gamma_ k(x,\tau)d\tau \| \leq K| t-s|^{\delta},\| \int^{t}_{s}[\Gamma_ k(x,\tau)-\Gamma_ k(x,\tau)]d\tau \| \leq K\| x-y\| \cdot | t-s|^{\gamma}$$ with $$1-\delta <\gamma \leq 1$$, and let there exist a bounded function h measurable in t and Lipschitzian in x such that $$\int^{t}_{s}\Gamma_ k(x,\tau)d\tau \to -\int^{t}_{s}h(x,\tau)d\tau$$ uniformly. Then the solutions of $$\dot x=g_ k(x,t)$$, $$x(0)=\tilde x$$ tend to the solutions of $$\dot x=h(x,t)$$, $$x(0)=\tilde x$$. The result explains some convergence phenomena not covered by the first author’s theory of generalized ODE [Czechosl. Math. J. 7(82), 418-449 (1957; Zbl 0090.300)], e.g. for equations of the type $\dot x=f_ 1(x,t)k^{\alpha}\cos (k^{\alpha +\beta}t+\vartheta_ 1)+f_ 2(x,t)k^{\beta}\cos (k^{\alpha +\beta}t+\vartheta_ 2)$ with $$0<\beta <2\alpha <4\beta$$ or, more particularly, $\dot x=f_ 1(x)k^{1/2}\cos (kt+\vartheta_ 1)+f_ 2(x)k^{1/2}\cos (kt+\vartheta_ 2);$ in this last case, the limit equation is $$\dot x=[f_ 1,f_ 2]\sin (\vartheta_ 2-\vartheta_ 1),$$ where $$[f_ 1,f_ 2]=Df_ 1\cdot f_ 2-Df_ 2\cdot f_ 1$$ is the Lie bracket.

### MSC:

 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations

Zbl 0090.300
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### References:

  J. Kurzweil and J. Jarnik,A convergence effect in ordinary differential equations. In: Asymptotic Methods of the Mathematical Physics. Dedicated to the 70th birthday of Acad. Ju. A. Mitropolskij. Kijev, to appear.  J. Kurzweil,Generalized ordinary differential equations and continuous dependence on a parameter. Czechoslovak Math. J.7 (82), 418-449 (1957). · Zbl 0090.30002
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