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.


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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[1] 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.
[2] 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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