Limit processes in ordinary differential equations (Q1821238)

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.