Asymptotic properties of solutions of nonlinear vector initial value problem on the infinite interval (Q580560)

We study initial value problems on the infinite interval: \[ (1)\quad dx/dt=f(t,x,y;\epsilon),\quad x(0,\epsilon)=\xi (\epsilon),\quad \epsilon (dy/dt)=g(t,x,y;\epsilon),\quad y(0,\epsilon)=\eta (\epsilon) \] where \(x,f\in E^ m\), \(y,g\in E^ n\), \(\epsilon\) are real small positive parameters, \(0\leq t<+\infty\). If \(g_ y(t)\) is nonsingular and under other assumptions, we prove that there are serial \((k+m^*)\)-dimensional manifolds \(\{S_ R(\epsilon)\}\in E^{m+n}\) such that (1) degenerates regularly provided \((\xi (\epsilon),\eta (\epsilon))\in S_ R(\epsilon)\). Besides, the R-order asymptotic expansions of solutions are constructed, and their errors are estimated.