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

Abstract

In this paper we study initial value problems on the infinite interval:

$$\left. {\begin{array}{*{20}c} {\frac{{dx}}{{dt}} = f(t, x, y; \varepsilon ),} & {x(0, \varepsilon ) = \xi (\varepsilon )} \\ {e\frac{{dy}}{{dt}} = g(t, x, y; \varepsilon ),} & {y(0, \varepsilon ) = \eta (\varepsilon )} \\ \end{array} } \right\}$$

((1.1))

where x, ƒ∈E ^m, y, g∈Eⁿ,ε are real small positive parameters,0ť+∞.On condition that g _y (t) is nonsingular and under other assumptions, we have proved that there are serial (k+m^*)-dimensional manifolds {S_R(ε)}∈E^m+n such that (1.1) degenerates regularly provided (ξ(ε), η(ε))∈S_R(ε).Besides, the R-order asymptotic expansions of solutions are constructed, and their errors are estimated.