Asymptotic behavior of the solutions of a certain system of differential equations in the neighborhood of a singular point
Consider the following system of ordinary differential equations \(g_ i(x)y'\!_ i=\alpha_ i(y_ i)[1+f_ i(x,y_ 1,...,y_ n)] y_ i(0+)=0\), \(i=1,...,n\) where \(\alpha_ i\in C^ 0(0,y_ 0]\), \(g_ i\in C^ 0(0,x_ 0]\), \(\alpha_ i>0\), \(0<y<y_ 0\), \(g_ i>0\), \(0<x<x_ 0\), and \[ \lim_{x\to 0+} \int^{x}_{0+} dt/g_ i(t)=\lim_{x\to 0+} \int^{x}_{0+} dt/\alpha_ i(t)=0,\quad i=1,...,m, \] \[ 0\leq m\leq n\lim_{x\to 0+} \int^{x}_{x_ 0} dt/g_ i(t)=\lim_{x\to 0+} \int^{x}_{y_ 0} dt/\alpha_ i(t)=-\infty,\quad i=m+1,...,n. \] Under some additional assumptions the behaviour of the solutions for this problem for \(x\to 0+\) is studied.
scientific article; zbMATH DE number 3855543