Some asymptotic properties of solutions of homogeneous linear systems of ordinary differential equations
Consider the system (1) \(x'=A(t)x\), where \(t\in I_ 1=(x_ 0- \varepsilon,\infty)\), \(-\infty<x_ 0<\infty\), \(\varepsilon>0\) and \(A\) is a square \(n\times n\) real matrix, \(A\in C^ 1(I_ 1)\). We say that the solution \(x(t)=(x_ 1(t),\ldots,x_ n(t))\) of (1) is \(\alpha\)-bounded on \(I=\langle x_ 0,\infty)\) if there exists a vector-function \(\alpha(t)=(\alpha_ 1(t),\ldots,\alpha_ n(t))\), \(\alpha_ i:I\to(0,\infty)\) such that \(| x_ i(t)|<\alpha_ i(t)\) for \(t\in I\) and \(i=1,2,\ldots,n\). Using a modification of the topological method of T. Ważewski, the author gives sufficient conditions for the existence at least a \(k\)-parametric class of \(\alpha\)-bounded on \(I\) solutions of (1), where \(\alpha\) is a suitable vector-function. These results are applied to the study of the existence of at least a \(k\)- parametric class of solutions of (1) satisfying \(\lim_{t\to\infty}x_ i(t)=0\), \(i=1,2,\ldots,n\).
scientific article; zbMATH DE number 62075