Some asymptotic properties of solutions of homogeneous linear systems of ordinary differential equqtions

Some asymptotic properties of solutions of homogeneous linear systems of ordinary differential equqtions

Peer-reviewed article not indexed in WoS or Scopus

Consider the system (1) $x'=A(t)x$, where $t\in I\sb 1=(x\sb 0- \varepsilon,\infty)$, $-\infty0$ and $A$ is a square $n\times n$ real matrix, $A\in C\sp 1(I\sb 1)$. We say that the solution $x(t)=(x\sb 1(t),\ldots,x\sb n(t))$ of (1) is $\alpha$-bounded on $I=\langle x\sb 0,\infty)$ if there exists a vector-function $\alpha(t)=(\alpha\sb 1(t),\ldots,\alpha\sb n(t))$, $\alpha\sb i:I\to(0,\infty)$ such that $\vert x\sb i(t)\vert<\alpha\sb i(t)$ 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 $\lim\sb{t\to\infty}x\sb i(t)=0$, $i=1,2,\ldots,n$.

Consider the system (1) $x'=A(t)x$, where $t\in I\sb 1=(x\sb 0- \varepsilon,\infty)$, $-\infty0$ and $A$ is a square $n\times n$ real matrix, $A\in C\sp 1(I\sb 1)$. We say that the solution $x(t)=(x\sb 1(t),\ldots,x\sb n(t))$ of (1) is $\alpha$-bounded on $I=\langle x\sb 0,\infty)$ if there exists a vector-function $\alpha(t)=(\alpha\sb 1(t),\ldots,\alpha\sb n(t))$, $\alpha\sb i:I\to(0,\infty)$ such that $\vert x\sb i(t)\vert<\alpha\sb i(t)$ 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 $\lim\sb{t\to\infty}x\sb i(t)=0$, $i=1,2,\ldots,n$.

asymptotic properties, homogenous linear systems, ordinary differential equations

asymptotic properties, homogenous linear systems, ordinary differential equations

DIBLÍK, J.

17.04.1992

0022-247X

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

165

1

United States of America

288

304

17