Detail publikace

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

Originální název

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

Anglický název

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

Jazyk

en

Originální abstrakt

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 $\lim\sb{t\to\infty}x\sb i(t)=0$, $i=1,2,\ldots,n$.

Anglický abstrakt

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 $\lim\sb{t\to\infty}x\sb i(t)=0$, $i=1,2,\ldots,n$.

BibTex


@article{BUT37345,
  author="Josef {Diblík}",
  title="Some asymptotic properties of solutions of homogeneous linear systems of ordinary differential equqtions",
  annote="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 $\lim\sb{t\to\infty}x\sb i(t)=0$, $i=1,2,\ldots,n$.",
  chapter="37345",
  number="1",
  volume="165",
  year="1992",
  month="april",
  pages="288--304",
  type="journal article - other"
}