Publication detail

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

DIBLÍK, J.

Original Title

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

English Title

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

Type

journal article - other

Language

en

Original Abstract

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

English abstract

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

Keywords

asymptotic properties, homogenous linear systems, ordinary differential equations

RIV year

1992

Released

17.04.1992

ISBN

0022-247X

Periodical

Journal of Mathematical Analysis and Application

Year of study

165

Number

1

State

US

Pages from

288

Pages to

304

Pages count

17

Documents

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"
}