Asymptotic behavior of the solutions of a partial differential equation
with piecewise constant argument
Abstract
In this paper we study the partial differential equation with piecewise
constant argument of the form : \[
\begin{array}{lll}
x_t(t,s)=&A(t)x(t,s)+B(t,s)x([t],s)+C(t,s)x(t,[s])+\\[0.5cm]
&D(t,s)x([t],[s])+f(x(t,[s])),\
\ t,s\in
\R^{+}=(0,\infty)
\end{array} \] where $A(t)$ is a
$k\times k$ invertible and continuous matrix function
on $\R^{+}$, $B(t,s)$, $C(t,s)$, $D(t,s)$
are $k \times k$ continuous and bounded matrix
functions on $\R^{+}\times
\R^{+}$, $[t]$, $[s]$ are the integral
parts of $t,s$ respectively and
$f:\R^k\rightarrow
\R^k$ is a continuous function. More precisely under
some conditions on the matrices $A(t)$, $B(t,s)$, $C(t,s)$,
$D(t,s)$ and the function $f$ we investigate the asymptotic
behaviour of the solutions of the above equation.
\end{abstract}