Asymptotic behavior of the solutions of a partial differential equation with piecewise constant argument.

In this paper, we study the partial differential equation with piecewise constant argument of the form: xt(t,s)=A(t)x(t,s)+B(t,s)x([t],s)+C(t,s)x(t,[s])+D(t,s)x([t],[s])+f(x(t,[s])),t,s∈IR+=(0,∞),$$ {\displaystyle \begin{array}{cc}\hfill {x}_t\left(t,s\right)=& \kern0.2em A(t)x\left(t,s\right)+B\left(t,s\right)x\left(\left[t\right],s\right)+C\left(t,s\right)x\left(t,\left[s\right]\right)\hfill \\ {}\hfill & +D\left(t,s\right)x\left(\left[t\right],\left[s\right]\right)+f\left(x\left(t,\left[s\right]\right)\right),t,s\in I\kern-0.2em {R}^{+}=\left(0,\infty \right),\hfill \end{array}} $$where A(t)$$ A(t) $$ is a k×k$$ k\times k $$ invertible and continuous matrix function on IR+$$ I\kern-0.2em {R}^{+} $$; B(t,s)$$ B\left(t,s\right) $$, C(t,s)$$ C\left(t,s\right) $$, and D(t,s)$$ D\left(t,s\right) $$ are k×k$$ k\times k $$ continuous and bounded matrix functions on IR+×IR+$$ I\kern-0.2em {R}^{+}\times I\kern-0.2em {R}^{+} $$; [t]$$ \left[t\right] $$ and [s]$$ \left[s\right] $$ are the integral parts of t$$ t $$ and s$$ s $$, respectively; and f:IRk→IRk$$ f:I\kern-0.2em {R}^k\to I\kern-0.2em {R}^k $$ is a continuous function. More precisely, under some conditions on the matrices A(t)$$ A(t) $$, B(t,s)$$ B\left(t,s\right) $$, C(t,s)$$ C\left(t,s\right) $$, and D(t,s)$$ D\left(t,s\right) $$ and the function f$$ f $$, we investigate the asymptotic behavior of the solutions of the above equation. [ABSTRACT FROM AUTHOR]