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Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences. (English) Zbl 0978.34039

The aim of this paper is to characterize the existence of almost-periodic, asymptotically almost-periodic, and pseudo almost-periodic solutions to differential equations with piecewise constant argument of the form:

$\frac{dx}{dt}=f\left(t,x\left(t\right),x\left(\left[t\right]\right),x\left(\left[t-1\right]\right),\cdots ,x\left(\left[t-k\right]\right)\right),\phantom{\rule{1.em}{0ex}}t\in J,\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $k$ is a positive integer, $f\in C\left(J×{\Omega },{ℝ}^{d}\right)$, $\left[·\right]$ denotes the greatest integer function.

There are used the following notations: $H\left(f\right)$ – the hull of $f\in C\left(ℝ,{ℝ}^{d}\right)$;

$AP\left(ℝ×{\Omega }\right)$ – the set of all almost-periodic functions in $t\in ℝ$ uniformly for $x$ in compact subsets of ${\Omega }$;

$AA{P}_{0}\left({ℝ}^{+}×{\Omega }\right)=\left\{f\in C\left({ℝ}^{+}×{\Omega },{ℝ}^{d}\right):{lim}_{t\to +\infty }f\left(t,x\right)=0$, uniformly for $x$ in compact subsets of ${\Omega }\right\}$;

$AAP\left({ℝ}^{+}×{\Omega }\right)$ – the set of all asymptotically almost-periodic functions;

$PA{P}_{0}\left(ℝ×{\Omega }\right)=\left\{\varphi \in C\left(ℝ×{\Omega },{ℝ}^{d}\right):m\left(|\varphi |\right)={lim}_{T\to +\infty }\frac{1}{2T}{\int }_{-T}^{T}|\varphi \left(t,x\right)|dt=0$ uniformly for $x$ in compact subsets of ${\Omega }\right\}$;

$PAP\left(ℝ×{\Omega }\right)$ – the set of all pseudo almost-periodic functions;

$AP\left(Z\right)$ – the set of all almost-periodic sequences;

$AAP\left({Z}^{+}\right)$ – the set of all asymptotically almost-periodic sequences;

$PAP\left(Z\right)$ – the set of all pseudo almost-periodic sequences.

A first result states that if $f\in AP\left(ℝ×{{\Omega }}_{0}\right)$ in equation (1) for a compact ${{\Omega }}_{0}\subset {\Omega }$, all equations

$\frac{dx}{dt}=g\left(t,x\left(t\right),x\left(\left[t\right]\right),x\left(\left[t-1\right]\right),\cdots ,x\left(\left[t-k\right]\right)\right),\phantom{\rule{1.em}{0ex}}t\in ℝ,$

with $g\in H\left(t\right)$ have unique solutions to initial value problems, where the initial value condition is $x\left(j\right)={x}_{j}$, $j=0,-1,-2,\cdots ,-k$, and $\varphi \left(t\right)$ is a solution to (1) with $\varphi {\left(ℝ\right)}^{k+2}\subset {{\Omega }}_{0}$, then $\varphi \in AP\left(ℝ\right)$ if and only if ${\left\{\varphi \left(n\right)\right\}}_{n\in Z}\in AP\left(Z\right)$.

Later, if $f\in PA{P}_{0}\left({ℝ}^{+}×{\Omega }\right)$ in equation (1) satisfying a Lipschitz condition on ${\Omega }$, and $\varphi \left(t\right)$ is a solution to (1) with $\varphi {\left(ℝ\right)}^{k+2}\subset {\Omega }$, then $\varphi \in AA{P}_{0}\left({ℝ}^{+}\right)$ if and only if ${\left\{\varphi \left(n\right)\right\}}_{n\in Z}\in PA{P}_{0}\left({Z}^{+}\right)$.

Another result states that if $f\in PAP\left(ℝ×{{\Omega }}_{0}\right)$ $\left(AAP\left({ℝ}^{+}×{{\Omega }}_{0}\right)\right)$ in equation (1) for a compact subset ${{\Omega }}_{0}\subset {\Omega }$, $f$ and its almost-periodic component ${f}_{1}$ satisfy a Lipschitz condition on ${{\Omega }}_{0}$, and $\varphi \left(t\right)$ is a solution to (1) with $\varphi {\left(ℝ\right)}^{k+2}\subset {{\Omega }}_{0}$, then $\varphi \in PAP\left(ℝ\right)$ $\left(AAP\left({ℝ}^{+}\right)\right)$ if and only if ${\left\{\varphi \left(n\right)\right\}}_{n\in Z}\in PAP\left(Z\right)$ $\left(AAP\left({Z}^{+}\right)\right)$.

MSC:
 34C27 Almost and pseudo-almost periodic solutions of ODE