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Zbl 0978.34039
Alonso, A.I.; Hong, Jialin; Obaya, R.
Almost periodic type solutions of differential equations with piecewise constant argument via almost periodic type sequences.
(English)
[J] Appl. Math. Lett. 13, No.2, 131-137 (2000). ISSN 0893-9659

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: $${dx\over dt}= f(t, x(t), x([t]), x([t- 1]),\dots, x([t- k])),\quad t\in J,\tag 1$$ where $k$ is a positive integer, $f\in C(J\times \Omega,\bbfR^d)$, $[\cdot]$ denotes the greatest integer function.\par There are used the following notations: $H(f)$ -- the hull of $f\in C(\bbfR, \bbfR^d)$;\par $AP(\bbfR\times \Omega)$ -- the set of all almost-periodic functions in $t\in\bbfR$ uniformly for $x$ in compact subsets of $\Omega$;\par $AAP_0(\bbfR^+\times \Omega)= \{f\in C(\bbfR^+\times \Omega,\bbfR^d): \lim_{t\to+\infty} f(t,x)= 0$, uniformly for $x$ in compact subsets of $\Omega\}$;\par $AAP(\bbfR^+\times \Omega)$ -- the set of all asymptotically almost-periodic functions;\par $PAP_0(\bbfR\times \Omega)= \{\varphi\in C(\bbfR\times \Omega, \bbfR^d): m(|\varphi|)= \lim_{T\to+\infty} {1\over 2T} \int^T_{-T} |\varphi(t,x)|dt= 0$ uniformly for $x$ in compact subsets of $\Omega\}$;\par $PAP(\bbfR\times \Omega)$ -- the set of all pseudo almost-periodic functions;\par $AP(Z)$ -- the set of all almost-periodic sequences;\par $AAP(Z^+)$ -- the set of all asymptotically almost-periodic sequences;\par $PAP(Z)$ -- the set of all pseudo almost-periodic sequences.\par A first result states that if $f\in AP(\bbfR\times \Omega_0)$ in equation (1) for a compact $\Omega_0\subset \Omega$, all equations $${dx\over dt}= g(t, x(t), x([t]), x([t- 1]),\dots, x([t- k])),\quad t\in\bbfR,$$ with $g\in H(t)$ have unique solutions to initial value problems, where the initial value condition is $x(j)= x_j$, $j= 0,-1,-2,\dots, -k$, and $\varphi(t)$ is a solution to (1) with $\varphi(\bbfR)^{k+2}\subset \Omega_0$, then $\varphi\in AP(\bbfR)$ if and only if $\{\varphi(n)\}_{n\in Z}\in AP(Z)$.\par Later, if $f\in PAP_0(\bbfR^+\times \Omega)$ in equation (1) satisfying a Lipschitz condition on $\Omega$, and $\varphi(t)$ is a solution to (1) with $\varphi(\bbfR)^{k+2}\subset\Omega$, then $\varphi\in AAP_0(\bbfR^+)$ if and only if $\{\varphi(n)\}_{n\in Z}\in PAP_0(Z^+)$.\par Another result states that if $f\in PAP(\bbfR\times \Omega_0)$ $(AAP(\bbfR^+\times \Omega_0))$ 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 $\varphi(t)$ is a solution to (1) with $\varphi(\bbfR)^{k+2}\subset \Omega_0$, then $\varphi\in PAP(\bbfR)$ $(AAP(\bbfR^+))$ if and only if $\{\varphi(n)\}_{n\in Z}\in PAP(Z)$ $(AAP(Z^+))$.
[Marian Ivanovici (Craiova)]
MSC 2000:
*34C27 Almost periodic solutions of ODE

Keywords: almost-periodic sequences; almost-periodic functions; asymptotically almost-periodic functions; pseudo almost-periodic functions

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