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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: $${dx\over dt}= f(t, x(t), x([t]), x([t- 1]),\dots, x([t- k])),\quad t\in J,\tag 1$$ where $k$ is a positive integer, $f\in C(J\times \Omega,\bbfR^d)$, $[\cdot]$ denotes the greatest integer function. There are used the following notations: $H(f)$ -- the hull of $f\in C(\bbfR, \bbfR^d)$; $AP(\bbfR\times \Omega)$ -- the set of all almost-periodic functions in $t\in\bbfR$ uniformly for $x$ in compact subsets of $\Omega$; $AAP_0(\bbfR^+\times \Omega)= \{f\in C(\bbfR^+\times \Omega,\bbfR^d): \lim_{t\to+\infty} f(t,x)= 0$, uniformly for $x$ in compact subsets of $\Omega\}$; $AAP(\bbfR^+\times \Omega)$ -- the set of all asymptotically almost-periodic functions; $PAP_0(\bbfR\times \Omega)= \{\varphi\in C(\bbfR\times \Omega, \bbfR^d): m(|\varphi|)= \lim_{T\to+\infty} {1\over 2T} \int^T_{-T} |\varphi(t,x)|dt= 0$ uniformly for $x$ in compact subsets of $\Omega\}$; $PAP(\bbfR\times \Omega)$ -- the set of all pseudo almost-periodic functions; $AP(Z)$ -- the set of all almost-periodic sequences; $AAP(Z^+)$ -- the set of all asymptotically almost-periodic sequences; $PAP(Z)$ -- the set of all pseudo almost-periodic sequences. 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 $\{\varphi(n)\}_{n\in Z}\in AP(Z)$. 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 $\{\varphi(n)\}_{n\in Z}\in PAP_0(Z^+)$. 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 $\{\varphi(n)\}_{n\in Z}\in PAP(Z)$ $(AAP(Z^+))$.

##### MSC:
 34C27 Almost and pseudo-almost periodic solutions of ODE
Full Text:
##### References:
 [1] Aftabizadeh, A. R.; Wiener, J.; Xu, J. M.: Oscillatory and periodic solutions of delay differential equations with piecewise constant argument. Proc. amer. Math. soc. 99, 673-679 (1987) · Zbl 0631.34078 [2] Cooke, K. L.; Wiener, J.: A survey of differential equations with piecewise constant argument. Lecture notes in mathematics 1475, 1-15 (1991) · Zbl 0737.34045 [3] Wiener, J.: Generalized solutions of functional differential equations. (1993) · Zbl 0874.34054 [4] Busenberg, S.; Cooke, K. L.: Models of vertically transmitted diseases with sequential-continuous dynamics. Nonlinear phenomena in mathematical sciences, 179-187 (1982) · Zbl 0512.92018 [5] Meisters, G. H.: On almost periodic solutions of a class of differential equations. Proc. amer. Math. soc. 10, 113-119 (1959) · Zbl 0092.30401 [6] Opial, Z.: Sur LES solutions presque-periodiques d’une classe d’equations differentielles. Ann. polon. Math. 9, 157-181 (1960/1961) · Zbl 0098.28803 [7] Fink, A. M.: Almost periodic differential equations. Lecture notes in mathematics 377 (1974) · Zbl 0325.34039 [8] Basit, B.; Zhang, C.: New almost periodic type functions and solutions of differential equations. Can. J. Math. 48, No. 6, 1138-1153 (1996) · Zbl 0880.43009 [9] Zhang, C.: Pseudo almost-periodic solutions of some differential equations. J. math. Anal. appl. 181, 62-76 (1994) · Zbl 0796.34029 [10] Zhang, C.: Integration of vector-valued pseudo-almost periodic functions. Proc. amer. Math. soc. 121, 167-174 (1994) · Zbl 0818.42003 [11] Dads, E. Ait; Arino, O.: Exponential dichotomy and existence of pseudo almost periodic solutions of some differential equations. Nonlinear analysis, TMA 27, No. 4, 369-386 (1996) · Zbl 0855.34055 [12] Dads, E. Ait; Ezzinbi, K.; Arino, O.: Pseudo almost periodic solutions for some differential equations in a Banach space. Nonlinear analysis, TMA 28, No. 7, 1141-1155 (1997) · Zbl 0874.34041 [13] J. Hong, R. Obaya and A. Sanz, Almost periodic type solutions of some differential equations with piecewise constant arguments, Nonlinear Analysis, TMA (to appear). · Zbl 0996.34062 [14] Yuan, R.; Hong, J.: The existence of almost periodic solutions for a class of differential equations with piecewise constant argument. Nonlinear analysis, TMA 28, No. 8, 1439-1450 (1997) · Zbl 0869.34038 [15] Hong, J.; Núñez, C.: On the almost periodic type difference equations. Mathl. comput. Modelling 28, No. 12, 21-31 (1998) · Zbl 0992.39003 [16] Carvalho, L. A. V.; Cooke, K. L.: A nonlinear equation with piecewise continuous argument. Differential and integral equations, No. 1, 359-367 (1988) · Zbl 0723.34061