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, \(f\in C(J\times \Omega,\mathbb{R}^d)\), \([\cdot]\) denotes the greatest integer function.
There are used the following notations: \(H(f)\) – the hull of \(f\in C(\mathbb{R}, \mathbb{R}^d)\);
\(AP(\mathbb{R}\times \Omega)\) – the set of all almost-periodic functions in \(t\in\mathbb{R}\) uniformly for \(x\) in compact subsets of \(\Omega\);
\(AAP_0(\mathbb{R}^+\times \Omega)= \{f\in C(\mathbb{R}^+\times \Omega,\mathbb{R}^d): \lim_{t\to+\infty} f(t,x)= 0\), uniformly for \(x\) in compact subsets of \(\Omega\}\);
\(AAP(\mathbb{R}^+\times \Omega)\) – the set of all asymptotically almost-periodic functions;
\(PAP_0(\mathbb{R}\times \Omega)= \{\varphi\in C(\mathbb{R}\times \Omega, \mathbb{R}^d): m(|\varphi|)= \lim_{T\to+\infty} {1\over 2T} \int^T_{-T} |\varphi(t,x)|dt= 0\) uniformly for \(x\) in compact subsets of \(\Omega\}\);
\(PAP(\mathbb{R}\times \Omega)\) – 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 \(f\in AP(\mathbb{R}\times \Omega_0)\) in equation (1) for a compact \(\Omega_0\subset \Omega\), all equations \[ {dx\over dt}= g(t, x(t), x([t]), x([t- 1]),\dots, x([t- k])),\quad t\in\mathbb{R}, \] 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 \(\varphi(\mathbb{R})^{k+2}\subset \Omega_0\), then \(\varphi\in AP(\mathbb{R})\) if and only if \(\{\varphi(n)\}_{n\in Z}\in AP(Z)\).
Later, if \(f\in PAP_0(\mathbb{R}^+\times \Omega)\) in equation (1) satisfying a Lipschitz condition on \(\Omega\), and \(\varphi(t)\) is a solution to (1) with \(\varphi(\mathbb{R})^{k+2}\subset\Omega\), then \(\varphi\in AAP_0(\mathbb{R}^+)\) if and only if \(\{\varphi(n)\}_{n\in Z}\in PAP_0(Z^+)\).
Another result states that if \(f\in PAP(\mathbb{R}\times \Omega_0)\) \((AAP(\mathbb{R}^+\times \Omega_0))\) 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 \(\varphi(\mathbb{R})^{k+2}\subset \Omega_0\), then \(\varphi\in PAP(\mathbb{R})\) \((AAP(\mathbb{R}^+))\) if and only if \(\{\varphi(n)\}_{n\in Z}\in PAP(Z)\) \((AAP(Z^+))\).