On the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument
It is shown that \[ x''(t)+ p(t) x(t)= q x([t])+ f(t)\tag{\(*\)} \] has a unique almost-periodic (ap) solution \(x\) on \(\mathbb{R}\), with Bohr spectrum \(\sigma_B(x)\subset \sigma_B(f)+ 2\pi\mathbb{Z}\), provided \(p:\mathbb{R}\to \mathbb{R}\) is continuous with period \(1\), \(0\neq q\) and real, \(f\) ap, \(q \int^1_0 v(s)\,ds\neq 1\) with \(v''+ pv= 0= v(0)\), \(v'(0)= 1\), and the roots \(z\) of a certain quadratic equation satisfy \(|z|\neq 1\). So, \(X\) is quasiperiodic if \(f\) is. If \(f\) has a rational period \(p/q\), then \(x\) has period \(q\). If \(f\) has an irrational period \(T\) with absolutely convergent Fourier series and \(f(-t)= -\overline{f(t)}\), then \(x\) is quasiperiodic with the basis \(1\), \(2\pi/T\). Similar results are obtained if in \((*)\) \(q x([t])\) is replaced by \(q x([t- 1])+ \mu g(t, x([t- 1,t])\) for small \(\mu\). Several explicit examples are discussed.
scientific article; zbMATH DE number 2126033