Nonoscillation of Mathieu's equation whose coefficient is a finite Fourier series approximating a square wave
In this paper, a parametric non-oscillation region is given for the Mathieu-type equation \[ x''+(-\alpha+\beta c(t))x=0,\qquad\alpha,\beta\in{\mathbb R},\tag{1} \] where \(c(t)\) is periodic with mean value zero, i.e., \[ \int_0^\omega c(t){\mathrm d}t=0\qquad(\omega\quad\text{is the period of}\quad c(t)). \] If \(\alpha=0\) and \(\beta\neq0\), then all non-trivial solutions (1.1) are oscillatory. It follows from Sturm's comparison theorem that if \(\alpha<0\) and \(\beta\neq0\), then all non-trivial solutions (1.1) are oscillatory. It is clear that {\parindent=0.7cm \begin{itemize}\item[{\(\bullet\)}] if \(\alpha<0\) and \(\beta=0\), then all non-trivial solutions (1.1) are oscillatory. \item[{\(\bullet\)}] if \(\alpha=\beta=0\), then all non-trivial solutions (1.1) are non-oscillatory. \end{itemize}} Thus, a necessary condition for all non-trivial solutions of (1.1) to be non-oscillatory is that \(\alpha\geq0\). Oscillation problem about a kind of Meissner's equation (for which \(c(t)\) is a piecewise constant function) is also discussed. The obtained result is proved by using Sturm's comparison theorem and phase plane analysis of the second-order equation \[ y''+a(t)y'+b(t)y=0, \] where \(a,b\in\mathrm C[0,\infty)\). Parametric non-oscillation region about \((\alpha,\beta)\) and some solution orbits are drawn to help to understand the result.
scientific article; zbMATH DE number 6920375
scientific article