Non-oscillation criterion for generalized Mathieu-type differential equations with bounded coefficients

Abstract:

The following equation is considered in this work: \begin{equation*} x'' + (-\alpha + \beta \cos (\rho t) + f(t))x = 0, \end{equation*} where the parameters $\alpha$ and $\beta$ are real numbers, the frequency $\rho$ is a positive real number, and $f\!: [0,\infty ) \to \mathbb {R}$ is a continuous bounded function, i.e., there exists a positive constant $f^*$ such that $|f(t)| \le f^*$ for $t\ge 0$. This equation is generally referred to as the Mathieu equation when $f^*=0$. This work proposes a non-oscillation theorem that can be applied even if $f^* \neq 0$. The required conditions are expressed by the parameters $\alpha$, $\beta$, $\rho$, and a positive constant $f^*$. The results obtained herein include those by Sugie and Ishibashi [Appl. Math. Comput. 346 (2019), pp. 491–499]. Further, the result can be proved using the phase plane analysis proposed by Sugie [Monatsh. Math. 186 (2018), pp. 721–743]. Finally, the simple non-oscillation and oscillation conditions of the generalized Mathieu equation are summarized.