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

The following equation is considered in this work: x'' + (−α + β cos (ρ t) + ƒ(t))x = 0, where the parameters α and β are real numbers, the frequency ρ is a positive real number, and ƒ: [0,∞) → R is a continuous bounded function, i.e., there exists a positive constant ƒ* such that |ƒ(t)| ≤ ƒ* for t ≥ 0. This equation is generally referred to as the Mathieu equation when ƒ* = 0. This work proposes a non-oscillation theorem that can be applied even if ƒ* ≠ 0. The required conditions are expressed by the parameters α, β, ρ, and a positive constant ƒ*. 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. [ABSTRACT FROM AUTHOR]