Partially extended oscillation and nonoscillation theorems for half‐linear Hill‐type differential equations with periodic damping

Abstract

This study addressed the oscillation problems of half‐linear differential equations with periodic damping. The solution space of any linear equation is homogeneous and additive. Generally, by contrast, the solution space of half‐linear differential equations is homogeneous but not additive. Numerous oscillation and nonoscillation theorems have been devised for half‐linear differential equations featuring periodic functions as coefficients. However, in certain cases, such as applying Mathieu‐type differential equations to control engineering, which is a typical example of the Hill equation, some oscillation theorems cannot be applied. In this study, we established oscillation and nonoscillation theorems for half‐linear Hill‐type differential equations with periodic damping. To prove the results, we used the Riccati technique and the composite function method, which focuses on the composite function of the indefinite integral of the coefficients of the target equation and an appropriate multivalued continuously differentiable function. Furthermore, we discuss the special case of the oscillation constant of a damped half‐linear Mathieu equation.