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Oscillation and nonoscillation of Hill’s equation with periodic damping. (English) Zbl 1039.34026

In this elegant and well-written paper, the authors study the second-order linear differential equation with damping \[ y^{\prime\prime}+p(t)y^{\prime}+q(t)=0,\qquad t\geq0,\tag{1} \] where \(p(t)\) and \(q(t)\) are continuous periodic functions of period \(T.\) It is a well-known fact that the second-order linear differential equation \[ u^{\prime\prime}+q(t)u=0, \tag{2} \] where \(q(t)\) is of mean value zero and does not vanish identically, is oscillatory. The main question raised in the paper is whether equation (2) is also oscillatory if \(p(t)\) does not vanish identically and is periodic with mean value zero. Two results on nonoscillation and oscillation with easy verifiable conditions are established. Interesting examples illustrating the results are discussed. The paper concludes with three open problems regarding possible extensions of the theorems proved in the paper.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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