Nonoscillation and oscillation of second-order linear dynamic equations via the sequence of functions technique

Abstract

We study nonoscillation/oscillation of the dynamic equation $${(rx^\Delta)}^{\Delta}(t) + p(t)x(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$ where \({t_0 \in \mathbb{T}}\), \({{\rm sup} \mathbb{T} = \infty}\), \({r \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, \mathbb{R}^+)}\), \({p \in {\rm C}_{\rm rd}([t_0, \infty)_{\mathbb{T}}, {\mathbb{R}^+_0})}\). By using the Riccati substitution technique, we construct a sequence of functions which yields a necessary and sufficient condition for the nonoscillation of the equation. In addition, our results are new in the theory of dynamic equations and not given in the discrete case either. We also illustrate applicability and sharpness of the main result with a general Euler equation on arbitrary time scales. We conclude the paper by extending our results to the equation $${(rx^\Delta)}^{\Delta}(t) + p(t)x^\sigma(t)= 0 \quad {\rm for} t \in[t_0, \infty)_{\mathbb{T}},$$ which is extensively discussed on time scales.