Oscillation of fourth-order delay dynamic equations
The authors consider the fourth order delay dynamic equation \[ \bigg(c(t)\,\big(b(t)\,(a(t)\,x^\Delta(t))^\Delta\big)^\Delta\bigg)^\Delta+p(t)\,x(\tau(t))=0 \tag{*} \] on a time scale \(\mathbb{T}\), which is unbounded above. The coefficients \(a(t)\), \(b(t)\), \(c(t)\) are positive and rd-continuous and the delay \(\tau(t)\) is rd-continuous from \(\mathbb{T}\) to \(\mathbb{T}\) with \(\tau(t)\leq t\) and \(\tau(t)\to\infty\) as \(t\to\infty\). The authors derive several new oscillation criteria for equation (\(*\)), which improve the existing results on this subject. The paper contains interesting comparison of the current oscillation results for various nonlinear differential, difference, and dynamic equations. Four particular examples show the application of the main results in the situations, where the known criteria cannot be applied. Reviewer's remark: The reviewer believes that the assumption on the positivity of the rd-continuous coefficients \(a(t)\), \(b(t)\), \(c(t)\) should be strengthened to the condition \[ \inf_{t\in[t_0,t_1]_{\mathbb{T}}} r(t)>0 \quad\text{for all }\;t_1\in[t_0,\infty)_{\mathbb{T}}, \quad r(\cdot)\in\{a(\cdot),\,b(\cdot),\,c(\cdot)\}. \tag{**} \] Condition (\(**\)) guarantees the rd-continuity of the functions \(1/r(t)\), hence it guarantees the existence of the integrals involving \(1/r(t)\), see also [\textit{R. Hilscher} and \textit{C. C. Tisdell}, Electron. J. Differ. Equ. 2008, Paper No. 68, 21 p. (2008; Zbl 1176.34059)]. The paper will be useful for researchers in the oscillation theory of differential, difference, and dynamic equations.
scientific article; zbMATH DE number 6419694
scientific article