Li, Qiaoluan; Guo, Fang Oscillation of solutions to impulsive dynamic equations on time scales. (English) Zbl 1179.34108 Electron. J. Differ. Equ. 2009, Paper No. 122, 7 p. (2009). Summary: We consider the system\[ (a(t)x^\Delta(t))^\Delta + p(t)x(\sigma(t)) = 0,\quad t\in\mathbb J_{\mathbb T} := [t_0,\infty) \cap\mathbb T,\quad t\neq t_k,\;k = 1,2,\dots, \]\[ x(t^+_k) = b_kx(t_k),\quad x^\Delta(t^+_k)= c_kx^\Delta(t_k),\quad k = 1,2,\dots,\tag{1} \]\[ x(t^+_0) = x(t_0),\quad x^\Delta(t^+_0) = x_(t_0), \]where \(\mathbb T\) is a time scales, unbounded-above, with \(t_k\in\mathbb T\), \(0\leq t_0 < t_1 < t_2 <\cdots <t_k <\cdots, \lim_{k\to\infty}t_k =\infty\) and \(y(t^+_k) = \lim_{h\to 0^+} y(t_k+h)\), \(y_(t^+_k) = \lim_{h\to 0^+} y_(t_k+h)\), which represent right limits of \(y(t)\), \(y^\Delta(t)\) at \(t = t_k\) in the sense of time scales. We can define \(y(t^-_k)\), \(y_(t^-_k)\) similarly.In this paper, we assume that \(a\in C_{\text{rd}}(\mathbb T,\mathbb R^+)\), \(p\in C_{\text{rd}}(\mathbb T,\mathbb R^+)\), \(b_k > 0\), \(c_k > 0\), \(d_k =\frac{c_k}{b_k}\), \(t_k\) are right dense, where \(C_{\text{rd}}\) denotes the set of rd-continuous functions, \(\sigma(t) := \inf\{s\in\mathbb T: s > t\}\), \(\mathbb R^+ = \{x : x > 0\}\).Using Riccati transformation techniques, we obtain some conditions for the oscillation of all solutions of (1). Cited in 2 Documents MSC: 34N05 Dynamic equations on time scales or measure chains 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A37 Ordinary differential equations with impulses Keywords:oscillation; time scales; impulsive dynamic equations PDFBibTeX XMLCite \textit{Q. Li} and \textit{F. Guo}, Electron. J. Differ. Equ. 2009, Paper No. 122, 7 p. (2009; Zbl 1179.34108) Full Text: EuDML Link