×

Oscillation of solutions to impulsive dynamic equations on time scales. (English) Zbl 1179.34108

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).

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
PDFBibTeX XMLCite
Full Text: EuDML Link