Comparison theorems for linear dynamic equations on time scales. (English) Zbl 1034.34042

Erbe, Lynn; Peterson, Allan; Řehák, Pavel

J. Math. Anal. Appl. 275, No. 1, 418-438 (2002).

The authors study comparison theorems for second-order linear dynamic equations on a time scale \[ \begin{aligned} [p(t)x^{\Delta}(t)]^{\Delta} &+ q(t)x^{\sigma}(t)=0, \tag{1}\\ [p(t)y^{\Delta}(t)]^{\Delta} &+ a^{\sigma}(t)q(t)y^{\sigma}(t)=0,\tag{2}\\ [p(t)z^{\Delta}(t)]^{\Delta} &+ a(t)q(t)z^{\sigma}(t)=0,\tag{3} \end{aligned} \] where \(p(t)>0\) and \(p,q,a\) are right-dense continuous on \(\mathbb{T}\). Three different comparison theorems are presented along with their coresponding corollaries, and, by examples, it is shown that they are all independent. A typical result is the following theorem: If \(a\in C^1_{rd}\), \(\liminf_{t\to\infty} \int_T^t q(s)\,\Delta s\geq 0\) but not identically zero for large \(T\), \(\int^\infty\frac{\Delta s}{p(s)}=\infty\) and \(0<a(t)\leq 1\), \(a^{\Delta}(t)\leq 0\). Then (1) is nonoscillatory implies (3) is nonoscillatory.
The obtained results extend comparison theorems for the continuous case and provide some new results in the discrete case.

Reviewer: Zuzana Došlá (Brno)

MSC:

34C11	Growth and boundedness of solutions to ordinary differential equations
39A12	Discrete version of topics in analysis

Keywords:

linear dynamic equation; oscillation; nonoscillation; comparison theorems

References:

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