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

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.


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