## 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.

### MSC:

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

### References:

  Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001  Erbe, L., Oscillation theorems for second order linear differential equations, Pacific J. Math., 35, 337-343 (1970) · Zbl 0185.15903  Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differential Equations Dynam. Systems, 1, 223-246 (1993) · Zbl 0868.39007  Erbe, L. H.; Peterson, A., Green’s functions and comparison theorems for differential equations on measure chains, Dynam. Contin. Discrete Impuls. Systems, 6, 121-137 (1999) · Zbl 0938.34027  Fink, A. M.; Mary, D. F.St., A generalized Sturm comparison theorem and oscillation coefficients, Monatsh. Math., 73, 207-212 (1969) · Zbl 0182.12301  Hilger, S., Analysis on measure chains-a unified approach to continuous and discrete calculus, Res. Math., 18, 18-56 (1990) · Zbl 0722.39001  Kelley, W.; Peterson, A., Difference Equations: An Introduction with Applications (1991), Academic Press  Kwong, M. K., On certain comparison theorems for second order linear oscillation, Proc. Amer. Math. Soc., 84, 539-542 (1982) · Zbl 0494.34022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.