## 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
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### References:

 [1] R. Agarwal, M. Bohner, Basic calculs on time scales and some of its applications, Preprint; R. Agarwal, M. Bohner, Basic calculs on time scales and some of its applications, Preprint · Zbl 0927.39003 [2] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001 [3] Erbe, L., Oscillation theorems for second order linear differential equations, Pacific J. Math., 35, 337-343 (1970) · Zbl 0185.15903 [4] Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differential Equations Dynam. Systems, 1, 223-246 (1993) · Zbl 0868.39007 [5] 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 [6] Fink, A. M.; Mary, D. F.St., A generalized Sturm comparison theorem and oscillation coefficients, Monatsh. Math., 73, 207-212 (1969) · Zbl 0182.12301 [7] Hilger, S., Analysis on measure chains-a unified approach to continuous and discrete calculus, Res. Math., 18, 18-56 (1990) · Zbl 0722.39001 [8] Kelley, W.; Peterson, A., Difference Equations: An Introduction with Applications (1991), Academic Press · Zbl 0733.39001 [9] Kwong, M. K., On certain comparison theorems for second order linear oscillation, Proc. Amer. Math. Soc., 84, 539-542 (1982) · Zbl 0494.34022 [10] P. Řehák, Half-linear dynamic equations on time scales: IVP and oscillatory properties, J. Nonlinear Funct. Anal. Appl., in print; P. Řehák, Half-linear dynamic equations on time scales: IVP and oscillatory properties, J. Nonlinear Funct. Anal. Appl., in print · Zbl 1037.34002
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