×

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
PDFBibTeX XMLCite
Full Text: DOI

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
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.