×

A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales. (English) Zbl 1009.34033

This is a very nice paper about oscillatory properties of the second-order dynamic equation \[ ( r(t)x^\Delta)^\Delta+c(t)x^\sigma=0 \tag{1} \] on an arbitrary time scale \({\mathbb{T}}\). Equation (1) contains as special cases the well known second-order Sturm-Liouville differential \((\mathbb{T}= \mathbb{R})\) and difference \((\mathbb{T}= \mathbb{Z})\) equations. A necessary and sufficient condition for the oscillation of equation (1) is established by transforming equation (1) into a scalar trigonometric system (other terminology is a self-reciprocal system). The classification of (1) to be oscillatory/nonoscillatory makes sense, since the time scales Sturmian separation theorem holds for equation (1). The main tool for the proof is the time scales trigonometric transformation. This transformation preserves the oscillatory behavior of transformed systems and generalizes the corresponding continuous-time \((\mathbb{T}= \mathbb{R})\) and discrete-time \((\mathbb{T}= \mathbb{Z})\) trigonometric transformations (the latter one obtained by M. Bohner and the first author [J. Differ. equations 163, No. 1, 113-129 (2000; Zbl 0956.39011)]). A further oscillation criterion for equation (1) is obtained via the Riccati technique.
This paper will be useful for researchers interested in (non)oscillatory behavior of differential, difference, and/or dynamic equations.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B24 Sturm-Liouville theory
39A10 Additive difference equations
39A11 Stability of difference equations (MSC2000)
39A12 Discrete version of topics in analysis

Citations:

Zbl 0956.39011
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Agarwal, R. P.; Bohner, M., Quadratic functionals for second order matrix equations on time scales, Nonlinear Anal., 33, 675-692 (1998) · Zbl 0938.49001
[2] Ahlbrandt, C. D.; Clark, S. L.; Hooker, J. W.; Patula, W. T., A discrete interpretation of Reid’s Roundabout theorem for generalized differential systems, Comput. Math. Appl., 28, 11-22 (1994) · Zbl 0805.93024
[3] Bohner, M., Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl., 199, 804-826 (1996) · Zbl 0855.39018
[4] Borůvka, O., Linear Differential Transformations of the Second Order (1971), Oxford University Press: Oxford University Press London · Zbl 0222.34002
[5] O. Došlý, Trigonometric transformation and oscillatory properties of second order difference equation, Proceedings of the Fourth International Conference on Difference Equations, Poznan, Poland, 27-31 August 1998, pp. 125-133, Gordon and Breach, Amsterdam 2000.; O. Došlý, Trigonometric transformation and oscillatory properties of second order difference equation, Proceedings of the Fourth International Conference on Difference Equations, Poznan, Poland, 27-31 August 1998, pp. 125-133, Gordon and Breach, Amsterdam 2000. · Zbl 0988.39009
[6] O. Došlý, R. Hilscher, Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales, J. Difference Equations Appl. 7 (2001) 265-295.; O. Došlý, R. Hilscher, Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales, J. Difference Equations Appl. 7 (2001) 265-295. · Zbl 0989.34027
[7] Erbe, L.; Hilger, S., Sturmian theory on measure chains, Differential Equations Dynamic Systems, 1, 223-246 (1993) · Zbl 0868.39007
[8] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Res. Math., 18, 18-56 (1990) · Zbl 0722.39001
[9] Hilger, S., Differential and difference calculus—unified!, Nonlinear Anal. TMA, 30, 2683-2694 (1997) · Zbl 0927.39002
[10] Hilger, S., Special functions, Laplace and Fourier transform on measure chains, Dynamic System Appl., 8, 471-488 (1999) · Zbl 0943.39006
[11] R. Hilscher, A unified approach to continuous and discrete linear Hamiltonian systems via the calculus on time scales, in: Ulmer Seminäre, Vol. 3, University of Ulm, 1998, pp. 215-249.; R. Hilscher, A unified approach to continuous and discrete linear Hamiltonian systems via the calculus on time scales, in: Ulmer Seminäre, Vol. 3, University of Ulm, 1998, pp. 215-249.
[12] Lakshmikantham, V.; Sivasundaram, S.; Kaymakcalan, B., Dynamic Systems on Measure Chains (1996), Kluwer Academic Publ: Kluwer Academic Publ Dordrecht, Boston, London · Zbl 0869.34039
[13] Mingarelli, A. B., Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions. Volterra-Stieltjes Integral Equations and Generalized Ordinary Differential Expressions, Lectures Notes in Mathematics, Vol. 989 (1983), Springer: Springer Berlin, Heidelberg, New York, Tokyo · Zbl 0516.45012
[14] Ráb, M., Kriterien für oscillation der Lösungen der Differentialgleichung \([p(x)y\)′]′+\(q(x)y=0\), Časopsis Pěst. Mat., 84, 335-370 (1957) · Zbl 0087.29505
[15] Reid, W. T., Generalized linear differential systems and related Riccati matrix integral equation, Illinois J. Math., 10, 701-722 (1966) · Zbl 0199.14001
[16] Swanson, C. A., Comparison and Oscillation Theorems for Linear Differential Equations (1968), Academic Press: Academic Press London · Zbl 0191.09904
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.