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Trench’s perturbation theorem for dynamic equations. (English) Zbl 1203.34151
Summary: We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and dominant solutions of the unperturbed equation. As the theory of time scales unifies continuous and discrete analysis, our results contain as special cases results for corresponding differential and difference equations by William F. Trench.

34N05Dynamic equations on time scales or measure chains
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A30Linear ODE and systems, general
Full Text: DOI EuDML
[1] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0978.39001
[2] M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001
[3] W. F. Trench, “Linear perturbations of a nonoscillatory second order equation,” Proceedings of the American Mathematical Society, vol. 97, no. 3, pp. 423-428, 1986. · Zbl 0594.34057 · doi:10.2307/2046231
[4] W. F. Trench, “Linear perturbations of a nonoscillatory second order differential equation. II,” Proceedings of the American Mathematical Society, vol. 131, no. 5, pp. 1415-1422, 2003. · Zbl 1032.34049 · doi:10.1090/S0002-9939-02-06682-0
[5] W. F. Trench, “Linear perturbations of a nonoscillatory second order difference equation,” Journal of Mathematical Analysis and Applications, vol. 255, no. 2, pp. 627-635, 2001. · Zbl 0984.39005 · doi:10.1006/jmaa.2000.7305
[6] V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002. · Zbl 0986.05001