Forced oscillation of second-order half-linear dynamic equations on time scales. (English) Zbl 1205.34134

Summary: We establish a new interval oscillation criterion for the second-order half-linear dynamic equation
\[ (r(t)[x^\Delta(t)]^\alpha)^\Delta+p(t)x^\alpha(\sigma(t))=f(t) \]
on a time scale \(\mathbb T\) which is unbounded. This criterion is an extension of the oscillation result for second order linear dynamic equation established by L. Erbe, A. Peterson and S. H. Saker [J. Difference Equ. Appl. 14, No. 10–11, 997–1009 (2008; Zbl 1168.34025)]. As an application, we obtain a sufficient condition for oscillation of the second-order half-linear differential equation
\[ ([x'(t)]^\alpha)'+c\sin tx^\alpha(t)=\cos t, \]
where \(\alpha=p/q\), \(p, q\) are odd positive integers.


34N05 Dynamic equations on time scales or measure chains
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34K11 Oscillation theory of functional-differential equations


Zbl 1168.34025
Full Text: DOI EuDML


[1] S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18-56, 1990. · Zbl 0722.39001 · doi:10.1007/BF03323153
[2] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0978.39001
[3] M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001
[4] M. Bohner and S. H. Saker, “Oscillation criteria for perturbed nonlinear dynamic equations,” Mathematical and Computer Modelling, vol. 40, no. 3-4, pp. 249-260, 2004. · Zbl 1112.34019 · doi:10.1016/j.mcm.2004.03.002
[5] M. Bohner and T. S. Hassan, “Oscillation and boundedness of solutions to first and second order forced functional dynamic equations with mixed nonlinearities,” Applicable Analysis and Discrete Mathematics, vol. 3, no. 2, pp. 242-252, 2009. · Zbl 1194.34174 · doi:10.2298/AADM0902242B
[6] R. P. Agarwal and A. Zafer, “Oscillation criteria for second-order forced dynamic equations with mixed nonlinearities,” Advances in Difference Equations, vol. 2009, Article ID 938706, 20 pages, 2009. · Zbl 1181.34099 · doi:10.1155/2009/938706
[7] L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for a forced second-order nonlinear dynamic equation,” Journal of Difference Equations and Applications, vol. 14, no. 10-11, pp. 997-1009, 2008. · Zbl 1168.34025 · doi:10.1080/10236190802332175
[8] V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002. · Zbl 0986.05001
[9] O. Do and P. , “Half-Linear Differential Equations,” North-Holland Mathematics Studies 202, Elsevier, Amsterdam, The Netherlands, 2005. · Zbl 1090.34001
[10] E. T. Wittaker and G. N. Watson, Modern Analysis, Cambridge University Press, New York, NY, USA, American edition, 1945.
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.