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Oscillation of second order nonlinear dynamic equations on time scales. (English) Zbl 1075.34028
The authors consider the nonlinear second order dynamic equation \[ (p(t)x^\Delta)^\Delta+q(t)(f\circ x^\sigma)=0,\tag{1} \] where \(p\) and \(q\) are positive, real-valued continuous functions, and the nonlinearity \(f:\mathbb{R}\to\mathbb{R}\) satisfies the sign condition \(xf(x)>0\) and the superlinearity condition \(f(x)>K x\) for some \(K>0\) and every \(x\neq 0\). Two cases, depending on the convergence of the integral \[ \int _1^\infty\frac 1{p(t)}\Delta t\tag{2} \] are discussed separately. New sufficient conditions involving the integral over the coefficients of equation (1) which guarantee that all solutions are oscillatory (in the case when (2) is divergent) or either oscillatory or convergent to zero (in the case of convergence of the integral (2)) are derived. The sharpness of these criteria is shown on the example of the Euler dynamic equation. The authors’ main tool is the Riccati transformation.

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
39A12 Discrete version of topics in analysis
39A10 Additive difference equations
Full Text: DOI
[1] R.P. Agarwal, M. Bohner, D. O’Regan and A. Peterson, Dynamic equations on time scales : A survey , J. Comput. Appl. Math. 141 (2002), 1-26. · Zbl 1020.39008 · doi:10.1016/S0377-0427(01)00432-0
[2] E. Ak\in, L. Erbe, B. Kaymakçalan and A. Peterson, Oscillation results for a dynamic equation on a time scale , J. Differ. Equations Appl. 7 (2001), 793-810. · Zbl 1002.39024 · doi:10.1080/10236190108808303
[3] M. Bohner, O. Došlý and W. Kratz, An oscillation theorem for discrete eigenvalue problems , Rocky Mountain J. Math. 33 (2003), 1233-1260. · Zbl 1060.39003 · doi:10.1216/rmjm/1181075460
[4] M. Bohner and G.Sh. Guseinov, Improper integrals on time scales , Dynam. Systems Appl. 12 (2003), 45-66. · Zbl 1058.39011
[5] M. Bohner and A. Peterson, Dynamic equations on time scales : An introduction with applications , Birkhäuser, Boston, 2001. · Zbl 0978.39001
[6] O. Došlý and S. Hilger, A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales , J. Comput. Appl. Math. 141 (2002), 147-158. · Zbl 1009.34033 · doi:10.1016/S0377-0427(01)00442-3
[7] O. Došlý and R. Hilscher, Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales , J. Differ. Equations Appl. 7 (2001), 265-295. · Zbl 0989.34027 · doi:10.1080/10236190108808273
[8] L. Erbe and A. Peterson, Positive solutions for a nonlinear differential equation on a measure chain , in Boundary value problems and related topics , Math. Comput. Modelling 32 (2000), 571-585. · Zbl 0963.34020 · doi:10.1016/S0895-7177(00)00154-0
[9] ——–, Riccati equations on a measure chain , in Proceedings of dynamic systems and applications (G.S. Ladde, N.G. Medhin and M. Sambandham, eds.), Vol. 3, Dynamic Publishers, Atlanta, GA, 2001, pp. 193-199. · Zbl 1008.34006
[10] ——–, Oscillation criteria for second order matrix dynamic equations on a time scale , J. Comput. Appl. Math. 141 (2002), 169-185. · Zbl 1017.34030 · doi:10.1016/S0377-0427(01)00444-7
[11] G.Sh. Guseinov and B. Kaymakçalan, On a disconjugacy criterion for second order dynamic equations on time scales , J. Comput. Appl. Math. 141 (2002), 187-196. · Zbl 1014.34023 · doi:10.1016/S0377-0427(01)00445-9
[12] S. Hilger, Analysis on measure chains - A unified approach to continuous and discrete calculus , Results Math. 18 (1990), 18-56. · Zbl 0722.39001 · doi:10.1007/BF03323153
[13] I.V. Kamenev, An integral criterion for oscillation of linear differential equations of second order , Mat. Zametki 23 (1978), 249-251. · Zbl 0386.34032
[14] H.J. Li, Oscillation criteria for second order linear differential equations , J. Math. Anal. Appl. 194 (1995), 312-321. · Zbl 0836.34033 · doi:10.1006/jmaa.1995.1295
[15] G. Zhang and S.S. Cheng, A necessary and sufficient oscillation condition for the discrete Euler equation , Panamer. Math. J. 9 (1999), 29-34. · Zbl 0960.39005
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