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Comparison and oscillatory behavior for certain second order nonlinear dynamic equations. (English) Zbl 1219.34115
The authors consider the second order nonlinear dynamic equation $$\left(a(x^{\Delta})^{\alpha}\right)^{\Delta}(t)+q(t)x^{\beta}(t)=0$$ on an arbitrary time scale $\Bbb T$, where $\alpha$ and $\beta$ are ratios of positive odd integers, $a$ and $q$ are positive rd-continuous functions on $\Bbb T$. They establish comparison results with the inequality $$\left(a(x^{\Delta})^{\alpha}\right)^{\Delta}(t)+q(t)x^{\beta}(t)\leq 0$$ which are applied to neutral equations. A necessary and sufficient condition is obtained for the oscillation property of second order equations on time scales.

34N05Dynamic equations on time scales or measure chains
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text: DOI
[1] Agarwal, R.P., Grace, S.R., O’Regan, D.: On the oscillation of certain second order difference equations. J. Differ. Equ. Appl. 9, 109--119 (2003) · Zbl 1039.39003
[2] Agarwal, R.P., Bohner, M., Saker, S.H.: Oscillation of second order delay dynamic equations. Can. Appl. Math. Q. 13, 1--17 (2005) · Zbl 1126.39003
[3] Akin-Bohner, E., Bohner, M., Saker, S.: Oscillation criteria for a certain class of second order Emden--Fowler dynamic equations. Electron. Trans. Numer. Anal. 27, 1--12 (2007) · Zbl 1177.34047
[4] Bohner, M., Saker, S.: Oscillation criteria for perturbed nonlinear dynamic equations. Math. Comput. Model. 40, 249--260 (2004) · Zbl 1112.34019 · doi:10.1016/j.mcm.2004.03.002
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[9] Grace, S.R., Bohner, M., Agarwal, R.P.: On the oscillation of second--order half-linear dynamic equations. J. Differ. Equ. Appl. 15, 451--460 (2009) · Zbl 1170.34023 · doi:10.1080/10236190802125371
[10] Hilger, S.: Analysis on measure chains: A unified approach to continuous and discrete calculus. Results Math. 18, 18--56 (1990) · Zbl 0722.39001
[11] Zhou, X., Yan, J.: Oscillatory and asymptotic properties of higher order nonlinear difference equations. Nonlinear Anal. 31, 493--502 (1998) · Zbl 0887.39003 · doi:10.1016/S0362-546X(97)00417-3
[12] Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equation. Kluwer Academic, Dordrecht (2000)
[13] Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht (2002) · Zbl 1091.34518
[14] Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Second Order Dynamic Equations. Taylor & Francis, London (2003) · Zbl 1043.34032
[15] Bohner, M., Peterson: Dynamic Equations on Time Scales, an Introduction with Applications. Birkhäuser, Boston (2001) · Zbl 0978.39001
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[17] Agarwal, R.P., Bohner, M., Rehak, P.: Half-linear dynamic equations. In: Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday, pp. 1--57. Kluwer Academic, Dordrecht (2003)
[18] Agarwal, R.P., Bohner, M., Grace, S.R., O’Regan, D.: Discrete Oscillation Theory. Hindawi (2005)