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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 \(\mathbb T\), where \(\alpha\) and \(\beta\) are ratios of positive odd integers, \(a\) and \(q\) are positive rd-continuous functions on \(\mathbb T\). 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.

MSC:

34N05 Dynamic equations on time scales or measure chains
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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References:

[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
[5] Bohner, M., Saker, S.: Oscillation of second order nonlinear dynamic equations on time scales. Rocky Mt. J. Math. 34, 1239–1254 (2004) · Zbl 1075.34028
[6] Erbe, L.: Oscillation criteria for second order linear equations on a time scale. Can. Appl. Math. Q. 9, 345–375 (2001) · Zbl 1050.39024
[7] Erbe, L., Peterson, A., Rehak, P.: Comparison theorems for linear dynamic equations on time scales. J. Math. Anal. Appl. 275, 418–438 (2002) · Zbl 1034.34042
[8] Grace, S.R., Agarwal, R.P., Bohner, M., O’Regan, D.: Oscillation of second–order strongly superlinear and strongly sublinear dynamic equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3463–3471 (2009) · Zbl 1221.34083
[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
[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
[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
[16] Bohner, M., Peterson: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003) · Zbl 1025.34001
[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)
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