×

Oscillation of third order nonlinear delay dynamic equations on time scales. (English) Zbl 1175.34086

Summary: This paper gives oscillation criteria for the third order nonlinear delay dynamic equation
\[ \left(a(t)\left\{\left[r(t)x^\Delta(t)\right]^\Delta\right\}^\gamma\right)^\Delta+f(t,x(\tau(t)))=0 \]
on a time scale \(\mathbb T\) where \(\gamma\geq 1\) is the quotient of odd positive integers, \(a\) and \(r\) are positive \(rd\)-continuous functions on \(\mathbb T\), and the so-called delay function \(\tau:\mathbb T\to\mathbb T\) satisfies \(\tau(t)\leq t\) for \(t\in\mathbb T\) and \(\lim_{t\to\infty}\tau(t)=\infty\) and \(f\in C(\mathbb T\times \mathbb R,\mathbb R)\). Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when \(\mathbb T=\mathbb R\) and \(\mathbb T=\mathbb N\) involve and improve some oscillation results for third order delay differential and difference equations; when \(\mathbb T=h\mathbb N\), \(\mathbb T=q^{\mathbb N_0}\) and \(\mathbb T=\mathbb N^2\) our oscillation results are essentially new. Some examples are given to illustrate the main results.

MSC:

34K11 Oscillation theory of functional-differential equations
39A10 Additive difference equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hilger, S., Analysis on measure chains — a unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990) · Zbl 0722.39001
[2] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications (2001), Birkhäuser: Birkhäuser Boston · Zbl 0978.39001
[3] Kac, V.; Chueng, P., Quantum Calculus, Universitext (2002)
[4] Bohner, M.; Peterson, A., Advances in Dynamic Equations on Time Scales (2003), Birkhäuser: Birkhäuser Boston · Zbl 1025.34001
[5] Došlý, O.; Hilger, E.; Agarwal, P. P.; Bohner, M.; O’Regan, D., A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, Dynamic Equations on Time Scales. Dynamic Equations on Time Scales, J. Comput. Appl. Math., 141, 1-2, 571-585 (2002), (special issue)
[6] Erbe, L.; Hassan, T. S.; Peterson, A., Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. Math. Comput., 203, 343-357 (2008) · Zbl 1162.39005
[7] L. Erbe, T.S. Hassan, A. Peterson, Oscillation criteria for nonlinear functional neutral dynamic equations on time scales, J. Difference. Equ. Appl. (in press); L. Erbe, T.S. Hassan, A. Peterson, Oscillation criteria for nonlinear functional neutral dynamic equations on time scales, J. Difference. Equ. Appl. (in press) · Zbl 1193.34135
[8] Erbe, L.; Hassan, T. S.; Peterson, A.; Saker, S. H., Oscillation criteria for half-linear delay dynamic equations on time scales, Nonlinear Dyn. Syst. Theory, 9, 1, 51-68 (2009) · Zbl 1173.34037
[9] L. Erbe, T.S. Hassan, A. Peterson, S.H. Saker, Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, Int. J. Diff. Equ. (in press); L. Erbe, T.S. Hassan, A. Peterson, S.H. Saker, Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, Int. J. Diff. Equ. (in press) · Zbl 1173.34037
[10] Hassan, T. S., Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl., 345, 176-185 (2008) · Zbl 1156.34022
[11] Erbe, L.; Peterson, A.; Saker, S. H., Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, J. Compu. Appl. Math., 181, 92-102 (2005) · Zbl 1075.39010
[12] Erbe, L.; Peterson, A.; Saker, S. H., Hille and Nehari type criteria for third order dynamic equations, J. Math. Anal. Appl., 329, 112-131 (2007) · Zbl 1128.39009
[13] Erbe, L.; Peterson, A.; Saker, S. H., Oscillation and asymptotic behavior a third-order nonlinear dynamic equation, Can. Appl. Math. Q., 14, 2, 129-147 (2006) · Zbl 1145.34329
[14] Gera, M.; Graef, J. R.; Gregus, M., On oscillatory and asymptotic properties of solutions of certain nonlinear third order differential equations, Nonlinear Anal., 32, 417-425 (1998) · Zbl 0945.34021
[15] Hardy, G. H.; Littlewood, J. E.; Polya, G., Inequalities (1988), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0634.26008
[16] Baculíková, B.; Elabbasy, E. M.; Saker, S. H.; Dž urina, J., Oscillation criteria for third order nonlinear differential equations, Math. Slovaka, 58, 201-220 (2008) · Zbl 1174.34052
[17] Zhang, B. G.; Deng, X., Oscillation of delay differential equations on time scales, Math. Comput. Modelling, 36, 1307-1318 (2002) · Zbl 1034.34080
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.