Oscillation of second-order delay dynamic equations on time scales. (English) Zbl 1180.34069

Sun, Shurong; Han, Zhenlai; Zhang, Chenghui

J. Appl. Math. Comput. 30, No. 1-2, 459-468 (2009); erratum ibid. 39, No. 1-2, 551-554 (2012).

Summary: By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations
\[ \bigl(p(t)\bigl(x^{\Delta}(t)\bigr)^{\gamma}\bigr)^{\Delta}+q(t)f\bigl(x\bigl(\tau(t)\bigr)\bigr)=0 \]
on a time scale \(\mathbb{T}\), here \(\gamma \geq 1\) is a quotient of odd positive integers with \(p\) and \(q\) real-valued positive rd-continuous functions defined on \(\mathbb{T}\). Our results improve and extend some results established by S. H. Saker [J. Comput. Appl. Math. 177, No. 2, 375–387 (2005; Zbl 1082.34032)] but also unify the oscillation of the second order nonlinear delay differential equation and the second order nonlinear delay difference equation.

MSC:

34K11	Oscillation theory of functional-differential equations
34N05	Dynamic equations on time scales or measure chains
39A10	Additive difference equations

Keywords:

oscillation; second order; delay dynamic equations; time scale

Citations:

Zbl 1082.34032

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]	Agarwal, R.P., Bohner, M., O’Regan, D., Peterson, A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141, 1–26 (2002) · Zbl 1020.39008
[3]	Bohner, M., Peterson, A.: Dynamic Equations on Time Scales, an Introduction with Applications. Birkhauser, Boston (2001) · Zbl 0978.39001
[4]	Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhauser, Boston (2003) · Zbl 1025.34001
[5]	Bohner, M., Saker, S.H.: Oscillation of second order nonlinear dynamic equations on time scales. Rocky Mt. J. Math. 34, 1239–1254 (2004) · Zbl 1075.34028
[6]	Erbe, L.H.: Oscillation results for second order linear equations on a time scale. J. Differ. Equ. Appl. 8, 1061–1071 (2002) · Zbl 1021.34012
[7]	Saker, S.H.: Oscillation criteria of second-order half-linear dynamic equations on time scales. J. Comput. Appl. Math. 177, 375–387 (2005) · Zbl 1082.34032
[8]	Agarwal, R.P., Bohner, M., Saker, S.H.: Oscillation of second order delay dynamic equations. Can. Appl. Math. Q. 13, 1–18 (2005) · Zbl 1126.39003
[9]	Erbe, L., Peterson, A., Saker, S.H.: Oscillation criteria for second-order nonlinear delay dynamic equations. J. Math. Anal. Appl. 333, 505–522 (2007) · Zbl 1125.34046
[10]	Han, Z., Sun, S., Shi, B.: Oscillation criteria for a class of second order Emden-Fowler delay dynamic equations on time scales. J. Math. Anal. Appl. 334, 847–858 (2007) · Zbl 1125.34047
[11]	Sahiner, Y.: Oscillation of second-order delay differential equations on time scales. Nonlinear Anal. TMA 63, 1073–1080 (2005) · Zbl 1224.34294
[12]	Zhang, B.G., Shanliang, Z.: Oscillation of second order nonlinear delay dynamic equations on time scales. Comput. Math. Appl. 49, 599–609 (2005) · Zbl 1075.34061