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Oscillation criteria for second-order nonlinear delay dynamic equations. (English) Zbl 1125.34046
The authors consider the second-order nonlinear delay dynamic equation $$\left(r(t)x^\Delta(t)\right)^\Delta +p(t)f(x(\tau(t))=0$$ on a time scale. By employing a generalized Riccati transformation of the form $$w(t):= \delta(t)\left[\frac{r(t)x^\Delta(t)}{x(t)} +r(t)a(t)\right],$$ they establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. The obtained results improve the well-known oscillation results for dynamic equations and include as special cases the oscillation results for differential equations. Some applications to special time scales $R, N, q^{N_{0}}$ with $q>1$ and four examples are also included to illustrate the main results.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 39A10 Additive difference equations
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##### References:
 [1] Agarwal, R. P.; Bohner, M.; O’regan, D.; Peterson, A.: Dynamic equations on time scales: A survey. J. comput. Appl. math. 141, No. 1 -- 2, 1-26 (2002) · Zbl 1020.39008 [2] 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 [3] Agarwal, R. P.; O’regan, D.; Saker, S. H.: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. J. math. Anal. appl. 300, 203-217 (2004) · Zbl 1062.34068 [4] Agarwal, R. P.; O’regan, D.; Saker, S. H.: Oscillation criteria for nonlinear perturbed dynamic equations of second-order on time scales. J. appl. Math. comput. 20, 133-147 (2006) · Zbl 1089.39001 [5] R.P. Agarwal, D. O’Regan, S.H. Saker, Properties of bounded solutions of nonlinear dynamic equations on time scales, Can. Appl. Math. Q., in press [6] E. Akin Bohner, M. Bohner, S.H. Saker, Oscillation criteria for a certain class of second order Emden -- Fowler dynamic equations, Electron. Trans. Numer. Anal., in press · Zbl 1177.34047 [7] Bohner, M.; Peterson, A.: Dynamic equations on time scales: an introduction with applications. (2001) · Zbl 0978.39001 [8] Bohner, M.; Saker, S. H.: Oscillation of second order nonlinear dynamic equations on time scales. Rocky mountain J. Math. 34, 1239-1254 (2004) · Zbl 1075.34028 [9] Bohner, M.; Saker, S. H.: Oscillation criteria for perturbed nonlinear dynamic equations. Math. comp. Modelling 40, 249-260 (2004) · Zbl 1112.34019 [10] Erbe, L.: Oscillation criteria for second order linear equations on a time scale. Can. appl. Math. Q. 9, 1-31 (2001) · Zbl 1050.39024 [11] Erbe, L.; Peterson, A.: Riccati equations on a measure chain. Proc. dyn. Syst. appl. 3, 193-199 (2001) · Zbl 1008.34006 [12] Erbe, L.; Peterson, A.: Boundedness and oscillation for nonlinear dynamic equations on a time scale. Proc. amer. Math. soc. 132, 735-744 (2004) · Zbl 1055.39007 [13] Erbe, L.; Peterson, A.; Saker, S. H.: Oscillation criteria for second-order nonlinear dynamic equations on time scales. J. London math. Soc. 67, 701-714 (2003) · Zbl 1050.34042 [14] Erbe, L.; Peterson, A.; Saker, S. H.: Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. J. comput. Appl. math. 181, 92-102 (2005) · Zbl 1075.39010 [15] Erbe, L.; Peterson, A.; Saker, S. H.: Kamenev-type oscillation criteria for second-order linear delay dynamic equations. Dynam. systems appl. 15, 65-78 (2006) · Zbl 1104.34026 [16] Hilger, S.: Analysis on measure chains --- a unified approach to continuous and discrete calculus. Results math. 18, 18-56 (1990) · Zbl 0722.39001 [17] Li, H. J.: Oscillation criteria for second order linear differential equations. J. math. Anal. appl. 194, 312-321 (1995) · Zbl 0829.34060 [18] Saker, S. H.: New oscillation criteria for second-order nonlinear dynamic equations on time scales. Nonlinear funct. Anal. appl. 11, 351-370 (2006) · Zbl 1126.34024 [19] Saker, S. H.: Oscillation of nonlinear dynamic equations on time scales. Appl. math. Comput. 148, 81-91 (2004) · Zbl 1045.39012 [20] 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 [21] S.H. Saker, Boundedness of solutions of second-order forced nonlinear dynamic equations, Rocky Mountain J. Math., in press · Zbl 1139.34030 [22] Saker, S. H.: Oscillation of second-order forced nonlinear dynamic equations on time scales. Electron. J. Qual. theory differ. Equ. 23, 1-17 (2005) · Zbl 1097.34027 [23] Sahiner, Y.: Oscillation of second-order delay differential equations on time scales. Nonlinear anal. 63, 1073-1080 (2005) [24] 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