## Oscillation criteria for some new generalized Emden-Fowler dynamic equations on time scales.(English)Zbl 1277.34125

Summary: By means of novel analytical techniques, we establish several new oscillation criteria for the generalized Emden-Fowler dynamic equation on a time scale $$\mathbb T$$, that is, $(r(t)|Z^\Delta(t)|^{\alpha - 1}Z^\Delta(t))^\Delta + f(t, x(\delta(t))) = 0$ with respect to the case $$\int^\infty_{t_0} r^{-1/\alpha}(s)\Delta s = \infty$$ and the case $$\int^\infty_{t_0} r^{-1/\alpha}(s)\Delta s < \infty$$, where $$Z(t) = x(t) + p(t)x(\tau(t))$$, $$\alpha$$ is a constant, $$|f(t, u)| \geqslant q(t)|u^\beta|$$, $$\beta$$ is a constant satisfying $$\alpha \leqslant \beta > 0$$, and $$r$$, $$p$$, and $$q$$ are real valued right-dense continuous nonnegative functions defined on $$\mathbb T$$.

### MSC:

 34N05 Dynamic equations on time scales or measure chains 34K11 Oscillation theory of functional-differential equations
Full Text:

### References:

 [1] Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results in Mathematics, 18, 1-2, 18-56, (1990) · Zbl 0722.39001 [2] Agarwal, R. P.; O’Regan, D.; Saker, S. H., Oscillation criteria for second-order nonlinear neutral delay dynamic equations, Journal of Mathematical Analysis and Applications, 300, 1, 203-217, (2004) · Zbl 1062.34068 [3] Agarwal, R. P.; Bohner, M.; Saker, S. H., Oscillation of second order delay dynamic equations, The Canadian Applied Mathematics Quarterly, 13, 1, 1-18, (2005) · Zbl 1126.39003 [4] Saker, S. H.; Agarwal, R. P.; O’Regan, D., Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales, Applicable Analysis, 86, 1, 1-17, (2007) · Zbl 1128.34042 [5] Saker, S. H.; O’Regan, D.; Agarwal, R. P., Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales, Acta Mathematica Sinica, 24, 9, 1409-1432, (2008) · Zbl 1153.34040 [6] Saker, S., Oscillation criteria of second-order half-linear dynamic equations on time scales, Journal of Computational and Applied Mathematics, 177, 2, 375-387, (2005) · Zbl 1082.34032 [7] Saker, S. H., Oscillation of second-order nonlinear neutral delay dynamic equations on time scales, Journal of Computational and Applied Mathematics, 187, 2, 123-141, (2006) · Zbl 1097.39003 [8] Saker, S. H., Oscillation of second-order neutral delay dynamic equations of Emden-Fowler type, Dynamic Systems and Applications, 15, 3-4, 629-644, (2006) [9] Saker, S. H., Oscillation criteria for a certain class of second-order neutral delay dynamic equations, Dynamics of Continuous, Discrete & Impulsive Systems. Series B, 16, 3, 433-452, (2009) · Zbl 1180.34067 [10] Saker, S. H.; Grace, S. R., Oscillation criteria for quasi-linear functional dynamic equations on time scales, Mathematica Slovaca, 62, 3, 501-524, (2012) · Zbl 1324.34192 [11] Saker, S. H.; O’Regan, D., New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution, Communications in Nonlinear Science and Numerical Simulation, 16, 1, 423-434, (2011) · Zbl 1221.34245 [12] Saker, S. H.; O’Regan, D., New oscillation criteria for second-order neutral dynamic equations on time scales via Riccati substitution, Hiroshima Mathematical Journal, 42, 1, 77-98, (2012) · Zbl 1252.34104 [13] Saker, S. H., Oscillation criteria for second-order quasilinear neutral functional dynamic equation on time scales, Nonlinear Oscillations, 13, 3, 407-428, (2010) · Zbl 1334.34196 [14] Hassan, T. S., Oscillation criteria for half-linear dynamic equations on time scales, Journal of Mathematical Analysis and Applications, 345, 1, 176-185, (2008) · Zbl 1156.34022 [15] Erbe, L.; Peterson, A.; Saker, S. H., Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, Journal of Computational and Applied Mathematics, 181, 1, 92-102, (2005) · Zbl 1075.39010 [16] Erbe, L.; Peterson, A.; Saker, S. H., Oscillation criteria for second-order nonlinear delay dynamic equations, Journal of Mathematical Analysis and Applications, 333, 1, 505-522, (2007) · Zbl 1125.34046 [17] Erbe, L.; Peterson, A.; Saker, S. H., Hille and Nehari type criteria for third-order dynamic equations, Journal of Mathematical Analysis and Applications, 329, 1, 112-131, (2007) · Zbl 1128.39009 [18] Erbe, L.; Hassan, T. S.; Peterson, A., Oscillation criteria for nonlinear damped dynamic equations on time scales, Applied Mathematics and Computation, 203, 1, 343-357, (2008) · Zbl 1162.39005 [19] Erbe, L.; Hassan, T. S.; Peterson, A., Oscillation criteria for nonlinear functional neutral dynamic equations on time scales, Journal of Difference Equations and Applications, 15, 11-12, 1097-1116, (2009) · Zbl 1193.34135 [20] Del Medico, A.; Kong, Q., Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain, Journal of Mathematical Analysis and Applications, 294, 2, 621-643, (2004) · Zbl 1056.34050 [21] Wu, H.-W.; Zhuang, R.-K.; Mathsen, R. M., Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations, Applied Mathematics and Computation, 178, 2, 321-331, (2006) · Zbl 1104.39009 [22] Grace, S. R.; Agarwal, R. P.; Kaymakçalan, B.; Sae-jie, W., Oscillation theorems for second order nonlinear dynamic equations, Journal of Applied Mathematics and Computing, 32, 1, 205-218, (2010) · Zbl 1198.34194 [23] Grace, S. R.; Bohner, M.; Agarwal, R. P., On the oscillation of second-order half-linear dynamic equations, Journal of Difference Equations and Applications, 15, 5, 451-460, (2009) · Zbl 1170.34023 [24] Şahiner, Y., Oscillation of second order delay differential equations on time scales, Nonlinear Analysis: Theory, Methods & Applications, 63, 1073-1080, (2005) · Zbl 1224.34294 [25] Chen, D.-X., Oscillation of second-order Emden-Fowler neutral delay dynamic equations on time scales, Mathematical and Computer Modelling, 51, 9-10, 1221-1229, (2010) · Zbl 1205.34127 [26] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales: An Introduction with Applications, x+358, (2001), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 0978.39001 [27] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities, xii+324, (1952), Cambridge, UK: Cambridge University Press, Cambridge, UK
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.