Oscillation of third-order quasilinear neutral dynamic equations on time scales with distributed deviating arguments. (English) Zbl 1427.34094
Summary: The aim of this paper is to give oscillation criteria for the third-order quasilinear neutral delay dynamic equation
\[ \left[r(t)\left([x(t)+p(t)x(\tau_0(t))]^{\Delta\Delta}\right)^{\gamma}\right]^{\Delta}+\int_c^d q_1(t)x^{\alpha}(\tau_{1}(t,\xi))\Delta\xi+\int_{c}^{d}q_{2}(t)x^{\beta}(\tau_{2}(t,\xi))\Delta\xi=0, \]
on a time scale \(\mathbb{T}\), where \(0<\alpha<\gamma<\beta\). By using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.
MSC:
| 34K11 | Oscillation theory of functional-differential equations |
| 34N05 | Dynamic equations on time scales or measure chains |
| 34K40 | Neutral functional-differential equations |
Keywords:
oscillation; third order quasilinear neutral dynamic equation with distributed deviating arguments; time scalesReferences:
| [1] | Agarwal, R. P.; Bohner, M.; O’Regan, D.; Peterson, A., Dynamic equations on time scales: a survey, Dynamic equations on time scales, J. Comput. Appl. Math., 141, 1-26 (2002) · Zbl 1020.39008 |
| [2] | 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 |
| [3] | Bohner, M.; Peterson, A., Dynamic equations on time scales, An introduction with applications, Birkhäuser Boston, Inc., Boston, MA (2001) · Zbl 0978.39001 |
| [4] | Candan, T., Oscillation of second-order nonlinear neutral dynamic equations on time scales with distributed deviating arguments, Comput. Math. Appl., 62, 4118-4125 (2011) · Zbl 1236.34111 |
| [5] | Candan, T., Asymptotic properties of solutions of third-order nonlinear neutral dynamic equations, Adv. Difference Equ., 2014, 1-10 (2014) · Zbl 1351.34107 |
| [6] | Grace, S. R.; Graef, J. R.; El-Beltagy, M. A., On the oscillation of third order neutral delay dynamic equations on time scales, Comput. Math. Appl., 63, 775-782 (2012) · Zbl 1247.34139 |
| [7] | Han, Z.-L.; Li, T.-X.; Sun, S.-R.; Zhang, C.-H., Oscillation behavior of third-order neutral Emden-Fowler delay dynamic equations on time scales, Adv. Difference Equ., 2010, 1-23 (2010) · Zbl 1192.34075 |
| [8] | Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities, Reprint of the 1952 edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge (1988) · Zbl 0634.26008 |
| [9] | Hilger, S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math., 18, 18-56 (1990) · Zbl 0722.39001 |
| [10] | Li, T.-X.; Han, Z.-L.; Sun, S.-R.; Zhao, Y., Oscillation results for third order nonlinear delay dynamic equations on time scales, Bull. Malays. Math. Sci. Soc., 34, 639-641 (2011) · Zbl 1235.34242 |
| [11] | Li, T.-X.; Han, Z.-L.; Zhang, C.-H.; Sun, Y., Oscillation criteria for third-order nonlinear delay dynamic equations on time scales, Bull. Math. Anal. Appl., 3, 52-60 (2011) · Zbl 1314.34181 |
| [12] | Saker, S. H.; Graef, J. R., Oscillation of third-order nonlinear neutral functional dynamic equations on time scales, Dynam. Systems Appl., 21, 583-606 (2012) · Zbl 1278.34107 |
| [13] | Şenel, M. T.; Utku, N., Oscillation behavior of third-order nonlinear neutral dynamic equations on time scales with distributed deviating arguments, Filomat, 28, 1211-1223 (2014) · Zbl 1474.34454 |
| [14] | Utku, N.; Şenel, M. T., Oscillation behavior of third-order quasilinear neutral delay dynamic equations on time scales, Filomat, 28, 1425-1436 (2014) · Zbl 1474.34551 |
| [15] | Zhang, C.-H.; Saker, S. H.; T.-X. Li, Oscillation of third-order neutral dynamic equations on time scales, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, 20, 333-358 (2013) · Zbl 1268.34193 |
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