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Oscillation criteria of even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals. (English) Zbl 07022304
Summary: We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals: \((p(t) \left|x^{\Delta^{n - 1}}(t)\right|^{\alpha - 1} x^{\Delta^{n - 1}}(t))^{\Delta} + f(t, x(\delta(t))) + \int_a^{\sigma(b)} k(t, s) \left|x(g(t, s))\right|^{\theta(s)} \operatorname{sn}(x(g(t, s))) \Delta \xi(s) = 0\), where \(t \in [t_0, \infty)_{\mathbb{T}} : = [t_0, \infty) \cap \mathbb{T}\), \(\mathbb{T}\) a time scale which is unbounded above, \(n \geqslant 2\) is even, \(\left|f(t, u) \left|\geqslant q(t)\right| u^\alpha\right|\), \(\alpha > 0\) is a constant, and \(\theta : [a, b]_{\mathbb{T}_1} \rightarrow \mathbb{R}\) is a strictly increasing right-dense continuous function; \(p, q : [t_0, \infty)_{\mathbb{T}} \rightarrow \mathbb{R}\), \(k : [t_0, \infty)_{\mathbb{T}} \times [a, b]_{\mathbb{T}_1} \rightarrow \mathbb{R}\), \(\delta : [t_0, \infty)_{\mathbb{T}} \rightarrow [t_0, \infty)_{\mathbb{T}}\), and \(g : [t_0, \infty)_{\mathbb{T}} \times [a, b]_{\ mathbb{T}_1} \rightarrow [t_0, \infty)_{\mathbb{T}}\) are right-dense continuous functions; \(\xi : [a, b]_{\mathbb{T}_1} \rightarrow \mathbb{R}\) is strictly increasing. Our results extend and supplement some known results in the literature.

MSC:
34 Ordinary differential equations
39 Difference and functional equations
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[1] Grace, S. R., On the oscillation of \(n\)th order dynamic equations on time-scales, Mediterranean Journal of Mathematics, 10, 1, 147-156, (2013) · Zbl 1262.34104
[2] Grace, S. R.; Agarwal, R. P.; Zafer, A., Oscillation of higher order nonlinear dynamic equations on time scales, Advances in Difference Equations, 2012, article 67, (2012) · Zbl 1293.34113
[3] Chen, D. X., Oscillation and asymptotic behavior for \(n\)th-order nonlinear neutral delay dynamic equations on time scales, Acta Applicandae Mathematicae, 109, 3, 703-719, (2010) · Zbl 1210.34132
[4] Mert, R., Oscillation of higher-order neutral dynamic equations on time scales, Advances in Difference Equations, 2012, article 68, (2012) · Zbl 1294.34086
[5] Erbe, L.; Jia, B.; Peterson, A., Oscillation of \(n\)th order superlinear dynamic equations on time scales, The Rocky Mountain Journal of Mathematics, 41, 2, 471-491, (2011) · Zbl 1216.34100
[6] Karpuz, B., Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations, Applied Mathematics and Computation, 215, 6, 2174-2183, (2009) · Zbl 1182.34113
[7] Karpuz, B., Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients, Electronic Journal of Qualitative Theory of Differential Equations, 9, 1-14, (2009) · Zbl 1184.34072
[8] Chen, D. X.; Qu, P. X., Oscillation of even order advanced type dynamic equations with mixed nonlinearities on time scales, Journal of Applied Mathematics and Computing, 44, 1-2, 357-377, (2014) · Zbl 1303.34073
[9] Grace, S. R., On the oscillation of higher order dynamic equations, Journal of Advanced Research, 4, 201-204, (2013)
[10] Erbe, L.; Karpuz, B.; Peterson, A. C., Kamenev-type oscillation criteria for higher-order neutral delay dynamic equations, International Journal of Difference Equations, 6, 1, 1-16, (2011)
[11] Sun, T.; Xi, H.; Yu, W., Asymptotic behaviors of higher order nonlinear dynamic equations on time scales, Journal of Applied Mathematics and Computing, 37, 1-2, 177-192, (2011) · Zbl 1368.34100
[12] Sun, T. X.; Xi, H. J.; Peng, X. F., Asymptotic behavior of solutions of higher-order dynamic equations on time scales, Advances in Difference Equations, 2011, (2011)
[13] Zhang, Z. G.; Dong, W. L.; Li, Q. L.; Liang, H. Y., Existence of nonoscillatory solutions for higher order neutral dynamic equations on time scales, Journal of Applied Mathematics and Computing, 28, 1-2, 29-38, (2008) · Zbl 1161.34037
[14] 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
[15] Bohner, M.; Peterson, A., Dynamic Equations on Time Scales, (2001), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 1021.34005
[16] Agarwal, R. P.; Bohner, M., Basic calculus on time scales and some of its applications, Results in Mathematics, 35, 1-2, 3-22, (1999) · Zbl 0927.39003
[17] 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
[18] Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities, (1952), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0047.05302
[19] Sun, Y., Interval oscillation criteria for second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals, Abstract and Applied Analysis, 2011, (2011) · Zbl 1226.26020
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