# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Nonoscillatory solutions for higher-order neutral dynamic equations on time scales. (English) Zbl 1205.34136
Summary: We study the higher-order neutral dynamic equation $$\{a(t)[(x(t)-p(t)x(\tau(t)))^{\Delta^m}]^\alpha\}^\Delta+f(t,x(\rho(t)))=0\text{ for }t\in[t_0,\infty)_{\Bbb T}$$ and obtain some necessary and sufficient conditions for the existence of nonoscillatory bounded solutions for this equation.

##### MSC:
 34N05 Dynamic equations on time scales or measure chains 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations
Full Text:
##### References:
 [1] S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18-56, 1990. · Zbl 0722.39001 · doi:10.1007/BF03323153 [2] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0978.39001 [3] V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, New York, NY, USA, 2002. · Zbl 0986.05001 [4] M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003. · Zbl 1025.34001 [5] M. Bohner and S. H. Saker, “Oscillation of second order nonlinear dynamic equations on time scales,” The Rocky Mountain Journal of Mathematics, vol. 34, no. 4, pp. 1239-1254, 2004. · Zbl 1075.34028 · doi:10.1216/rmjm/1181069797 [6] L. Erbe, “Oscillation results for second-order linear equations on a time scale,” Journal of Difference Equations and Applications, vol. 8, no. 11, pp. 1061-1071, 2002. · Zbl 1021.34012 · doi:10.1080/10236190290015317 [7] T. S. Hassan, “Oscillation criteria for half-linear dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 176-185, 2008. · Zbl 1156.34022 · doi:10.1016/j.jmaa.2008.04.019 [8] R. P. Agarwal, M. Bohner, and S. H. Saker, “Oscillation of second order delay dynamic equations,” The Canadian Applied Mathematics Quarterly, vol. 13, no. 1, pp. 1-17, 2005. · Zbl 1126.39003 [9] M. Bohner, B. Karpuz, and Ö. Öcalan, “Iterated oscillation criteria for delay dynamic equations of first order,” Advances in Difference Equations, vol. 2008, Article ID 458687, 12 pages, 2008. · Zbl 1160.39302 · doi:10.1155/2008/458687 · eudml:55317 [10] L. Erbe, A. Peterson, and S. H. Saker, “Oscillation criteria for second-order nonlinear delay dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 505-522, 2007. · Zbl 1125.34046 · doi:10.1016/j.jmaa.2006.10.055 [11] Z. Han, B. Shi, and S. Sun, “Oscillation criteria for second-order delay dynamic equations on time scales,” Advances in Difference Equations, vol. 2007, Article ID 70730, 16 pages, 2007. · Zbl 1147.39002 · doi:10.1155/2007/70730 · eudml:54599 [12] Z. Han, S. Sun, and B. Shi, “Oscillation criteria for a class of second-order Emden-Fowler delay dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 334, no. 2, pp. 847-858, 2007. · Zbl 1125.34047 · doi:10.1016/j.jmaa.2007.01.004 [13] Y. \cSahiner, “Oscillation of second-order delay differential equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5-7, pp. e1073-e1080, 2005. · doi:10.1016/j.na.2005.01.062 [14] Y. \cSahíner, “Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales,” Advances in Difference Equations, vol. 2006, Article ID 65626, 9 pages, 2006. · Zbl 1139.39302 · doi:10.1155/ADE/2006/65626 [15] B. G. Zhang and Z. Shanliang, “Oscillation of second-order nonlinear delay dynamic equations on time scales,” Computers & Mathematics with Applications, vol. 49, no. 4, pp. 599-609, 2005. · Zbl 1075.34061 · doi:10.1016/j.camwa.2004.04.038 [16] S. R. Grace, R. P. Agarwal, B. Kaymak\ccalan, and W. Sae-jie, “On the oscillation of certain second order nonlinear dynamic equations,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 273-286, 2009. · Zbl 1185.34041 · doi:10.1016/j.mcm.2008.12.007 [17] L. Erbe, A. Peterson, and S. H. Saker, “Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales,” Journal of Computational and Applied Mathematics, vol. 181, no. 1, pp. 92-102, 2005. · Zbl 1075.39010 · doi:10.1016/j.cam.2004.11.021 [18] L. Erbe, A. Peterson, and S. H. Saker, “Hille and Nehari type criteria for third-order dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 112-131, 2007. · Zbl 1128.39009 · doi:10.1016/j.jmaa.2006.06.033 [19] L. Erbe, A. Peterson, and S. H. Saker, “Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation,” The Canadian Applied Mathematics Quarterly, vol. 14, no. 2, pp. 129-147, 2006. · Zbl 1145.34329 [20] B. Karpuz, Ö. Öcalan, and R. Rath, “Necessary and sufficient conditions for the oscillatory and asymptotic behaviour of solutions to neutral delay dynamic equations,” Electronic Journal of Differential Equations, vol. 2009, no. 64, pp. 1-15, 2009. · Zbl 1165.39006 · emis:journals/EJDE/Volumes/2009/64/abstr.html · eudml:130346 [21] Z.-Q. Zhu and Q.-R. Wang, “Existence of nonoscillatory solutions to neutral dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 751-762, 2007. · Zbl 1128.34043 · doi:10.1016/j.jmaa.2007.02.008 [22] B. Karpuz and Ö. Öcalan, “Necessary and sufficient conditions on asymptotic behaviour of solutions of forced neutral delay dynamic equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3063-3071, 2009. · Zbl 1188.34120 · doi:10.1016/j.na.2009.01.218 [23] B. Karpuz, “Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2174-2183, 2009. · Zbl 1182.34113 · doi:10.1016/j.amc.2009.08.013 [24] B. Karpuz, “Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2009, no. 34, pp. 1-14, 2009. · Zbl 1184.34072 · emis:journals/EJQTDE/2009/200934.html · eudml:230745 [25] T. Li, Z. Han, S. Sun, and D. Yang, “Existence of nonoscillatory solutions to second-order neutral delay dynamic equations on time scales,” Advances in Difference Equations, vol. 2009, Article ID 562329, 10 pages, 2009. · Zbl 1178.34119 · doi:10.1155/2009/562329 · eudml:45769 [26] Z. Zhang, W. Dong, Q. Li, and H. Liang, “Existence of nonoscillatory solutions for higher order neutral dynamic equations on time scales,” Journal of Applied Mathematics and Computing, vol. 28, no. 1-2, pp. 29-38, 2008. · Zbl 1161.34037 · doi:10.1007/s12190-008-0067-y [27] Z. Zhang, W. Dong, Q. Li, and H. Liang, “Positive solutions for higher order nonlinear neutral dynamic equations on time scales,” Applied Mathematical Modelling, vol. 33, no. 5, pp. 2455-2463, 2009. · Zbl 1185.34117 · doi:10.1016/j.apm.2008.07.011 [28] R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in Mathematics, vol. 35, no. 1-2, pp. 3-22, 1999. · Zbl 0927.39003 · doi:10.1007/BF03322019 [29] D.-X. Chen, “Oscillation and asymptotic behavior for nth-order nonlinear neutral delay dynamic equations on time scales,” Acta Applicandae Mathematicae, vol. 109, no. 3, pp. 703-719, 2010. · Zbl 1210.34132 · doi:10.1007/s10440-008-9341-0