×

Nonoscillation of second-order dynamic equations with several delays. (English) Zbl 1217.34139

Summary: The existence of nonoscillatory solutions for the second-order dynamic equation
\[ (A_0,x^\Delta)^\Delta(t)+\sum_{i\in[1,n]_{\mathbb N}}A_i(t)x(\alpha_i(t))=0\text{ for }t\in [t_0,\infty)_{\mathbb T} \]
is investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows us to obtain most known nonoscillation results for second-order delay differential equations in the case \(A_0(t)\equiv 1\) for \(t\in [t_0,\infty)_{\mathbb R}\) and for second-order nondelay difference equations \((\alpha_i(t)=t+1\) for \(t\in [0,\infty)_{\mathbb N})\). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary \(A_0\) and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced.

MSC:

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

References:

[1] R. P. Agarwal, M. Bohner, S. R. Grace, and D. O’Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, New York, NY, USA, 2005. · Zbl 1084.39001
[2] L. H. Erbe, Q. K. Kong, and B. G. Zhang, Oscillation Theory for Functional-Differential Equations, vol. 190 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1995. · Zbl 0821.34067
[3] I. Gy\Hori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991. · Zbl 0780.34048
[4] G. S. Ladde, V. K. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, vol. 110 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1987. · Zbl 0622.34071
[5] A. D. My\vskis, Linear Differential Equations with Retarded Argument, Izdat. “Nauka”, Moscow, Russia, 1972.
[6] S. B. Norkin, Differential Equations of the Second Order with Retarded Argument. Some Problems of the Theory of Vibrations of Systems with Retardation, vol. 3 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1972. · Zbl 0234.34080
[7] C. A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, vol. 4 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1968. · Zbl 0191.09904
[8] L. Berezansky and E. Braverman, “Some oscillation problems for a second order linear delay differential equation,” Journal of Mathematical Analysis and Applications, vol. 220, no. 2, pp. 719-740, 1998. · Zbl 0915.34064 · doi:10.1006/jmaa.1997.5879
[9] E. Braverman and B. Karpuz, “Nonoscillation of first-order dynamic equations with several delays,” Advances in Difference Equations, vol. 2010, Article ID 873459, 22 pages, 2010. · Zbl 1216.34099
[10] W. Leighton, “On self-adjoint differential equations of second order,” Journal of the London Mathematical Society, vol. 27, pp. 37-47, 1952. · Zbl 0048.06503 · doi:10.1112/jlms/s1-27.1.37
[11] W. J. Coles, “A simple proof of a well-known oscillation theorem,” Proceedings of the American Mathematical Society, vol. 19, p. 507, 1968. · Zbl 0155.12802 · doi:10.2307/2035563
[12] A. Wintner, “A criterion of oscillatory stability,” Quarterly of Applied Mathematics, vol. 7, pp. 115-117, 1949. · Zbl 0032.34801
[13] A. Kneser, “Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen,” Mathematische Annalen, vol. 42, no. 3, pp. 409-435, 1893. · doi:10.1007/BF01444165
[14] E. Hille, “Non-oscillation theorems,” Transactions of the American Mathematical Society, vol. 64, pp. 234-252, 1948. · Zbl 0031.35402 · doi:10.2307/1990500
[15] J. Deng, “Oscillation criteria for second-order linear differential equations,” Journal of Mathematical Analysis and Applications, vol. 271, no. 1, pp. 283-287, 2002. · Zbl 1023.34027 · doi:10.1016/S0022-247X(02)00061-6
[16] C. Huang, “Oscillation and nonoscillation for second order linear differential equations,” Journal of Mathematical Analysis and Applications, vol. 210, no. 2, pp. 712-723, 1997. · Zbl 0895.34031 · doi:10.1006/jmaa.1997.5572
[17] R. N. Rath, J. G. Dix, B. L. S. Barik, and B. Dihudi, “Necessary conditions for the solutions of second order non-linear neutral delay difference equations to be oscillatory or tend to zero,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 60907, 16 pages, 2007. · Zbl 1148.39010 · doi:10.1155/2007/60907
[18] S. Z. Chen and L. H. Erbe, “Riccati techniques and discrete oscillations,” Journal of Mathematical Analysis and Applications, vol. 142, no. 2, pp. 468-487, 1989. · Zbl 0686.39001 · doi:10.1016/0022-247X(89)90015-2
[19] S. S. Cheng, T. C. Yan, and H. J. Li, “Oscillation criteria for second order difference equation,” Funkcialaj Ekvacioj, vol. 34, no. 2, pp. 223-239, 1991. · Zbl 0773.39001
[20] P. , “Oscillation and nonoscillation criteria for second order linear difference equations,” Fasciculi Mathematici, no. 31, pp. 71-89, 2001. · Zbl 0999.39006
[21] B. G. Zhang and Y. Zhou, “Oscillation and nonoscillation for second-order linear difference equations,” Computers & Mathematics with Applications, vol. 39, no. 1-2, pp. 1-7, 2000. · Zbl 0973.39007 · doi:10.1016/S0898-1221(99)00308-9
[22] Y. Zhou and B. G. Zhang, “Oscillations of delay difference equations in a critical state,” Computers & Mathematics with Applications, vol. 39, no. 7-8, pp. 71-80, 2000. · Zbl 0958.39017 · doi:10.1016/S0898-1221(00)00066-3
[23] X. H. Tang, J. S. Yu, and D. H. Peng, “Oscillation and nonoscillation of neutral difference equations with positive and negative coefficients,” Computers & Mathematics with Applications, vol. 39, no. 7-8, pp. 169-181, 2000. · Zbl 0958.39016 · doi:10.1016/S0898-1221(00)00073-0
[24] B. G. Zhang and S. S. Cheng, “Oscillation criteria and comparison theorems for delay difference equations,” Fasciculi Mathematici, no. 25, pp. 13-32, 1995. · Zbl 0830.39005
[25] J. Baoguo, L. Erbe, and A. Peterson, “A Wong-type oscillation theorem for second order linear dynamic equations on time scales,” Journal of Difference Equations and Applications, vol. 16, no. 1, pp. 15-36, 2010. · Zbl 1193.34179 · doi:10.1080/10236190802409312
[26] O. Do and S. Hilger, “A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 147-158, 2002. · Zbl 1009.34033 · doi:10.1016/S0377-0427(01)00442-3
[27] L. Erbe, “Oscillation criteria for second order linear equations on a time scale,” The Canadian Applied Mathematics Quarterly, vol. 9, no. 4, p. 345-375 (2002), 2001. · Zbl 1050.39024
[28] L. Erbe, A. Peterson, and P. , “Comparison theorems for linear dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 275, no. 1, pp. 418-438, 2002. · Zbl 1034.34042 · doi:10.1016/S0022-247X(02)00390-6
[29] A. Zafer, “On oscillation and nonoscillation of second-order dynamic equations,” Applied Mathematics Letters, vol. 22, no. 1, pp. 136-141, 2009. · Zbl 1163.39301 · doi:10.1016/j.aml.2008.03.003
[30] L. Erbe, A. Peterson, and C. C. Tisdell, “Basic existence, uniqueness and approximation results for positive solutions to nonlinear dynamic equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 7, pp. 2303-2317, 2008. · Zbl 1158.34025 · doi:10.1016/j.na.2007.08.010
[31] B. Karpuz, “Existence and uniqueness of solutions to systems of delay dynamic equations on time scales,” International Journal of Computer Mathematics, vol. 10, no. M11, pp. 48-58, 2011.
[32] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. · Zbl 0978.39001
[33] 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
[34] 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
[35] J. R. Yan, “Oscillatory properties of second-order differential equations with an “integralwise small” coefficient,” Acta Mathematica Sinica, vol. 30, no. 2, pp. 206-215, 1987. · Zbl 0635.34030
[36] M. Bohner and M. Ünal, “Kneser’s theorem in q-calculus,” Journal of Physics A, vol. 38, no. 30, pp. 6729-6739, 2005. · Zbl 1080.39023 · doi:10.1088/0305-4470/38/30/008
[37] L. Berezansky and E. Braverman, “On oscillation of a second order impulsive linear delay differential equation,” Journal of Mathematical Analysis and Applications, vol. 233, no. 1, pp. 276-300, 1999. · Zbl 0926.34054 · doi:10.1006/jmaa.1999.6297
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.