# 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)
Oscillation of second-order sublinear impulsive differential equations. (English) Zbl 1223.34049
Summary: Oscillation criteria obtained by Kusano and Onose (1973) and by Belohorec (1969) are extended to second-order sublinear impulsive differential equations of Emden-Fowler type: $$x''(t) + p(t)|x(\tau(t))|^{\alpha - 1}x(\tau(t)) = 0,\quad t\ne\theta_k;$$ $$\Delta x'(t)|_{t=\theta_k} + q_k|x(\tau(\theta_k))|^{\alpha-1}x(\tau(\theta_{k})) = 0;\quad \Delta x(t)|_{t=\theta_{k}} = 0,\ (0 < \alpha < 1)$$ by considering the cases $\tau(t) \leq t$ and $\tau(t) = t$, respectively. Examples are inserted to show how impulsive perturbations greatly affect the oscillation behavior of the solutions.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A37 Differential equations with impulses
Full Text:
##### References:
 [1] T. Kusano and H. Onose, “Nonlinear oscillation of a sublinear delay equation of arbitrary order,” Proceedings of the American Mathematical Society, vol. 40, pp. 219-224, 1973. · Zbl 0268.34075 · doi:10.2307/2038666 [2] H. E. Gollwitzer, “On nonlinear oscillations for a second order delay equation,” Journal of Mathematical Analysis and Applications, vol. 26, pp. 385-389, 1969. · Zbl 0169.11401 · doi:10.1016/0022-247X(69)90161-9 [3] V. N. Sevelo and O. N. Odaric, “Certain questions on the theory of the oscillation (non-oscillation) of the solutions of second order differential equations with retarded argument,” Ukrainskii Matematicheskii Zhurnal, vol. 23, pp. 508-516, 1971 (Russian). · Zbl 0238.34115 · doi:10.1007/BF01085475 [4] S. Belohorec, “Two remarks on the properties of solutions of a nonlinear differential equation,” Acta Facultatis Rerum Naturalium Universitatis Comenianae/Mathematica, vol. 22, pp. 19-26, 1969. · Zbl 0271.34045 [5] D. D. Bainov, Yu. I. Domshlak, and P. S. Simeonov, “Sturmian comparison theory for impulsive differential inequalities and equations,” Archiv der Mathematik, vol. 67, no. 1, pp. 35-49, 1996. · Zbl 0856.34033 · doi:10.1007/BF01196165 [6] K. Gopalsamy and B. G. Zhang, “On delay differential equations with impulses,” Journal of Mathematical Analysis and Applications, vol. 139, no. 1, pp. 110-122, 1989. · Zbl 0687.34065 · doi:10.1016/0022-247X(89)90232-1 [7] J. Yan, “Oscillation properties of a second-order impulsive delay differential equation,” Computers & Mathematics with Applications, vol. 47, no. 2-3, pp. 253-258, 2004. · Zbl 1050.34098 · doi:10.1016/S0898-1221(04)90022-3 [8] C. Yong-shao and F. Wei-zhen, “Oscillations of second order nonlinear ODE with impulses,” Journal of Mathematical Analysis and Applications, vol. 210, no. 1, pp. 150-169, 1997. · doi:10.1006/jmaa.1997.5378 [9] Z. He and W. Ge, “Oscillations of second-order nonlinear impulsive ordinary differential equations,” Journal of Computational and Applied Mathematics, vol. 158, no. 2, pp. 397-406, 2003. · Zbl 1042.34063 · doi:10.1016/S0377-0427(03)00474-6 [10] C. Huang, “Oscillation and nonoscillation for second order linear impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 214, no. 2, pp. 378-394, 1997. · Zbl 0895.34031 · doi:10.1006/jmaa.1997.5572 [11] J. Luo, “Second-order quasilinear oscillation with impulses,” Computers & Mathematics with Applications, vol. 46, no. 2-3, pp. 279-291, 2003. · Zbl 1063.34004 · doi:10.1016/S0898-1221(03)90031-9 [12] A. Özbekler and A. Zafer, “Sturmian comparison theory for linear and half-linear impulsive differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5-7, pp. e289-e297, 2005. · Zbl 1159.34306 · doi:10.1016/j.na.2005.01.087 [13] A. Özbekler and A. Zafer, “Picone’s formula for linear non-selfadjoint impulsive differential equations,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 410-423, 2006. · Zbl 1100.34012 · doi:10.1016/j.jmaa.2005.06.019 [14] G. Ballinger and X. Liu, “Permanence of population growth models with impulsive effects,” Mathematical and Computer Modelling, vol. 26, no. 12, pp. 59-72, 1997. · Zbl 1185.34014 · doi:10.1016/S0895-7177(97)00240-9 [15] Z. Lu, X. Chi, and L. Chen, “Impulsive control strategies in biological control of pesticide,” Theoretical Population Biology, vol. 64, no. 1, pp. 39-47, 2003. · Zbl 1100.92071 · doi:10.1016/S0040-5809(03)00048-0 [16] J. Sun, F. Qiao, and Q. Wu, “Impulsive control of a financial model,” Physics Letters A, vol. 335, no. 4, pp. 282-288, 2005. · Zbl 1123.91325 · doi:10.1016/j.physleta.2004.12.030 [17] S. Tang and L. Chen, “Global attractivity in a “food-limited” population model with impulsive effects,” Journal of Mathematical Analysis and Applications, vol. 292, no. 1, pp. 211-221, 2004. · Zbl 1062.34055 · doi:10.1016/j.jmaa.2003.11.061 [18] S. Tang, Y. Xiao, and D. Clancy, “New modelling approach concerning integrated disease control and cost-effectivity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 3, pp. 439-471, 2005. · Zbl 1078.92059 · doi:10.1016/j.na.2005.05.029 [19] V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989. · Zbl 0719.34002 [20] A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995. · doi:10.1142/9789812798664 [21] M. Akhmetov and R. Sejilova, “The control of the boundary value problem for linear impulsive integro-differential systems,” Journal of Mathematical Analysis and Applications, vol. 236, no. 2, pp. 312-326, 1999. · Zbl 0943.93007 · doi:10.1006/jmaa.1999.6428 [22] D. Bainov and V. Covachev, Impulsive Differential Equations with a Small Parameter, vol. 24 of Series on Advances in Mathematics for Applied Sciences, World Scientific, River Edge, NJ, USA, 1994. · Zbl 0828.34001