International Journal of Differential Equations

Volume 2010, Article ID 598068, 14 pages

http://dx.doi.org/10.1155/2010/598068

Review Article

## Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations

^{1}Department of Mathematics, University of Gjirokastra, 6002 Gjirokastra, Albania^{2}Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 2 December 2009; Accepted 9 January 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 L. K. Kikina and I. P. Stavroulakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- J. S. Bradley, “Oscillation theorems for a second-order delay equation,”
*Journal of Differential Equations*, vol. 8, pp. 397–403, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Erbe, “Oscillation criteria for second order nonlinear delay equations,”
*Canadian Mathematical Bulletin*, vol. 16, pp. 49–56, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. H. Erbe, Q. 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. View at MathSciNet - K. Gopalsamy,
*Stability and Oscillations in Delay Differential Equations of Population Dynamics*, vol. 74 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992. View at MathSciNet - I. Győri and G. Ladas,
*Oscillation Theory of Delay Differential Equations*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991. View at MathSciNet - J. K. Hale,
*Theory of Functional Differential Equations*, Springer, New York, NY, USA, 1997. - E. Hille, “Non-oscillation theorems,”
*Transactions of the American Mathematical Society*, vol. 64, pp. 234–252, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Kneser, “Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen,”
*Mathematische Annalen*, vol. 42, no. 3, pp. 409–435, 1893. View at Publisher · View at Google Scholar · View at MathSciNet - R. G. Koplatadze, “Criteria for the oscillation of solutions of differential inequalities and second-order equations with retarded
argument,”
*Trudy Instituta Prikladnoj Matematiki Imeni I. N. Vekua*, vol. 17, pp. 104–121, 1986 (Russian). View at Google Scholar · View at MathSciNet - R. Koplatadze, “On oscillatory properties of solutions of functional-differential equations,”
*Memoirs on Differential Equations and Mathematical Physics*, vol. 3, pp. 1–179, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Koplatadze, G. Kvinikadze, and I. P. Stavroulakis, “Properties $A$ and $B$ of $n$th order linear differential equations with deviating argument,”
*Georgian Mathematical Journal*, vol. 6, no. 6, pp. 553–566, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Koplatadze, G. Kvinikadze, and I. P. Stavroulakis, “Oscillation of second order linear delay differential equations,”
*Functional Differential Equations*, vol. 7, no. 1-2, pp. 121–145, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. S. Ladde, V. 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. View at MathSciNet - A. D. Myshkis,
*Linear Differential Equations with Retarded Argument*, Nauka, Moscow, Russia, 2nd edition, 1972. View at MathSciNet - S. B. Norkin,
*Differential Equations of the Second Order with Retarded Arguments*, Nauka, Moscow, Russia, 1965. View at MathSciNet - C. Sturm, “Sur les équations différentielles linéaires du second ordre,”
*Journal de Mathématiques Pures et Appliquées*, vol. 1, pp. 106–186, 1836. View at Google Scholar - 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. View at MathSciNet - P. Waltman, “A note on an oscillation criterion for an equation with a functional argument,”
*Canadian Mathematical Bulletin*, vol. 11, pp. 593–595, 1968. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. J. Wei, “Oscillation of second order delay differential equation,”
*Annals of Differential Equations*, vol. 4, no. 4, pp. 473–478, 1988. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. S. W. Wong, “Second order oscillation with retarded arguments,” in
*Ordinary Differential Equations (Proc. Conf., Math. Res. Center, Naval Res. Lab., Washington, D. C., 1971)*, pp. 581–596, Academic Press, New York, NY, USA, 1972. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. R. Yan, “Oscillatory property for second order linear delay differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 122, no. 2, pp. 380–384, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal, E. Thandapani, and P. J. Y. Wong, “Oscillations of higher-order neutral difference equations,”
*Applied Mathematics Letters*, vol. 10, no. 1, pp. 71–78, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Grzegorczyk and J. Werbowski, “Oscillation of higher-order linear difference equations,”
*Computers & Mathematics with Applications*, vol. 42, no. 3–5, pp. 711–717, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Koplatadze, “Oscillation of linear difference equations with deviating arguments,”
*Computers & Mathematics with Applications*, vol. 42, no. 3–5, pp. 477–486, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Koplatadze, G. Kvinikadze, and I. P. Stavroulakis, “Oscillation of second-order linear difference equations with deviating arguments,”
*Advances in Mathematical Sciences and Applications*, vol. 12, no. 1, pp. 217–226, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Wyrwińska, “Oscillation criteria of a higher order linear difference equation,”
*Bulletin of the Institute of Mathematics. Academia Sinica*, vol. 22, no. 3, pp. 259–266, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Domshlak, “Oscillatory properties of linear difference equations with continuous time,”
*Differential Equations and Dynamical Systems*, vol. 1, no. 4, pp. 311–324, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Golda and J. Werbowski, “Oscillation of linear functional equations of the second order,”
*Funkcialaj Ekvacioj*, vol. 37, no. 2, pp. 221–227, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Nowakowska and J. Werbowski, “Oscillation of linear functional equations of higher order,”
*Archivum Mathematicum*, vol. 31, no. 4, pp. 251–258, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Shen, “Comparison theorems for the oscillation of difference equations with continuous arguments and applications,”
*Chinese Science Bulletin*, vol. 41, no. 18, pp. 1506–1510, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Shen and I. P. Stavroulakis, “An oscillation criteria for second order functional equations,”
*Acta Mathematica Scientia. Series B*, vol. 22, no. 1, pp. 56–62, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. Shen and I. P. Stavroulakis, “Sharp conditions for nonoscillation of functional equations,”
*Indian Journal of Pure and Applied Mathematics*, vol. 33, no. 4, pp. 543–554, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Yan and F. Zhang, “Oscillation for system of delay difference equations,”
*Journal of Mathematical Analysis and Applications*, vol. 230, no. 1, pp. 223–231, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. G. Zhang, J. Yan, and S. K. Choi, “Oscillation for difference equations with continuous variable,”
*Computers & Mathematics with Applications*, vol. 36, no. 9, pp. 11–18, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Zhang, J. Yan, and A. Zhao, “Oscillation criteria for a difference equation,”
*Indian Journal of Pure and Applied Mathematics*, vol. 28, no. 9, pp. 1241–1249, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. G. Sficas and I. P. Stavroulakis, “Oscillation criteria for first-order delay equations,”
*The Bulletin of the London Mathematical Society*, vol. 35, no. 2, pp. 239–246, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Á. Elbert and I. P. Stavroulakis, “Oscillations of first order differential equations with deviating arguments, University of Ioannina, T. R. N0 172 1990,” in
*Recent Trends in Differential Equations*, vol. 1 of*World Scientific Series in Applicable Analysis*, pp. 163–178, World Scientific, River Edge, NJ, USA, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Á. Elbert and I. P. Stavroulakis, “Oscillation and nonoscillation criteria for delay differential equations,”
*Proceedings of the American Mathematical Society*, vol. 123, no. 5, pp. 1503–1510, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Fukagai and T. Kusano, “Oscillation theory of first order functional-differential equations with deviating arguments,”
*Annali di Matematica Pura ed Applicata*, vol. 136, pp. 95–117, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Jaroš and I. P. Stavroulakis, “Oscillation tests for delay equations,”
*The Rocky Mountain Journal of Mathematics*, vol. 29, no. 1, pp. 197–207, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Kon, Y. G. Sficas, and I. P. Stavroulakis, “Oscillation criteria for delay equations,”
*Proceedings of the American Mathematical Society*, vol. 128, no. 10, pp. 2989–2997, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Koplatadze and G. Kvinikadze, “On the oscillation of solutions of first-order delay differential inequalities and equations,”
*Georgian Mathematical Journal*, vol. 1, no. 6, pp. 675–685, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Kozakiewicz, “Conditions for the absence of positive solutions of first order differential inequalities with deviating arguments,” in
*Proceedings of the 4th International Colloquium on Differential Equations (Plovdiv, 1993)*, pp. 157–161, VSP, Utrecht, The Netherlands. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. K. Kwong, “Oscillation of first-order delay equations,”
*Journal of Mathematical Analysis and Applications*, vol. 156, no. 1, pp. 274–286, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. T. Li, “Oscillations of delay differential equations with variable coefficients,”
*Journal of Mathematical Analysis and Applications*, vol. 192, no. 1, pp. 312–321, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Li, “Oscillation of first order delay differential equations,”
*Proceedings of the American Mathematical Society*, vol. 124, no. 12, pp. 3729–3737, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Ch. G. Philos and Y. G. Sficas, “An oscillation criterion for first order linear delay differential equations,”
*Canadian Mathematical Bulletin*, vol. 41, no. 2, pp. 207–213, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-C. Wang, I. P. Stavroulakis, and X.-Z. Qian, “A survey on the oscillation of solutions of first order linear differential equations with deviating arguments,”
*Applied Mathematics E-Notes*, vol. 2, pp. 171–191, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet