Abstract and Applied Analysis

Existence of Oscillatory Solutions of Second Order Delay Differential Equations with Distributed Deviating Arguments

Youjun Liu, Jianwen Zhang, and Jurang Yan

Full-text: Open access

Abstract

Under weaker hypothesis, we use the Schauder-Tychonoff theorem to obtain new sufficient condition for the global existence of oscillatory solutions for forced second order nonlinear delay differential equations with distributed deviating arguments.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 731021, 6 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512197

Digital Object Identifier
doi:10.1155/2013/731021

Mathematical Reviews number (MathSciNet)
MR3139468

Zentralblatt MATH identifier
07095295

Citation

Liu, Youjun; Zhang, Jianwen; Yan, Jurang. Existence of Oscillatory Solutions of Second Order Delay Differential Equations with Distributed Deviating Arguments. Abstr. Appl. Anal. 2013 (2013), Article ID 731021, 6 pages. doi:10.1155/2013/731021. https://projecteuclid.org/euclid.aaa/1393512197


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References

  • A. D. Myshkis, Linear Differential Equations with Delayed Argument, Nauka, Moscow, Russia, 1972.
  • S. B. Norkin, Differential Equations of the Second Order with Retarded Argument, vol. 31, American Mathematical Society, Providence, RI, USA, 1972, Translations of Mathematical Monographs.
  • N. V. Shevelo, Oscillation of Solutions of Differential Equations with Retarded Argument, Naukova dumka, Kyiv, Ukrania, 1978.
  • R. P. Agarwal, L. Berezansky, E. Braverman, and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, NY, USA, 2012.
  • 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.
  • M. G. Shmul'yan, “On the oscillating solutions of a linear second order differential equation with retarding argument,” Differentsial'nye Uravnenia, vol. 31, pp. 622–629, 1995.
  • A. L. Skubachevskii, “Oscillating solutions of a second order homogeneous linear differential equation with time-lag,” Differentsial'nye Uravnenia, vol. 11, pp. 462–469, 1975.
  • N. V. Azbelev, “The zeros of the solutions of a second order linear differential equation with retarded argument,” Differentsial'nye uravnenia, vol. 7, pp. 1147–1157, 1971.
  • Yu. I. Domshlak, “Comparison theorems of Sturm type for first- and second-order differential equations with sign-variable deviations of the argument,” Ukrainskii Matematicheskii Zhurnal, vol. 34, no. 2, pp. 158–163, 1982.
  • A. I. Domoshnitskiĭ, “Extension of the Sturm theorem to equations with retarded argument,” Differentsial'nye Uravnenia, vol. 19, no. 9, pp. 1475–1482, 1983.
  • A. Domoshnitsky, “Sturm's theorem for equations with delayed argument,” Georgian Mathematical Journal, vol. 1, no. 3, pp. 299–309, 1993.
  • A. Domoshnitsky, “Unboundedness of solutions and instability of differential equations of the second order with delayed argument,” Differential and Integral Equations, vol. 14, no. 5, pp. 559–576, 2001.
  • D. Xia, Z. Wu, S. Yan, and W. Shu, Real Variable Function and Functional Analysis, Higher Education Press, Beijing, China, 1978, (Chinese).
  • S. M. Labovskiĭ, “A condition for the nonvanishing of the Wronskian of a fundamental system of solutions of a linear differential equation with retarded argument,” Differentsial'nye Uravnenia, vol. 10, pp. 426–430, 1974.
  • R. Koplatadze, “On oscillatory properties of solutions of functional-differential equations,” Memoirs on Differential Equations and Mathematical Physics, vol. 3, pp. 1–177, 1994.
  • I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations, The Clarendon Press, Oxford, UK, 1991.
  • L. H. Erbe, Q. Kong, and B. G. Zhang, Oscillation Theorem for Functional Differential Equations, vol. 190, Marcel Dekker, New York, NY, USA, 1995.
  • J. Yan, “Existence of oscillatory solutions of forced second order delay differential equations,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1455–1460, 2011.
  • W.-T. Li and P. Zhao, “Oscillation theorems for second-order nonlinear differential equations with damped term,” Mathematical and Computer Modelling, vol. 39, no. 4-5, pp. 457–471, 2004.
  • W.-T. Li and X. Li, “Oscillation criteria for second-order nonlinear differential equations with integrable coefficient,” Applied Mathematics Letters, vol. 13, no. 8, pp. 1–6, 2000.
  • W.-T. Li, “Oscillation of certain second-order nonlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 217, no. 1, pp. 1–14, 1998.
  • W.-T. Li and R. P. Agarwal, “Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 245, no. 1, pp. 171–188, 2000.
  • P. Wang, “Oscillation criteria for second-order neutral equations with distributed deviating arguments,” Computers & Mathematics with Applications, vol. 47, no. 12, pp. 1935–1946, 2004.
  • Z. Xu and P. Weng, “Oscillation of second order neutral equations with distributed deviating argument,” Journal of Computational and Applied Mathematics, vol. 202, no. 2, pp. 460–477, 2007.