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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 175479, 8 pages
http://dx.doi.org/10.1155/2013/175479
Research Article

On the Existence and Stability of Periodic Solutions for a Nonlinear Neutral Functional Differential Equation

1School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
2School of Mathematics and Computer Sciences, Yichun University, Yichun 336000, China
3Key Laboratory of Mathematics and Interdisciplinary Science of Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China

Received 31 January 2013; Accepted 1 March 2013

Academic Editor: Chuangxia Huang

Copyright © 2013 Yueding Yuan and Zhiming Guo. 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

  1. R. K. Brayton, “Bifurcation of periodic solutions in a nonlinear difference-differential equations of neutral type,” Quarterly of Applied Mathematics, vol. 24, pp. 215–224, 1966. View at MathSciNet
  2. R. K. Brayton, “Nonlinear oscillations in a distributed network,” Quarterly of Applied Mathematics, vol. 24, pp. 289–301, 1967.
  3. R. K. Brayton and W. L. Miranker, “A stability theory for nonlinear mixed initial boundary value problems,” Archive for Rational Mechanics and Analysis, vol. 17, pp. 358–376, 1964. View at MathSciNet
  4. R. K. Brayton and J. K. Moser, “A theory of nonlinear networks. I,” Quarterly of Applied Mathematics, vol. 22, pp. 1–33, 1964. View at MathSciNet
  5. K. L. Cooke and D. W. Krumme, “Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 24, pp. 372–387, 1968. View at MathSciNet
  6. J. M. Ferreira, “On the stability of a distributed network,” SIAM Journal on Mathematical Analysis, vol. 17, no. 1, pp. 38–45, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  7. W. Krawcewicz, J. Wu, and H. Xia, “Global Hopf bifurcation theory for condensing fields and neutral equations with applications to lossless transmission problems,” The Canadian Applied Mathematics Quarterly, vol. 1, no. 2, pp. 167–220, 1993. View at MathSciNet
  8. O. Lopes, “Forced oscillations in nonlinear neutral differential equations,” SIAM Journal on Applied Mathematics, vol. 29, pp. 196–207, 1975. View at MathSciNet
  9. O. Lopes, “Stability and forced oscillations,” Journal of Mathematical Analysis and Applications, vol. 55, no. 3, pp. 686–698, 1976. View at MathSciNet
  10. S. Siqueira Ceron and O. Lopes, “α-contractions and attractors for dissipative semilinear hyperbolic equations and systems,” Annali di Matematica Pura ed Applicata. Serie Quarta, vol. 160, pp. 193–206, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  11. S. Lu and W. Ge, “On the existence of periodic solutions for neutral functional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 7, pp. 1285–1306, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  12. W. L. Miranker, “Periodic solutions of the wave equation with a nonlinear interface condition,” International Business Machines Corporation, vol. 5, pp. 2–24, 1961. View at MathSciNet
  13. M. Slemrod, “Nonexistence of oscillations in a nonlinear distributed network,” Journal of Mathematical Analysis and Applications, vol. 36, pp. 22–40, 1971. View at MathSciNet
  14. J. J. Wei and S. G. Ruan, “Stability and global Hopf bifurcation for neutral differential equations,” Acta Mathematica Sinica. Chinese Series, vol. 45, no. 1, pp. 93–104, 2002 (Chinese). View at MathSciNet
  15. L. Ding and Z. Li, “Periodicity and stability in neutral equations by Krasnoselskii's fixed point theorem,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1220–1228, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  16. T. A. Burton, “Perron-type stability theorems for neutral equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 55, no. 3, pp. 285–297, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  17. V. B. Kolmanovskiĭ and V. R. Nosov, Stability of functional-differential equations, vol. 180 of Mathematics in Science and Engineering, Academic Press, London, UK, 1986. View at MathSciNet
  18. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1993. View at MathSciNet
  19. T.-R. Ding and C. Li, Ordinary Differential Equations Tutorial, Higher Education Press, Bejing, China, 2004.
  20. J. K. Hale, “Functional differential equations,” in Analytic Theory of Differential Equations, vol. 183 of Lecture Notes in Mathematics, p. 2, Springer, Berlin, Germany, 1971. View at MathSciNet
  21. D. X. Xia, Z. R. Wu, S. Z. Yan, and W. C. Shu, Real Variable Function and Functional Analysis, Higher Education Press, Beijing, China, 2nd edition, 2010.
  22. D. R. Smart, Fixed Point Theorems, Cambridge University Press, London, UK, 1974. View at MathSciNet
  23. V. G. Angelov, “Lossy transmission lines terminated by R-loads with exponential V-I characteristics,” Nonlinear Analysis: Real World Applications, vol. 8, no. 2, pp. 579–589, 2007. View at Publisher · View at Google Scholar · View at MathSciNet