## Abstract and Applied Analysis

### On the Existence of Positive Periodic Solutions for Second-Order Functional Differential Equations with Multiple Delays

#### Abstract

The existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation with multiple delays $u″(t)+a(t)u(t)=f(t,u(t),u(t-{\tau }_{1}(t)),\dots ,u(t-{\tau }_{n}(t)))$, where $a(t)\in C({\Bbb R})$ is a positive ω-periodic function, $f:{\Bbb R}{\times}{[0,+\infty )}^{n+1}\to [0,+\infty )$ is a continuous function which is ω-periodic in t, and ${\tau }_{1}(t),\dots ,{\tau }_{n}(t)\in C({\Bbb R},[0,+\infty ))$ are ω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 929870, 13 pages.

Dates
First available in Project Euclid: 7 May 2014

https://projecteuclid.org/euclid.aaa/1399486653

Digital Object Identifier
doi:10.1155/2012/929870

Mathematical Reviews number (MathSciNet)
MR3004880

Zentralblatt MATH identifier
1260.34136

#### Citation

Li, Qiang; Li, Yongxiang. On the Existence of Positive Periodic Solutions for Second-Order Functional Differential Equations with Multiple Delays. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 929870, 13 pages. doi:10.1155/2012/929870. https://projecteuclid.org/euclid.aaa/1399486653

#### References

• B. Liu, “Periodic solutions of a nonlinear second-order differential equation with deviating argument,” Journal of Mathematical Analysis and Applications, vol. 309, no. 1, pp. 313–321, 2005.
• J.-W. Li and S. S. Cheng, “Periodic solutions of a second order forced sublinear differential equation with delay,” Applied Mathematics Letters, vol. 18, no. 12, pp. 1373–1380, 2005.
• Y. Wang, H. Lian, and W. Ge, “Periodic solutions for a second order nonlinear functional differential equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 110–115, 2007.
• J. Wu and Z. Wang, “Two periodic solutions of second-order neutral functional differential equations,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 677–689, 2007.
• Y. Wu, “Existence nonexistence and multiplicity of periodic solutions for a kind of functional differential equation with parameter,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 1, pp. 433–443, 2009.
• C. Guo and Z. Guo, “Existence of multiple periodic solutions for a class of second-order delay differential equations,” Nonlinear Analysis. Real World Applications, vol. 10, no. 5, pp. 3285–3297, 2009.
• B. Yang, R. Ma, and C. Gao, “Positive periodic solutions of delayed differential equations,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4538–4545, 2011.
• Y. Li, “Positive periodic solutions of second-order differential equations with delays,” Abstract and Applied Analysis, vol. 2012, Article ID 829783, 13 pages, 2012.
• Y. X. Li, “Positive periodic solutions of nonlinear second order ordinary differential equations,” Acta Mathematica Sinica, vol. 45, no. 3, pp. 481–488, 2002.
• Y. Li, “Positive periodic solutions of first and second order ordinary differential equations,” Chinese Annals of Mathematics B, vol. 25, no. 3, pp. 413–420, 2004.
• Y. Li and H. Fan, “Existence of positive periodic solutions for higher-order ordinary differential equations,” Computers & Mathematics with Applications, vol. 62, no. 4, pp. 1715–1722, 2011.
• F. Li and Z. Liang, “Existence of positive periodic solutions to nonlinear second order differential equations,” Applied Mathematics Letters, vol. 18, no. 11, pp. 1256–1264, 2005.
• B. Liu, L. Liu, and Y. Wu, “Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 7-8, pp. 3337–3345, 2010.
• P. J. Torres, “Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem,” Journal of Differential Equations, vol. 190, no. 2, pp. 643–662, 2003.
• K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, NY, USA, 1985.
• D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5, Academic Press, New York, NY, USA, 1988.