Abstract and Applied Analysis

Positive Periodic Solutions of Second-Order Differential Equations with Delays

Yongxiang Li

Abstract

The existence results of positive ω-periodic solutions are obtained for the second-order differential equation with delays $-{u}^{″}+a(t)=f(t,u(t-{\tau }_{1}),...,u(t-{\tau }_{n}))$, where $a\in C(\Bbb R,(0,\infty ))$ is a ω-periodic function, $f:\Bbb R{\times}{[0,\infty )}^{n}\to [0,\infty )$ is a continuous function, which is ω-periodic in $t$, and ${\tau }_{1},{\tau }_{2},...,{\tau }_{n}$ are positive constants. Our discussion is based on the fixed point index theory in cones.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 829783, 13 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495766

Digital Object Identifier
doi:10.1155/2012/829783

Mathematical Reviews number (MathSciNet)
MR2947690

Zentralblatt MATH identifier
1246.34037

Citation

Li, Yongxiang. Positive Periodic Solutions of Second-Order Differential Equations with Delays. Abstr. Appl. Anal. 2012 (2012), Article ID 829783, 13 pages. doi:10.1155/2012/829783. https://projecteuclid.org/euclid.aaa/1355495766

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. X. 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. J. Guo and Z. M. Guo, “Existence of multiple periodic solutions for a class of second-order delay differential equations,” Nonlinear Analysis, vol. 10, no. 5, pp. 3285–3297, 2009.
• W. S. Cheung, J. Ren, and W. Han, “Positive periodic solution of second-order neutral functional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 9, pp. 3948–3955, 2009.
• X. Lv, P. Yan, and D. Liu, “Anti-periodic solutions for a class of nonlinear second-order Rayleigh equations with delays,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 11, pp. 3593–3598, 2010.
• F. M. Atici and G. Sh. Guseinov, “On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions,” Journal of Computational and Applied Mathematics, vol. 132, no. 2, pp. 341–356, 2001.
• 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.
• Y. X. Li, “Positive periodic solutions of nonlinear second order ordinary differential equations,” Acta Mathematica Sinica, vol. 45, no. 3, pp. 481–488, 2002 (Chinese).
• 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.
• 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.
• J. R. Graef, L. Kong, and H. Wang, “Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem,” Journal of Differential Equations, vol. 245, no. 5, pp. 1185–1197, 2008.
• K. Deimling, Nonlinear Functional Analysis, Springer, NewYork, NY, USA, 1985.
• D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, NewYork, NY, USA, 1988.