Positive periodic solutions of second-order differential equations with delays. (English) Zbl 1246.34037

Li, Yongxiang

Abstr. Appl. Anal. 2012, Article ID 829783, 13 p. (2012).

Summary: The existence results of positive \(\omega \)-periodic solutions are obtained for the second-order differential equation with delays \(-u'' + a(t) = f(t, u(t - \tau_1), \dots, u(t - \tau_n))\), where \(a \in C(\mathbb R,(0, \infty))\) is a \(\omega\)-periodic function, \(f : \mathbb R \times [0, \infty)^n \rightarrow [0, \infty)\) is a continuous function, which is \(\omega\)-periodic in \(t\), and \(\tau_1, \tau_2, \dots, \tau_n\) are positive constants. Our discussion is based on the fixed point index theory in cones.

MSC:

34C25

Periodic solutions to ordinary differential equations

Keywords:

positive periodic solutions; second-order differential equations

References:

[1]	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. · Zbl 1126.34362 · doi:10.1016/j.jmaa.2005.01.045
[2]	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. · Zbl 1097.34050 · doi:10.1016/j.aml.2005.02.031
[3]	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. · Zbl 1151.34056 · doi:10.1016/j.aml.2006.02.028
[4]	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. · Zbl 1118.34063 · doi:10.1016/j.jmaa.2006.06.084
[5]	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. · Zbl 1166.34038 · doi:10.1016/j.na.2007.12.011
[6]	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. · Zbl 1190.34083 · doi:10.1016/j.nonrwa.2008.10.023
[7]	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. · Zbl 1185.34093 · doi:10.1016/j.na.2009.02.064
[8]	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. · Zbl 1222.34085 · doi:10.1016/j.cnsns.2010.01.002
[9]	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. · Zbl 0993.34022 · doi:10.1016/S0377-0427(00)00438-6
[10]	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. · Zbl 1032.34040 · doi:10.1016/S0022-0396(02)00152-3
[11]	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). · Zbl 1018.34046
[12]	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. · Zbl 1073.34041 · doi:10.1142/S025295990400038X
[13]	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. · Zbl 1088.34038 · doi:10.1016/j.aml.2005.02.014
[14]	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. · Zbl 1203.34028 · doi:10.1016/j.jde.2008.06.012
[15]	K. Deimling, Nonlinear Functional Analysis, Springer, NewYork, NY, USA, 1985. · Zbl 0559.47040
[16]	D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, NewYork, NY, USA, 1988. · Zbl 0661.47045