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Positive periodic solutions of delayed differential equations. (English) Zbl 1248.34104

Summary: In this paper, we are concerned with the existence, multiplicity and nonexistence of positive ?-periodic solutions of the following equation

u '' (t)+a(t,u)u(t)=λb(t)f(u(t-τ(t))),t,

where a(·,·)C(×, + ) is a ω-periodc function with respect to the first variable, b(·)C(,[0,)), τ(·)C(,) are ω-periodic functions, fC([0,),[0,)) and f(s)>0 for s>0, λ>0 is a parameter. The proof of our main result is based upon fixed point index theory.

MSC:
34K13Periodic solutions of functional differential equations
47N20Applications of operator theory to differential and integral equations
References:
[1]Chu, J.; Torres, P. J.; Zhang, M.: Periodic solution of second order non-autonomous singular dynamical systems, J. differ. Eqs. 239, 196-212 (2007) · Zbl 1127.34023 · doi:10.1016/j.jde.2007.05.007
[2]Franco, D.; Torres, P. J.: Periodic solution of singular systems without the strong force condition, Proc. am. Math. soc. 136, 1229-1236 (2008) · Zbl 1129.37033 · doi:10.1090/S0002-9939-07-09226-X
[3]Jiang, D.; Chu, J.; Zhang, M.: Multiplicity of positive periodic solution to superlinear repulsive singular equation, J. differ. Eqs. 211, 282-302 (2005) · Zbl 1074.34048 · doi:10.1016/j.jde.2004.10.031
[4]Torres, P. J.; Zhang, M.: A monotone scheme for a nonlinear second order equation based on a generalized anti-maximum principle, Mathematische nachrichten 251, 101-107 (2003) · Zbl 1024.34030 · doi:10.1002/mana.200310033
[5]Freedman, H. I.; Wu, J.: Periodic solutions of single-species models with periodic delay, SIAM J. Math. anal. 23, 689-701 (1992) · Zbl 0764.92016 · doi:10.1137/0523035
[6]Wu, J.; Wang, Z.: Positive periodic solutions of second-order nonlinear differential systems with two parameters, Comput. math. Appl. 56, 43-54 (2008) · Zbl 1145.34333 · doi:10.1016/j.camwa.2007.07.017
[7]Cheng, S.; Zhang, G.: Existence of positive periodic solutions for non-autonomous functional differential equations, Electron. J. Differ. eqs. 59, 1-8 (2001) · Zbl 1003.34059 · doi:emis:journals/EJDE/Volumes/2001/59/abstr.html
[8]Torres, P. J.: Existence of one-signed periodic solution of some second order differential equations via a Krasnoselskii fixed point theorem, J. differ. Eqs. 190, 643-662 (2003) · Zbl 1032.34040 · doi:10.1016/S0022-0396(02)00152-3
[9]R.Y. Ma, C.H. Gao, R.P. Chen, Existence of positive solutions of nonlinear second-order periodic boundary value problems, Boundary value problems, Article ID 626054, p. 18, 2010. · Zbl 1219.34055 · doi:10.1155/2010/626054
[10]Guo, C. J.; Guo, Z. M.: Existence of multiple periodic solutions for a class of second-order delay differential equations, Nonlinear anal. Real world appl. 10, 3285-3297 (2009) · Zbl 1190.34083 · doi:10.1016/j.nonrwa.2008.10.023
[11]Ye, D.; Fan, M.; Wang, H.: Periodic solutions for scalar functional differential equations, Nonlinear anal. 62, 1157-1181 (2005) · Zbl 1089.34056 · doi:10.1016/j.na.2005.03.084
[12]Li, Y.; Fan, X.; Zhao, L.: Positive periodic solutions of functional differential equations with impulses and a parameter, Comput. math. Appl. 56, 2556-2560 (2008) · Zbl 1165.34401 · doi:10.1016/j.camwa.2008.05.007
[13]Jin, Z. L.; Wang, H. Y.: A note on positive periodic solutions of delayed differential equations, Appl. math. Lett. 23, 581-584 (2010) · Zbl 1194.34130 · doi:10.1016/j.aml.2010.01.015
[14]M. Zhang, Optimal conditions for maximum and antimaximum principle of the periodic solutions problem, Boundary value problems, Article ID 410986, p. 26, 2010. · Zbl 1200.34001 · doi:10.1155/2010/410986
[15]Wang, H. Y.: Positive periodic solutions of functional differential systems, J. differ. Eqs. 202, 354-366 (2004) · Zbl 1064.34052 · doi:10.1016/j.jde.2004.02.018