## Existence nonexistence and multiplicity of periodic solutions for a kind of functional differential equation with parameter.(English)Zbl 1166.34038

The author obtains some results on the existence, nonexistence and multiplicity of positive periodic solutions for a family of functional periodic differential equations of second order of the form
$u''(t)+a(t)u(t)=\lambda f(t,u(t-\tau_0(t)),u(t-\tau_1(t)),\dots,u(t-\tau_n(t))).$
The main tool used is the Krasnoselkii fixed point theorem in cones. For closely related results, see, e.g. [H. Y. Wang, J. Differ. Equations 202, No. 2, 354–366 (2004; Zbl 1064.34052)], [Y. Wu, Nonlinear Anal., Theory Methods Appl. 68, No. 7(A), 1954–1962 (2008; Zbl 1146.34049)].

### MSC:

 34K13 Periodic solutions to functional-differential equations 47H10 Fixed-point theorems 47N20 Applications of operator theory to differential and integral equations

### Citations:

Zbl 1064.34052; Zbl 1146.34049
Full Text:

### References:

 [1] Leela, S., Monotone method for second order periodic boundary value problem, Nonlinear anal. TMA, 7, 349-355, (1983) · Zbl 0524.34023 [2] Henderson, J.; Wang, H., Positive solutions for nonlinear eigenvalue problems, J. math. anal. appl., 208, 252-259, (1997) · Zbl 0876.34023 [3] Zhang, Z.; Wang, J., On existence and multiplicity of positive solution to periodic boundary value problem for singular nonlinear second order differential equations, J. math. anal. appl., 281, 99-107, (2003) · Zbl 1030.34024 [4] Cheng, S.; Zhang, G., Existence of positive periodic solutions for non-autonomous functional differential equations, Electron. J. differential equations, 59, 1-8, (2001) [5] Wang, H., Positive periodic solutions of functional differential equations, J. differential equations, 202, 354-366, (2004) · Zbl 1064.34052 [6] Wan, A.; Jiang, D.; Xu, X., A new existence theory for positive periodic solutions to functional differential equations, Comput. math. appl., 47, 1257-1262, (2004) · Zbl 1073.34082 [7] Li, Y., Positive periodic solutions of first and second order ordinary differential equations, Chinese ann. math., 25 B, 3, 413-420, (2004) · Zbl 1073.34041 [8] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag New York · Zbl 0559.47040 [9] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045 [10] Li, Y., Positive periodic solutions of nonlinear second order ordinary differential equations, Acta math. sinica, 45, 3, 481-488, (2002), (in Chinese) · Zbl 1018.34046 [11] Torres, P.J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. differential equations, 190, 643-662, (2003) · Zbl 1032.34040 [12] Jiang, D.; Wei, J., Existence of positive periodic solutions of non-autonomous functional differential equations, Chinese ann. math. A, 20, 6, 715-720, (1999), (in Chinese) · Zbl 0948.34046 [13] Mackey, M.C.; Glass, L., Oscillations and chaos in physiological control systems, Sciences, 197, 287-289, (1997) · Zbl 1383.92036 [14] Kuang, Y., Delay differential equations with application in population dynamics, (1993), Academic Press New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.