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Existence of maximal and minimal periodic solutions for first-order functional differential equations. (English) Zbl 1186.34092

Summary: One important question in population models is whether periodic solutions exist and whether they are bounded between minimal and maximal solutions. This paper deals with the existence of maximal and minimal periodic solutions for the periodic solutions of a first-order functional differential equation

$y{}^{\text{'}}\left(t\right)=-a\left(t\right)y\left(t\right)+f\left(t,y\left(t-\tau \left(t\right)\right)\right)$

by using the method of lower and upper solutions.

##### MSC:
 34K13 Periodic solutions of functional differential equations 47N20 Applications of operator theory to differential and integral equations
##### References:
 [1] Cheng, S. S.; Zhang, G.: Existence of positive periodic solutions for non-autonomous functional differential equations, Electron. J. Differential equations 2001, No. 59, 1-8 (2001) [2] Zhang, G.; Cheng, S. S.: Positive periodic solutions of nonautonomous functional differential equations depending on a parameter, Abstr. appl. Anal. 7, No. 5, 256-269 (2002) · Zbl 1007.34066 · doi:10.1155/S1085337502000878 [3] Jiang, D. Q.; Wei, J. J.: Existence of positive periodic solutions of nonautonomous functional differential equations, Chinese ann. Math. 20, No. 6, 715-720 (1999) · Zbl 0948.34046 [4] Kang, S. G.; Zhang, G.: Existence of nontrivial periodic solutions for first order functional differential equations, Appl. math. Lett. 18, 101-107 (2005) · Zbl 1075.34064 · doi:10.1016/j.aml.2004.07.018 [5] Kang, S. G.; Zhang, G.; Cheng, S. S.: Periodic solutions of a class of integral equations, Topol. methods nonlinear anal. 22, 245-252 (2003) · Zbl 1042.45002 [6] Shi, B.; Zhang, D. C.; Gai, M. J.: Theory and applications of differential equations, (2005)