# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Existence of positive periodic solutions for two kinds of neutral functional differential equations. (English) Zbl 1149.34040
The following two classes of neutral functional differential equations $$\frac{d}{dt} [ x(t) - c\,x(t-\tau) ] = -a(t) x(t) + f(t,x(t-\tau(t)))$$ and $$\frac{d}{dt} \left[ x(t) - c \int_{-\infty}^0 Q(r) x(t+r) dr \right] = -a(t) x(t) + b(t) \int_{-\infty}^0 Q(r) f(t,x(t+r)) dr$$ are considered, where $a$, $b \in C({\Bbb R},(0,\infty))$, $\tau \in C({\Bbb R},{\Bbb R})$, $f \in C({\Bbb R} \times {\Bbb R},{\Bbb R})$, and $a(t)$, $b(t)$, $\tau(t)$, $f(t,\cdot)$ are $\omega$-periodic functions, $\omega>0$ and $\vert c\vert <1$. Sufficient conditions for the existence of a positive $\omega$-periodic solution are obtained. The proof is based on Krasnoselskii’s fixed point theorem. The results obtained here are applied to various mathematical models.

##### MSC:
 34K13 Periodic solutions of functional differential equations 34K40 Neutral functional-differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text:
##### References:
 [1] Gopalsamy, K.: Stability and oscillation in delay differential equations of population dynamics. (1992) · Zbl 0752.34039 [2] Gurney, W. S. C.; Blythe, S. P.; Nisbet, R. M.: Nicholson’s blowflies revisited. Nature 287, 17-20 (1980) [3] Gil, M. I.: Existence and stability of periodic solutions of semilinear neutral type systems. Dyn. discrete contin. Syst. 7, 809-820 (2001) · Zbl 1021.34056 [4] Joseph, W.; So, H.; Yu, J.: Global attractivity and uniform persistence in Nicholson’s blowflies. Differential equations dynam. Systems 1, 11-18 (1994) · Zbl 0869.34056 [5] Jiang, D.; Wei, J.: Existence of positive periodic solutions for Volterra integro-differential equations. Acta math. Sci. 21B4, 553-560 (2002) · Zbl 1035.45003 [6] Li, Y.: Existence and global attractivity of a positive periodic solution of a class of delay differential equation. Sci. China 41A3, 273-284 (1998) · Zbl 0955.34057 [7] Luo, J.; Yu, J.: Global asymptotic stability of nonautonomous mathematical ecological equations with distributed deviating arguments. Acta math. Sinica 41, 1273-1282 (1998) · Zbl 1027.34088 [8] Li, Z.; Wang, X.: Existence of positive periodic solutions for neutral functional differential equations. Electron. J. Differential equations 34, 1-8 (2006) · Zbl 1099.34063 [9] Weng, P.: Existence and global attractivity of periodic solution of integrodifferential equation in population dynamics. Acta appl. Math. 4, 427-434 (1996) · Zbl 0886.45005 [10] Weng, P.; Liang, M.: The existence and behavior of periodic solution of hematopoiesis model. Math. appl. 4, 434-439 (1995) · Zbl 0949.34517 [11] Wan, A.; Jiang, D.: Existence of positive periodic solutions for functional differential equations. Kyushu J. Math. 1, 193-202 (2002) · Zbl 1012.34068 [12] 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 [13] You, B.: Ordinary differential equation complementary curriculum. (1982)