# Article

Full entry | PDF   (0.2 MB)
Keywords:
neutral delay differential equation; positive periodic solution; cone; fixed point index
Summary:
This paper deals with the existence of positive $\omega$-periodic solutions for the neutral functional differential equation with multiple delays $$(u(t)-cu(t-\delta ))'+a(t) u(t)=f(t, u(t-\tau _1), \cdots , u(t-\tau _n)).$$ The essential inequality conditions on the existence of positive periodic solutions are obtained. These inequality conditions concern with the relations of $c$ and the coefficient function $a(t)$, and the nonlinearity $f(t, x_1,\cdots , x_n)$. Our discussion is based on the perturbation method of positive operator and fixed point index theory in cones.
References:
[1] Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985). DOI 10.1007/978-3-662-00547-7 | MR 0787404 | Zbl 0559.47040
[2] Freedman, H. I., Wu, J.: Periodic solutions of single-species models with periodic delay. SIAM J. Math. Anal. 23 (1992), 689-701. DOI 10.1137/0523035 | MR 1158828 | Zbl 0764.92016
[3] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Notes and Reports in Mathematics in Science and Engineering 5. Academic Press, Boston (1988). MR 0959889 | Zbl 0661.47045
[4] Hale, J. K.: Theory of Functional Differential Equations. Applied Mathematical Sciences. Vol. 3. Springer, New York (1977). DOI 10.1007/978-1-4612-9892-2 | MR 0508721 | Zbl 0352.34001
[5] Hatvani, L., Krisztin, T.: On the existence of periodic solutions for linear inhomogeneous and quasilinear functional differential equations. J. Differ. Equations 97 (1992), 1-15. DOI 10.1016/0022-0396(92)90080-7 | MR 1161308 | Zbl 0758.34054
[6] Kang, S., Zhang, G.: Existence of nontrivial periodic solutions for first order functional differential equations. Appl. Math. Lett. 18 (2005), 101-107. DOI 10.1016/j.aml.2004.07.018 | MR 2121560 | Zbl 1075.34064
[7] Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Mathematics in Science and Engineering 191. Academic Press, Boston (1993). MR 1218880 | Zbl 0777.34002
[8] Luo, Y., Wang, W., Shen, J.: Existence of positive periodic solutions for two kinds of neutral functional differential equations. Appl. Math. Lett. 21 (2008), 581-587. DOI 10.1016/j.aml.2007.07.009 | MR 2412382 | Zbl 1149.34040
[9] Mallet-Paret, J., Nussbaum, R. D.: Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation. Ann. Math. Pure Appl. (4) 145 (1986), 33-128. DOI 10.1007/BF01790539 | MR 0886709 | Zbl 0617.34071
[10] Serra, E.: Periodic solutions for some nonlinear differential-equations of neutral type. Nonlinear Anal., Theory Methods Appl. 17 (1991), 139-151. DOI 10.1016/0362-546X(91)90217-O | MR 1118073 | Zbl 0735.34066
[11] Wan, A., Jiang, D.: Existence of positive periodic solutions for functional differential equations. Kyushu J. Math. 56 (2002), 193-202. DOI 10.2206/kyushujm.56.193 | MR 1898353 | 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 (2004), 1257-1262. DOI 10.1016/S0898-1221(04)90120-4 | MR 2070981 | Zbl 1073.34082

Partner of