Opuscula Math. 39, no. 6 (2019), 839-862
https://doi.org/10.7494/OpMath.2019.39.6.839

Opuscula Mathematica

# On existence and global attractivity of periodic solutions of nonlinear delay differential equations

Chuanxi Qian
Justin Smith

Abstract. Consider the delay differential equation with a forcing term $\tag{$$\ast$$} x'(t)=-f(t,x(t))+g(t,x(t-\tau ))+r(t), \quad t \geq 0$ where $$f(t,x): [0,\infty) \times [0,\infty) \to \mathbb{R}$$, $$g(t,x): [0,\infty) \times [0,\infty) \to [0,\infty)$$ are continuous functions and $$\omega$$-periodic in $$t$$, $$r(t): [0,\infty) \to\mathbb{R}$$ is a continuous function and $$\tau \in (0,\infty)$$ is a positive constant. We first obtain a sufficient condition for the existence of a unique nonnegative periodic solution $$\tilde{x}(t)$$ of the associated unforced differential equation of Eq. ($$\ast$$) $\tag{$$\ast\ast$$} x'(t)=-f(t,x(t))+g(t,x(t-\tau)), \quad t \geq 0.$ Then we obtain a sufficient condition so that every nonnegative solution of the forced equation ($$\ast$$) converges to this nonnegative periodic solution $$\tilde{x}(t)$$ of the associated unforced equation($$\ast\ast$$). Applications from mathematical biology and numerical examples are also given.

Keywords: delay differential equation, periodic solution, global attractivity.

Mathematics Subject Classification: 34K13, 34K20, 34K25.

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1. R.P. Agarwal, L. Berezansky, E. Braverman, A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, 2012.
2. L. Berezansky, E. Braverman, A. Domoshnitsky, First order functional differential equations: nonoscillation and positivity of Green's functions, Funct. Diff. Equ. 15 (2008), 57-94.
3. L. Berezansky, E. Braverman, L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model. 34 (2010), 1405-1417.
4. L. Berezansky, E. Braverman, L. Idels, Mackey-Glass model of hematopoiesis with monotone feedback revisited, Appl. Math. Comput. 219 (2013), 4892-4907.
5. F. Brauer, C. Castillo-Chávez, Mathematical models in population biology and epidemiology, Texts in Applied Mathematics, vol. 40, Springer-Verlag, New York, 2001.
6. E. Braverman, S.H. Saker, On a difference equation with exponentially decreasing nonlinearity, Discrete Dyn. Nat. Soc. (2011), Art. ID 147926, 17 pp.
7. H.-S. Ding, M.-X. Ji, Pseudo-almost periodic solutions for a discrete Nicholson's blowflies model with harvesting term, Adv. Difference Equ. (2016), Paper no. 289, 11 pp.
8. A. Domoshnitsky, Maximum principles and nonoscillation intervals for first order Volterra functional differential equations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 15 (2008), 769-814.
9. S.A. Gusarenko, A.I. Domoshnitskii, Asymptotic and oscillation properties of first order linear scalar functional-differential equations, Differentsial'nye Uravneniya 25 (1989), 2090-2103.
10. I. Györi, G. Ladas, Oscillation Theory of Delay Differential Equations, Oxford University Press, New York, 1991.
11. D.D. Hai, C. Qian, On global convergence of forced nonlinear delay differential equations and applications, Differ. Equ. Appl. 9 (2017), 13-28.
12. G. Kiss, G. Röst, Controlling Mackey-Glass chaos, Chaos 27 (2017), 114321, 7 pp.
13. M. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science 197 (1977), 287-289.
14. R. Olach, Positive periodic solutions of delay differential equations, Appl. Math. Lett. 26 (2013), 1141-1145.
15. S. Padhi, S. Pati, R. Kumar, Positive periodic solutions of Nicholson's blowflies model with harvesting, PanAmerican Math. Journal 24 (2014), 15-26.
16. C. Qian, Global attractivity of periodic solutions in a delay differential equation, Comm. in Appl. Anal. 18 (2014), 253-260.
17. A. Wan, D. Jiang, X. Xu, A new existence theory for positive periodic solutions to functional differential equations, Comput. Math. Appl. 47 (2004), 1257-1262.
18. W. Wang, Positive periodic solutions of delayed Nicholson's blowflies models with a nonlinear density-dependent mortality term, Appl. Math. Model. 36 (2012), 4708-4713.
• Communicated by Alexander Domoshnitsky.
• Revised: 2019-08-08.
• Accepted: 2019-08-12.
• Published online: 2019-11-22. 