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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• Communicated by Alexander Domoshnitsky.
• Revised: 2019-08-08.
• Accepted: 2019-08-12.
• Published online: 2019-11-22.