Opuscula Math. 39, no. 6 (2019), 839-862

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.
  • Received: 2019-02-26.
  • Revised: 2019-08-08.
  • Accepted: 2019-08-12.
  • Published online: 2019-11-22.
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Cite this article as:
Chuanxi Qian, Justin Smith, On existence and global attractivity of periodic solutions of nonlinear delay differential equations, Opuscula Math. 39, no. 6 (2019), 839-862, https://doi.org/10.7494/OpMath.2019.39.6.839

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