Periodicity and continuous dependence in iterative differential equations

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Abstract

In this work, we study the existence, uniqueness and continuous dependence of periodic solutions of the iterative differential equation

$$\begin{aligned} x^{\prime }(t)=\sum \limits _{m=1}^{N}\sum \limits _{l=1}^{\infty }C_{l,m}(t)\left( x^{[m]}(t)\right) ^{l}+\frac{d}{dt}g\left( t,x^{[1]}(t),x^{[2]} (t),\ldots ,x^{[N]}(t)\right) +h(t). \end{aligned}$$

Using Schauder’s fixed point theorem, we obtain the existence of periodic solution and by the contraction mapping principle we obtain the uniqueness. An example is given to illustrate this work. The results obtained here extend the work of Zhao and Feckan [14].

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Acknowledgements

The authors would like to thank the anonymous referee for his valuable comments.

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Correspondence to Abdelouaheb Ardjouni.

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Mansouri, B., Ardjouni, A. & Djoudi, A. Periodicity and continuous dependence in iterative differential equations. Rend. Circ. Mat. Palermo, II. Ser 69, 561–576 (2020). https://doi.org/10.1007/s12215-019-00420-5

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Keywords

  • Fixed point
  • Periodic solutions
  • Stability
  • Iterative differential equations

Mathematics Subject Classification

  • Primary 34K13
  • 34A34
  • Secondary 34K30
  • 34L30