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Green’s Function for Periodic Solutions in Alternately Advanced and Delayed Differential Systems

Abstract

In this paper we investigate the existence of the periodic solutions of a nonlinear differential equation with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use the Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

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Acknowledgments

The author thanks the referees very much for their valuable suggestions which made this paper much improved. This research was in part supported by FGI 05-16 DIUMCE.

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Correspondence to Kuo-Shou Chiu.

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This research was in part supported by FGI 05-16 DIUMCE.

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Chiu, KS. Green’s Function for Periodic Solutions in Alternately Advanced and Delayed Differential Systems. Acta Math. Appl. Sin. Engl. Ser. 36, 936–951 (2020). https://doi.org/10.1007/s10255-020-0975-7

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  • DOI: https://doi.org/10.1007/s10255-020-0975-7

Keywords

  • piecewise constant arguments
  • Green’s function
  • periodic solutions
  • hybrid equations
  • fixed point theorems

2000 MR Subject Classification

  • 34A36
  • 34B27
  • 34K13
  • 37C25