Green's function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications

Abstract

In this paper we investigate the existence of the periodic solutions of a nonlinear impulsive differential system with piecewise alternately advanced and retarded arguments, in short IDEPCAG, 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 system are obtained. In the process we use the Green's function for impulsive periodic solutions and convert the given the IDEPCAG into an equivalent integral equation system. 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 impulsive differential systems. We also use the contraction mapping principle to show the existence of a unique impulsive periodic solution. Appropriate examples are given to show the feasibility of our results.

Keywords

• Impulsive differential equation
• Piecewise constant arguments of generalized type
• Green's function
• Periodic solutions
• Fixed point theorems

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