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**Existence of periodic solutions for integrodifferential impulsive periodic system on Banach space.**
*(English)*
Zbl 1160.45006

Summary: This paper deals with a class of integrodifferential impulsive periodic systems on a Banach space. Using the impulsive periodic evolution operator given by us, the \(T_{0}\)-periodic PC-mild solution is introduced and a suitable Poincaré operator is constructed. By virtue of the generalized new Gronwall lemma with impulse and \(B\)-norm, an estimate on the PC-mild solutions is derived. Showing the continuity and compactness of the Poincaré operator, we utilize Horn’s fixed point theorem to prove the existence of \(T_{0}\)-periodic PC-mild solutions when the PC-mild solutions are bounded and ultimate bounded. This extends the study of periodic solutions of integrodifferential periodic system without impulse to integrodifferential periodic system with impulse on general Banach spaces. At last, an example is given for demonstration.

### MSC:

45N05 | Abstract integral equations, integral equations in abstract spaces |

45J05 | Integro-ordinary differential equations |

45M15 | Periodic solutions of integral equations |

45G10 | Other nonlinear integral equations |

### Keywords:

periodic solution; integrodifferential impulsive periodic systems; Banach space; impulsive periodic evolution operator; Poincaré operator; Horn’s fixed point theorem
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\textit{J. Wang} et al., Abstr. Appl. Anal. 2008, Article ID 939062, 19 p. (2008; Zbl 1160.45006)

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