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Periodic solutions of semilinear impulsive periodic system on Banach space. (English) Zbl 1238.34079
Summary: A class of semilinear impulsive periodic systems on Banach space is considered. Using impulsive periodic evolution operator, the $$T_{0}$$-periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of the Poincaré operator and using a new generalized Gronwall inequality with mixed type integral operators, we utilize the Leray-Schauder fixed point theorem to prove the existence of the $$T_{0}$$-periodic PC-mild solutions. Our method is much different from methods of other papers. An example illustrates the applicability of our results.

##### MSC:
 34C25 Periodic solutions to ordinary differential equations 34A37 Ordinary differential equations with impulses 47H10 Fixed-point theorems 34G20 Nonlinear differential equations in abstract spaces 47D06 One-parameter semigroups and linear evolution equations 47N20 Applications of operator theory to differential and integral equations
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