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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.

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
Full Text: DOI
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