Periodic solutions of semilinear impulsive periodic system on Banach space. (English) Zbl 1238.34079
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 12, e-Suppl., e1344-e1353 (2009).
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 |
Keywords:
semilinear impulsive periodic system; \(T_{0}\)-periodic PC-mild solution; Leray-Schauder fixed point theorem; generalized Gronwall’s inequalityReferences:
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