Periodic solutions of a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space

JinRong Wang, Guizhou University, Guiyang, Guizhou, P. R. China
X. Xiang, Guizhou University, Guiyang, Guizhou, P. R. China
W. Wei, Guizhou University, Guiyang, Guizhou, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 4. (2009), pp. 1-17.

Abstract: This paper deals with a class of integrodifferential impulsive periodic systems with time-varying generating operators on Banach space. Using impulsive periodic evolution operator given by us, the suitable $T_{0}$-periodic $PC$-mild solution is introduced and $Poincar\acute{e}$ operator is constructed. Showing the compactness of $Poincar\acute{e}$ operator and using a new generalized Gronwall's inequality with impulse, mixed type integral operators and $B$-norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of $T_{0}$-periodic $PC$-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration.

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