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The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space

Abstract

This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, the -periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of Poincaré operator and using a new generalized Gronwall's inequality with impulse, mixed type integral operators and -norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of -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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Correspondence to Jin Rong Wang.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Wang, J.R., Xiang, X., Wei, W. et al. The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space. J Inequal Appl 2008, 430521 (2008). https://doi.org/10.1155/2008/430521

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Keywords

  • Banach Space
  • Periodic Solution
  • Integral Operator
  • Point Theorem
  • Fixed Point Theorem