The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space
This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, the T0-periodic PC-mild solution is introduced and suitable Poincaré operator is constructed. Showing the compactness of Poincaré operator a...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2008-12-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | http://dx.doi.org/10.1155/2008/430521 |
| _version_ | 1819054612146028544 |
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| author | Qian Chen W. Wei X. Xiang JinRong Wang |
| author_facet | Qian Chen W. Wei X. Xiang JinRong Wang |
| author_sort | Qian Chen |
| collection | DOAJ |
| description | This paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, the T0-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 B-norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of T0-periodic PC-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration. |
| first_indexed | 2024-12-21T12:54:23Z |
| format | Article |
| id | doaj.art-f9b7922fff7c4db48b443f0e97fb7f92 |
| institution | Directory Open Access Journal |
| issn | 1025-5834 1029-242X |
| language | English |
| last_indexed | 2024-12-21T12:54:23Z |
| publishDate | 2008-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-f9b7922fff7c4db48b443f0e97fb7f922022-12-21T19:03:24ZengSpringerOpenJournal of Inequalities and Applications1025-58341029-242X2008-12-01200810.1155/2008/430521The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach SpaceQian ChenW. WeiX. XiangJinRong WangThis paper deals with a class of integrodifferential impulsive periodic systems on Banach space. Using impulsive periodic evolution operator given by us, the T0-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 B-norm given by us, we utilize Leray-Schauder fixed point theorem to prove the existence of T0-periodic PC-mild solutions. Our method is much different from methods of other papers. At last, an example is given for demonstration.http://dx.doi.org/10.1155/2008/430521 |
| spellingShingle | Qian Chen W. Wei X. Xiang JinRong Wang The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space Journal of Inequalities and Applications |
| title | The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space |
| title_full | The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space |
| title_fullStr | The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space |
| title_full_unstemmed | The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space |
| title_short | The Generalized Gronwall Inequality and Its Application to Periodic Solutions of Integrodifferential Impulsive Periodic System on Banach Space |
| title_sort | generalized gronwall inequality and its application to periodic solutions of integrodifferential impulsive periodic system on banach space |
| url | http://dx.doi.org/10.1155/2008/430521 |
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