Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy
Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies |f(t,x)|≤μe−ε|t|. However, this condition is very restricted. There are few functions satisfying |f(t,x)|≤μe−ε|t|. In som...
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| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/725098 |
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| author | Yong-Hui Xia Xiaoqing Yuan Kit Ian Kou Patricia J. Y. Wong |
| author_facet | Yong-Hui Xia Xiaoqing Yuan Kit Ian Kou Patricia J. Y. Wong |
| author_sort | Yong-Hui Xia |
| collection | DOAJ |
| description | Nonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies |f(t,x)|≤μe−ε|t|. However, this condition is very restricted. There are few functions satisfying |f(t,x)|≤μe−ε|t|. In some sense, this assumption is not reasonable enough. More suitable assumption should be |f(t,x)|≤μ. To the best of the authors' knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption |f(t,x)|≤μ. In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper. |
| format | Article |
| id | doaj-art-c60d0d2915f54c04945608f1f1b90e3a |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-c60d0d2915f54c04945608f1f1b90e3a2025-02-03T01:07:21ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/725098725098Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential DichotomyYong-Hui Xia0Xiaoqing Yuan1Kit Ian Kou2Patricia J. Y. Wong3Department of Mathematics, Zhejiang Normal University, Jinhua 321004, ChinaDepartment of Mathematics, Zhejiang Normal University, Jinhua 321004, ChinaDepartment of Mathematics, Faculty of Science and Technology, University of Macau, Avenue Padre Tomás Pereira Taipa, Macau, ChinaSchool of Electrical and Electronic Engineering, Nanyang Technological University, 639798, SingaporeNonuniform exponential dichotomy has been investigated extensively. The essential condition of these previous results is based on the assumption that the nonlinear term satisfies |f(t,x)|≤μe−ε|t|. However, this condition is very restricted. There are few functions satisfying |f(t,x)|≤μe−ε|t|. In some sense, this assumption is not reasonable enough. More suitable assumption should be |f(t,x)|≤μ. To the best of the authors' knowledge, there is no paper considering the existence and uniqueness of solution to the perturbed nonautonomous system with a relatively conservative assumption |f(t,x)|≤μ. In this paper, we prove that if the nonlinear term is bounded, the perturbed nonautonomous system with nonuniform exponential dichotomy has a unique solution. The technique employed to prove Theorem 4 is the highlight of this paper.http://dx.doi.org/10.1155/2014/725098 |
| spellingShingle | Yong-Hui Xia Xiaoqing Yuan Kit Ian Kou Patricia J. Y. Wong Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy Abstract and Applied Analysis |
| title | Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy |
| title_full | Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy |
| title_fullStr | Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy |
| title_full_unstemmed | Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy |
| title_short | Existence and Uniqueness of Solution for Perturbed Nonautonomous Systems with Nonuniform Exponential Dichotomy |
| title_sort | existence and uniqueness of solution for perturbed nonautonomous systems with nonuniform exponential dichotomy |
| url | http://dx.doi.org/10.1155/2014/725098 |
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