Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times
In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the...
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MDPI AG
2024-02-01
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| Online Access: | https://www.mdpi.com/2075-1680/13/2/126 |
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| author | Huifu Xia Yunfei Peng Peng Zhang |
| author_facet | Huifu Xia Yunfei Peng Peng Zhang |
| author_sort | Huifu Xia |
| collection | DOAJ |
| description | In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the continuous dependence and Gâteaux differentiable of the solution relative to the initial value are also presented in a new topology sense. For the autonomous impulsive system, the periodicity of the solution is given. As an application, the properties of the solution for a type of controlled nonlinear ordinary differential equation with impulses at variable times is obtained. These results are a foundation to study optimal control problems of systems governed by differential equations with impulses at variable times. |
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| format | Article |
| id | doaj.art-41f6e77a6b474cbdbf28fd0cef8f5db7 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-07T22:42:28Z |
| publishDate | 2024-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-41f6e77a6b474cbdbf28fd0cef8f5db72024-02-23T15:07:29ZengMDPI AGAxioms2075-16802024-02-0113212610.3390/axioms13020126Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable TimesHuifu Xia0Yunfei Peng1Peng Zhang2School of Mathematics and Statistics, Guizhou University, Guiyang 550025, ChinaSchool of Mathematics and Statistics, Guizhou University, Guiyang 550025, ChinaDepartment of Mathematics, Zunyi Normal University, Zunyi 563006, ChinaIn this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the continuous dependence and Gâteaux differentiable of the solution relative to the initial value are also presented in a new topology sense. For the autonomous impulsive system, the periodicity of the solution is given. As an application, the properties of the solution for a type of controlled nonlinear ordinary differential equation with impulses at variable times is obtained. These results are a foundation to study optimal control problems of systems governed by differential equations with impulses at variable times.https://www.mdpi.com/2075-1680/13/2/126differential equationimpulses at variable timesexistencequalitative theorypulse phenomena |
| spellingShingle | Huifu Xia Yunfei Peng Peng Zhang Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times Axioms differential equation impulses at variable times existence qualitative theory pulse phenomena |
| title | Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times |
| title_full | Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times |
| title_fullStr | Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times |
| title_full_unstemmed | Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times |
| title_short | Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times |
| title_sort | existence and properties of the solution of nonlinear differential equations with impulses at variable times |
| topic | differential equation impulses at variable times existence qualitative theory pulse phenomena |
| url | https://www.mdpi.com/2075-1680/13/2/126 |
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