Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump
No previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain <i>V</i>-jump process and an uncertain canonical process. First of all, for the one-dimension...
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| Format: | Article |
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MDPI AG
2020-05-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/12/5/765 |
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| author | Zhifu Jia Xinsheng Liu Cunlin Li |
| author_facet | Zhifu Jia Xinsheng Liu Cunlin Li |
| author_sort | Zhifu Jia |
| collection | DOAJ |
| description | No previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain <i>V</i>-jump process and an uncertain canonical process. First of all, for the one-dimensional case, we give two types of uncertain FDEs with jump that are symmetric in terms of form. The next, for the multidimensional case, when the coefficients of the equations satisfy Lipschitz condition and linear growth condition, we establish an existence and uniqueness theorems of uncertain FDEs with jump of Riemann-Liouville type by Banach fixed point theorem. A symmetric proof in terms of form is suitable to the Caputo type. When the coefficients do not satisfy the Lipschitz condition and linear growth condition, we just prove an existence theorem of the Caputo type equation by Schauder fixed point theorem. In the end, we present an application about uncertain interest rate model. |
| first_indexed | 2024-03-10T20:00:06Z |
| format | Article |
| id | doaj.art-316b574803584bcfaf7abce42e829187 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-10T20:00:06Z |
| publishDate | 2020-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-316b574803584bcfaf7abce42e8291872023-11-19T23:35:26ZengMDPI AGSymmetry2073-89942020-05-0112576510.3390/sym12050765Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with JumpZhifu Jia0Xinsheng Liu1Cunlin Li2State Key Laboratory of Mechanics Control of Mechanical Structures, Institute of Nano Science and Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, ChinaState Key Laboratory of Mechanics Control of Mechanical Structures, Institute of Nano Science and Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, ChinaNingxia Key Laboratory of Intelligent Information and Big Data Processing, Governance and Social Management Research Center of Northwest Ethnic Regions, North Minzu University, Yinchuan 750021, ChinaNo previous study has involved uncertain fractional differential equation (FDE, for short) with jump. In this paper, we propose the uncertain FDEs with jump, which is driven by both an uncertain <i>V</i>-jump process and an uncertain canonical process. First of all, for the one-dimensional case, we give two types of uncertain FDEs with jump that are symmetric in terms of form. The next, for the multidimensional case, when the coefficients of the equations satisfy Lipschitz condition and linear growth condition, we establish an existence and uniqueness theorems of uncertain FDEs with jump of Riemann-Liouville type by Banach fixed point theorem. A symmetric proof in terms of form is suitable to the Caputo type. When the coefficients do not satisfy the Lipschitz condition and linear growth condition, we just prove an existence theorem of the Caputo type equation by Schauder fixed point theorem. In the end, we present an application about uncertain interest rate model.https://www.mdpi.com/2073-8994/12/5/765uncertain fractional differential equationsV-jump processexistence and uniquenessbanach fixed point theoremschauder fixed point theorem |
| spellingShingle | Zhifu Jia Xinsheng Liu Cunlin Li Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump Symmetry uncertain fractional differential equations V-jump process existence and uniqueness banach fixed point theorem schauder fixed point theorem |
| title | Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump |
| title_full | Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump |
| title_fullStr | Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump |
| title_full_unstemmed | Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump |
| title_short | Fixed Point Theorems Applied in Uncertain Fractional Differential Equation with Jump |
| title_sort | fixed point theorems applied in uncertain fractional differential equation with jump |
| topic | uncertain fractional differential equations V-jump process existence and uniqueness banach fixed point theorem schauder fixed point theorem |
| url | https://www.mdpi.com/2073-8994/12/5/765 |
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