On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem
In this paper, we investigate a class of boundary value problems involving Caputo fractional derivative $ {{}^C\mathcal{D}^{\alpha}_{a}} $ of order $ \alpha \in (2, 3) $, and the usual derivative, of the form $ \begin{equation*} ({{}^C\mathcal{D}^{\alpha}_{a}}x)(t)+p(t)x'(t)+q(t)x(t) = g(t),...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-07-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2022505?viewType=HTML |
| _version_ | 1811309122179563520 |
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| author | Luís P. Castro Anabela S. Silva |
| author_facet | Luís P. Castro Anabela S. Silva |
| author_sort | Luís P. Castro |
| collection | DOAJ |
| description | In this paper, we investigate a class of boundary value problems involving Caputo fractional derivative $ {{}^C\mathcal{D}^{\alpha}_{a}} $ of order $ \alpha \in (2, 3) $, and the usual derivative, of the form
$ \begin{equation*} ({{}^C\mathcal{D}^{\alpha}_{a}}x)(t)+p(t)x'(t)+q(t)x(t) = g(t), \quad a\leq t\leq b, \end{equation*} $
for an unknown $ x $ with $ x(a) = x'(a) = x(b) = 0 $, and $ p, \; q, \; g\in C^2([a, b]) $. The proposed method uses certain integral inequalities, Banach's Contraction Principle and Krasnoselskii's Fixed Point Theorem to identify conditions that guarantee the existence and uniqueness of the solution (for the problem under study) and that allow the deduction of Ulam-Hyers and Ulam-Hyers-Rassias stabilities. |
| first_indexed | 2024-04-13T09:36:35Z |
| format | Article |
| id | doaj.art-6feaae2a298c40cb9ef7a95ec67cadad |
| institution | Directory Open Access Journal |
| issn | 1551-0018 |
| language | English |
| last_indexed | 2024-04-13T09:36:35Z |
| publishDate | 2022-07-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj.art-6feaae2a298c40cb9ef7a95ec67cadad2022-12-22T02:52:05ZengAIMS PressMathematical Biosciences and Engineering1551-00182022-07-011911108091082510.3934/mbe.2022505On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problemLuís P. Castro0Anabela S. Silva1CIDMA–Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, PortugalCIDMA–Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, PortugalIn this paper, we investigate a class of boundary value problems involving Caputo fractional derivative $ {{}^C\mathcal{D}^{\alpha}_{a}} $ of order $ \alpha \in (2, 3) $, and the usual derivative, of the form $ \begin{equation*} ({{}^C\mathcal{D}^{\alpha}_{a}}x)(t)+p(t)x'(t)+q(t)x(t) = g(t), \quad a\leq t\leq b, \end{equation*} $ for an unknown $ x $ with $ x(a) = x'(a) = x(b) = 0 $, and $ p, \; q, \; g\in C^2([a, b]) $. The proposed method uses certain integral inequalities, Banach's Contraction Principle and Krasnoselskii's Fixed Point Theorem to identify conditions that guarantee the existence and uniqueness of the solution (for the problem under study) and that allow the deduction of Ulam-Hyers and Ulam-Hyers-Rassias stabilities. https://www.aimspress.com/article/doi/10.3934/mbe.2022505?viewType=HTMLboundary value problemcaputo fractional derivativefixed pointulam-hyers stabilityulam-hyers-rassias stability |
| spellingShingle | Luís P. Castro Anabela S. Silva On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem Mathematical Biosciences and Engineering boundary value problem caputo fractional derivative fixed point ulam-hyers stability ulam-hyers-rassias stability |
| title | On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem |
| title_full | On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem |
| title_fullStr | On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem |
| title_full_unstemmed | On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem |
| title_short | On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem |
| title_sort | on the solution and ulam hyers rassias stability of a caputo fractional boundary value problem |
| topic | boundary value problem caputo fractional derivative fixed point ulam-hyers stability ulam-hyers-rassias stability |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2022505?viewType=HTML |
| work_keys_str_mv | AT luispcastro onthesolutionandulamhyersrassiasstabilityofacaputofractionalboundaryvalueproblem AT anabelassilva onthesolutionandulamhyersrassiasstabilityofacaputofractionalboundaryvalueproblem |