On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem

Luís P. Castro; Anabela S. Silva; Luís P. Castro; Anabela S. Silva

doi:10.3934/mbe.2022505

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 11: 10809-10825. doi: 10.3934/mbe.2022505

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On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem

Luís P. Castro ^,,
Anabela S. Silva

CIDMA–Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal

Academic Editor: Feliz Manuel Barrão Minhós

Received: 16 June 2022 Revised: 20 July 2022 Accepted: 21 July 2022 Published: 28 July 2022

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.
- boundary value problem,
- Caputo fractional derivative,
- fixed point,
- Ulam-Hyers stability,
- Ulam-Hyers-Rassias stability
Citation: Luís P. Castro, Anabela S. Silva. On the solution and Ulam-Hyers-Rassias stability of a Caputo fractional boundary value problem[J]. Mathematical Biosciences and Engineering, 2022, 19(11): 10809-10825. doi: 10.3934/mbe.2022505

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Abstract

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.

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