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Mathematics 2019, 7(4), 333; https://doi.org/10.3390/math7040333

Hyers–Ulam Stability and Existence of Solutions for Differential Equations with Caputo–Fabrizio Fractional Derivative

1
Department of Mathematics, Guizhou University, Guiyang 550025, China
2
Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia
3
Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia
4
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 TK33 Galway, Ireland
5
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
*
Author to whom correspondence should be addressed.
Received: 14 March 2019 / Revised: 30 March 2019 / Accepted: 1 April 2019 / Published: 5 April 2019
PDF [263 KB, uploaded 5 April 2019]

Abstract

In this paper, the Hyers–Ulam stability of linear Caputo–Fabrizio fractional differential equation is established using the Laplace transform method. We also derive a generalized Hyers–Ulam stability result via the Gronwall inequality. In addition, we establish existence and uniqueness of solutions for nonlinear Caputo–Fabrizio fractional differential equations using the generalized Banach fixed point theorem and Schaefer’s fixed point theorem. Finally, two examples are given to illustrate our main results.
Keywords: Caputo–Fabrizio fractional differential equations; Hyers–Ulam stability Caputo–Fabrizio fractional differential equations; Hyers–Ulam stability
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Liu, K.; Fečkan, M.; O’Regan, D.; Wang, J. Hyers–Ulam Stability and Existence of Solutions for Differential Equations with Caputo–Fabrizio Fractional Derivative. Mathematics 2019, 7, 333.

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