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Mathematics 2019, 7(4), 333;

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

Department of Mathematics, Guizhou University, Guiyang 550025, China
Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia
Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia
School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, H91 TK33 Galway, Ireland
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]


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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