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\(E_\alpha\)-Ulam type stability of fractional order ordinary differential equations. (English) Zbl 1296.34035
Summary: In this paper, the concepts of \(\mathbb E_\alpha\)-Ulam-Hyers stability, generalized \(\mathbb E_\alpha\)-Ulam-Hyers stability, \(\mathbb E_\alpha\)-Ulam-Hyers-Rassias stability and generalized \(\mathbb E_\alpha\)-Ulam-Hyers-Rassias stability for fractional order ordinary differential equations are raised. Without loss of generality, \(\mathbb E_\alpha\)-Ulam-Hyers-Rassias stability result is derived by using a singular integral inequality of Gronwall type. Two examples are also provided to illustrate our results.

MSC:
34A08 Fractional ordinary differential equations and fractional differential inclusions
34D10 Perturbations of ordinary differential equations
45N05 Abstract integral equations, integral equations in abstract spaces
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