Abstract
In this paper, the Schaefer's fixed-point theorem is used to investigate the existence of solutions to nonlocal initial value problems for implicit differential equations with Hilfer–Hadamard fractional derivative. Then the Ulam stability result is obtained by using Banach contraction principle. An example is given to illustrate the applications of the main result.
Highlights
Fractional differential equations (FDEs) have been applied in many fields such as physics, mechanics, chemistry, engineering etc
There has been a significant development in ordinary differential equations involving fractional-order derivatives, one can see the monographs of Hilfer [19], Kilbas [16] and Podlubny [18] and the references therein
Some mathematicians have considered FDEs depending on the Hadamard fractional derivative [2, 6, 7]
Summary
Fractional differential equations (FDEs) have been applied in many fields such as physics, mechanics, chemistry, engineering etc. There has been a significant development in ordinary differential equations involving fractional-order derivatives, one can see the monographs of Hilfer [19], Kilbas [16] and Podlubny [18] and the references therein. Some mathematicians have considered FDEs depending on the Hadamard fractional derivative [2, 6, 7]. We adopt some ideas in [24] to establish an equivalent mixed-type integral equation. We study different types of Ulam stability: Ulam– Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability and generalized Ulam–Hyers–Rassias stability for the IDEs with Hilfer–Hadamard fractional derivative. The Ulam–Hyers stability for FDEs with Hilfer fractional derivative was investigated in [1, 22].
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