Geodesic
On the theory of $\psi $-hilfer nonlocal Cauchy problem
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 2, pp. 159-175

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In this paper, we derive the representation formula of the solution for $\psi $-Hilfer fractional differential equation with constant coefficient in the form of Mittag-Leffler function by using Picard's successive approximation. Moreover, by using some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer, we introduce new results of some qualitative properties of solution such as existence and uniqueness. The generalized Gronwall inequality lemma is used in analyze $\mathrm{E}_{\alpha}$-Ulam-Hyers stability. Finally, one example to illustrate the obtained results.
Keywords: fractional differential equations, fractional derivatives, $\mathrm{E}_{\alpha}$-Ulam-Hyers stability, fixed point theorem. 
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     title = {On the theory of $\psi $-hilfer nonlocal {Cauchy} problem},
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Mohammed A. Almalahi; Satish K. Panchal. On the theory of $\psi $-hilfer nonlocal Cauchy problem. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 2, pp. 159-175. http://geodesic.mathdoc.fr/item/JSFU_2021_14_2_a3/