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In this paper, we propose a generalized Gronwall inequality through thefractional integral with respect to another function. The Cauchy-type problemfor a nonlinear differential equation involving the $\psi$-Hilfer fractionalderivative and the existence and uniqueness of solutions are discussed.Finally, through generalized Gronwall inequality, we prove the continuousdependence of data on the Cauchy-type problem.

A Gronwall inequality and the Cauchy-type problem by means of $\psi$-Hilfer operator