Some existence and stability results of Hilfer-Hadamard fractional implicit differential fractional equation in a weighted space

Abstract

This paper studies a nonlinear fractional implicit differential equation (FIDE) with boundary conditions involving a Hilfer-Hadamard type fractional derivative. We establish the equivalence between the Cauchy-type problem (FIDE) and its mixed type integral equation through a variety of tools of some properties of fractional calculus and weighted spaces of continuous functions. The existence and uniqueness of solutions are obtained. Further, the Ulam-Hyers and Ulam-Hyers-Rassias stability are discussed. The arguments in the analysis rely on Schaefer fixed point theorem, Banach contraction principle and generalized Gronwall inequality. At the end, an illustrative example will be introduced to justify our results.