Some Existence and Stability Results of Hilfer-Hadmard Fractional Implicit Differential Equation in a Weighted Space
2021
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.
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