Discontinuity, Nonlinearity, and Complexity
Some Existence and Stability Results of HilferHadmard Fractional Implicit
Differential Equation in a Weighted Space
Discontinuity, Nonlinearity, and Complexity 10(2) (2021) 207225  DOI:10.5890/DNC.2021.06.004
Laxman A. Palve , Mohammed S. Abdo, Satish K. Panchal
Department of Mathematics, Dr.Babasaheb Ambedkar Marathwada University,
Aurangabad, (M.S), 431001, India
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Abstract
This paper studies a nonlinear fractional implicit
differential equation (FIDE) with boundary conditions involving a
HilferHadamard type fractional derivative. We establish the equivalence
between the Cauchytype 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 UlamHyers and UlamHyersRassias
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.
Acknowledgments
The authors would like to thank the referees for their careful reading of
the manuscript and insightful comments, which helped improve the quality of
the paper. The first author is grateful to the UGC, New Delhi for the award
of National Fellowship for Persons with Disabilities
No.F./201415/RGNF201415DOBCMAH84864.\newline
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