ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

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

Discontinuity, Nonlinearity, and Complexity 10(2) (2021) 207--225 | DOI:10.5890/DNC.2021.06.004

Laxman A. Palve , Mohammed S. Abdo, Satish K. Panchal

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.

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./2014-15/RGNF-2014-15D-OBC-MAH-84864.\newline

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