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Open AccessArticle
On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces
by
Mieczysław Cichoń
Mieczysław Cichoń 1,*,†
,
Hussein A. H. Salem
Hussein A. H. Salem 2,†
and
Wafa Shammakh
Wafa Shammakh 3,†
1
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
2
Department of Mathematics and Computer Science, Faculty of Sciences, Alexandria University, Alexandria 5424041, Egypt
3
College of Science, Department of Mathematics and Statistics, University of Jeddah, Jeddah 21493, Saudi Arabia
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Mathematics 2024, 12(17), 2631; https://doi.org/10.3390/math12172631 (registering DOI)
Submission received: 30 July 2024
/
Revised: 22 August 2024
/
Accepted: 23 August 2024
/
Published: 24 August 2024
Abstract
As claimed in many papers, the equivalence between the Caputo-type fractional differential problem and the corresponding integral forms may fail outside the spaces of absolutely continuous functions, even in Hölder spaces. To avoid such an equivalence problem, we define a “new” appropriate fractional integral operator, which is the right inverse of the Caputo derivative on some Hölder spaces of critical orders less than 1. A series of illustrative examples and counter-examples substantiate the necessity of our research. As an application, we use our method to discuss the BVP for the Langevin fractional differential equation , for and some critical orders , combined with appropriate initial or boundary conditions, and with general classes of -tempered Hilfer problems with -tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied.
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MDPI and ACS Style
Cichoń, M.; Salem, H.A.H.; Shammakh, W.
On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces. Mathematics 2024, 12, 2631.
https://doi.org/10.3390/math12172631
AMA Style
Cichoń M, Salem HAH, Shammakh W.
On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces. Mathematics. 2024; 12(17):2631.
https://doi.org/10.3390/math12172631
Chicago/Turabian Style
Cichoń, Mieczysław, Hussein A. H. Salem, and Wafa Shammakh.
2024. "On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces" Mathematics 12, no. 17: 2631.
https://doi.org/10.3390/math12172631
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