On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces

Mieczysław Cichoń (), Hussein A. H. Salem and Wafa Shammakh
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Mieczysław Cichoń: Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland
Hussein A. H. Salem: Department of Mathematics and Computer Science, Faculty of Sciences, Alexandria University, Alexandria 5424041, Egypt
Wafa Shammakh: Department of Mathematics and Statistics, College of Science, University of Jeddah, Jeddah 21493, Saudi Arabia

Mathematics, 2024, vol. 12, issue 17, 1-23

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 d ψ β , μ d t β d ψ α , μ d t α + λ x ( t ) = f ( t , x ( t ) ) , t ∈ [ a , b ] , λ ∈ R , for f ∈ C [ a , b ] × R and some critical orders β , α ∈ ( 0 , 1 ) , combined with appropriate initial or boundary conditions, and with general classes of ψ -tempered Hilfer problems with ψ -tempered fractional derivatives. The BVP for fractional differential problems of the Bagley–Torvik type was also studied.

Keywords: fractional calculus; tempered derivative; Hölder space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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