1. On the Equivalence between Differential and Integral Forms of Caputo-Type Fractional Problems on Hölder Spaces.
- Author
-
Cichoń, Mieczysław, Salem, Hussein A. H., and Shammakh, Wafa
- Subjects
- *
DIFFERENTIAL forms , *FRACTIONAL calculus , *FRACTIONAL integrals , *INTEGRAL operators , *CONTINUOUS functions - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF