Riemann-Liouville fractional integrals and derivatives on Morrey spaces and applications to a Cauchy-type problem. (English) Zbl 07923033
Summary: We investigate the boundedness and compactness of Riemann-Liouville integral operators on Morrey spaces, a class of nonseparable function spaces. Instead of adopting dual or maximal viewpoints in integrable function spaces, our approach is based on the compactness of the truncated Riemann-Liouville fractional integrals, leveraging a criterion for strongly pre-compact sets. By constructing a truncated Marchaud fractional derivative function, we characterize the solution to Abel’s equation on Morrey spaces. Utilizing the fixed-point theorem, we establish the existence and uniqueness of solutions to a Cauchy-type problem for fractional differential equations. Additionally, we provide an illustrative example to demonstrate the sufficiency of the conditions presented in our main result.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 42B35 | Function spaces arising in harmonic analysis |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
Keywords:
Morrey space; Riemann-Liouville fractional integral; fractional differential equation; fixed-point theoremReferences:
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