John R. Graef, Bellabes Zineb, Boussekkine Naima, and Kadda Maazouz
Existence Results for Nonlinear Implicit Fractional Differential Equations Involving the Katugampola Derivative with Variable Order
Dynamic Systems and Applications 34 (2025)   151-164

https://doi.org/10.46719/dsa2025.34.08

ABSTRACT.
The authors introduce a fractional derivative with a variable order based on the Katugampola fractional derivative and integral. They show how to handle an implicit Dirichlet boundary value problem by applying a fixed point theorem for the sum of a contraction and compact maps due to Krasnosel’skii. They show that solutions to the problem exist and are unique. An example illustrates the results.

AMS (MOS) Subject Classification. 26A33, 34A08.

Key Words and Phrases. Katugampola derivative, variable order fractional operators, fixed point theorem.