Existence and stability of Langevin equations with two Hilfer-Katugampola fractional derivatives

Abstract

In this note, we debate the existence, uniqueness and stability results for
a general class of Langevin equations. We suggest the generalization via the Hilfer- Katugampola fractional derivative. We introduce some conditions for existence and uniqueness of solutions. We utilize the concept of fixed point theorems (Krasnoselskii fixed point theorem (KFPT), Banach contraction principle (BCP)). Moreover, we illustrate definitions of the Ulam type stability. These definitions generalize the fractional
Ulam stability.