# Delayed Langevin type equations with two fractional derivatives

## TL;DR

This paper introduces a new class of delayed Mittag-Leffler functions to explicitly solve linear fractional time-delay Langevin equations with two Riemann-Liouville derivatives, establishing existence, uniqueness, and stability of solutions.

## Contribution

It develops explicit solutions for fractional Langevin equations with delays using novel delayed Mittag-Leffler functions, extending the analytical tools for such equations.

## Key findings

- Explicit solutions derived using delayed Mittag-Leffler functions
- Proved existence and uniqueness of solutions
- Established Ulam-Hyers stability results

## Abstract

In this paper, we introduce a delayed Mittag-Leffler type function. With the help of the delayed Mittag-Leffler type functions, we give an explicit formula of solutions to linear nonhomogeneous fractional time-delay Langevin equations involving two Riemann-Liouville fractional derivatives. The existence and uniqueness of solutions are obtained by using an estimation of delayed Mittag-Leffler type functions in terms of exponential functions and a weighted norm via fixed point theorems. Further, we present Ulam--Hyers stability results.