Existence and stability analysis for Φ-Caputo generalized proportional fractional differential Langevin equation

This paper aims to investigate the existence, uniquness and stability results for a new class of Φ-Caputo generalized proportional fractional GPF differential Langevin equation. We present and discuss some of the characteristics of the generalized proportional fractional derivative. By making use of Krasnoselskii fixed point theorem and the Banach contraction principle, we established the existence and uniqueness results and also we investigate different kinds of stability such as Ulam-Hyers and generalized Ulam-Hyers stability. As application, we give an example to demonstrate our theoretical result