Investigation of Langevin equation in terms of generalized proportional fractional derivatives with respect to another function

Abstract

The current work concerns the existence and uniqueness results for a nonlinear Langevin equation involving two generalized proportional fractional operators with respect to another function. The main results are proved by means of Krasnoselskii’s fixed point theorem and the Banach contraction principle. An example is set forth to make efficient our main results.