Existence theory on the Caputo-type fractional differential Langevin hybrid inclusion with variable coefficient

This paper investigates a hybrid Langevin inclusion involving the generalized Caputo fractional derivative with a variable coefficient. The proposed Langevin inclusion is based on the ρ-Caputo fractional derivative, which extends the well-known fractional derivatives by incorporating a variable coefficient that depends on different values of the function ρ. By applying Dhage’s fixed point theorem, we establish the needed conditions for the existence of a solution. Based on this theorem, we investigate the problem by focusing on some cases for variable coefficients. To validate our theoretical findings, we provide a detailed example that highlights the practical significance and applicability of the proposed approach.