New Existence Theory for Fractional Langevin Equations

Objectives This work investigates the existence and uniqueness of solutions for Caputo fractional initial value problems of Langevin equations involving two fractional orders. It also aims to extend and complement existing results in the literature by proving that a larger class of such problems admits a unique solution compared to those previously reported. Material and Methods The analysis is carried out using Banach’s contraction principle together with weighted norms. The Banach fixed-point theorem is further employed to derive approximation results for the solutions. Results The results demonstrate that a larger class of fractional initial value problems of Langevin equations admits a unique solution compared to previously reported results. Additionally, approximation results are obtained, and illustrative examples are provided to support the theoretical findings. Conclusion The results extend the scope of existing work on fractional Langevin equations by establishing uniqueness for a broader class of problems. The application of fixed-point theory and weighted norms proves to be an effective framework for analysing such systems and deriving both theoretical and approximation results.