Existence, and Ulam's types stability of higher-order fractional Langevin equations on a star graph

A study was conducted on the existence of solutions for a class of nonlinear Caputo type higher-order fractional Langevin equations with mixed boundary conditions on a star graph with $k+1$ nodes and $k$ edges. By applying a variable transformation, a system of fractional differential equations with mixed boundary conditions and different domains was converted into an equivalent system with identical boundary conditions and domains. Subsequently, the existence and uniqueness of solutions were verified using Krasnoselskii's fixed point theorem and Banach's contraction principle. In addition, the stability results of different types of solutions for the system were further discussed. Finally, two examples are illustrated to reinforce the main study outcomes.