Forward and Inverse Problems for a Langevin-Type Fractional Equation Involving Non-Local Time Condition

The paper addresses the formulation and analysis of direct and inverse problems for a Langevin-type fractional differential equation under a non-local condition imposed on the time variable. An additional condition for solving the inverse problem has the form $u(t_0)=\omega$. The existence and uniqueness of a solution to the problems are established, and stability inequalities are derived. Criteria ensuring the uniqueness of the solution are identified.