On the Cauchy problem for the Langevin-type fractional equation

In this article, the Cauchy problem for the Langevin-type time-fractional equation Dβ t (Dtαu(t)) + Dtβ(Au(t)) = f(t); (0 < t ≤ T) is studied. Here α; β 2 (0; 1), Dtα; Dtβ is the Caputo derivative and A is an unbounded self-adjoint operator in a separable Hilbert space. Under certain conditions, we establish the existence and uniqueness of the solution and provide an explicit representation of it using eigenfunction expansions