Abstract
In the paper, we consider the problem of finding a solution to a fractional order equation of the form \(D^{\alpha}_{t}u(t)+A\left(D^{\alpha}_{t}u(t)\right)+A^{2}\left(D^{\alpha}_{t}u(t)\right)+Au(t)=f\) , \(0<\alpha<1\) , and \(0<t<T\) , satisfying the non-local condition \(u(T)=au(+0)+\varphi\) . Here \(a\) and \(T\) are given numbers, \(A:H\rightarrow H\) be a self-adjoint, unbounded, and positive operator defined on a separable Hilbert space \(H\) . In this work, we examine the role of the parameter \(a\) in determining the existence and uniqueness of solutions to the associated problem. Furthermore, we consider the inverse problem of reconstructing the right-hand side of the equation based on additional information about the solution.
