Let \(E\) be a complex Banach space, \(A\) a closed linear operator on \(E\) and let \(\mu\) be a scalar function of bounded variation on \([0, T]\) such that \(t=0\) and \(t=T\) are variation points of \(d\mu(t)\). The author considers the Cauchy problem \[ \frac{du(t)}{dt} = A u(t), \quad t \in [0, T], \qquad \int_0^T u(t)\,d\mu(t) = u_1 \in E, \] and associates the characteristic function \(L(\lambda)= \int_0^T e^{\lambda t}\,d\mu(t)\), \(\lambda \in {\mathbb C}\). The main result of the paper states that if the Cauchy problem has a generalized solution on \([0, T]\), then this solution is unique if and only if no zero of \(L(\lambda)\) is an eigenvalue of \(A\). The author presents examples and connections related to the main result, motivating his investigation. He also deduces that if \(B\) is a closed linear operator on \(E\), then the Cauchy problem \[ B \frac{du(t)}{dt} = A u(t),\quad t \in [0, T],\qquad \int_0^T u(t)\,d\mu(t) = 0, \] has only the trivial solution \(u =0\) if and only if no zero of \(L(\lambda)\) is an eigenvalue of \(A - \lambda B\).
The paper is nicely written and gives interesting answers concerning the uniqueness of solutions of nonlocal problems.