Summary: Given \(T > 0\) the author considers the inverse problem in a Banach space \(E\) \[ du(t)/dt = Au(t) + \Phi(t) f, \quad 0 \leq t \leq T, \]
\[ u(0) = u_{0}, u(T) = u_{1}, \quad u_{0}, u_{1} \in D(A), \] where the element \(f \in E\) is unknown. The main result may be written as follows: Let \(E=L^{1}(X, {\mu})\) and let \(A\) be the infinitesimal generator of a \(C_{0}\) semigroup \(U(t)\) on \(L^{1} (X,{\mu})\) satisfying \(\|U(t)\|< 1\) for \(t > 0\). Let \(\Phi(t)\) be defined by \[ (\Phi(t))(x) = {\phi}(x,t)f(x) \] with \({\phi}\in C^{1}([0,T]; L^{\infty}(X,{\mu}))\). Suppose that \({\phi}(x,t) \geq 0\), \(\partial{\phi} (x,t)/ \partial t \geq 0\) and \({\mu}-\inf{\phi}(x,T) > 0\). Then for each pair \(u_{0},u_{1} \in D(A)\) the inverse problem has a unique solution \(f \in L^{1}(X,{\mu})\), i. e., there exists a unique \(f \in L^{1}(X, {\mu})\) such that the corresponding function \[ u(t)=U(t)u_{0}+ \int_{0}^{t}U(t-s) \Phi(s)f ds, \quad 0 \leq t \leq T, \] satisfies the final condition \(u(T)=u_{1}\). Moreover, \(\|f\|\leq C(\|Au_{0}\|+\|Au_{1}\|)\) with the constant \(C > 0\) computable in explicit form.
To illustrate the results the author presents three examples: a linear inhomogeneous system of ODEs, the heat equation in \(\mathbb{R}^n\), and the one-dimensional transport equation.