Summary: In a Banach space \(E\) we consider the problem of finding a solution of the equation \(du(t)/dt=Au(t)\), \(t\geq 0\), subject to the relation \(\int_0^{\infty}\omega (t)u(t)dt=g\). We assume that the linear operator \(A\) with dense domain \(D(A)\) generates a \(C_0\) semigroup \(U(t)\). We show that if \(\|U(t)\|\leq Me^{-\alpha t}\) for \(t\geq 0\) with constants \(M\geq 1\), \(\alpha >0\), and \(\omega (t)\) is a nonnegative nonincreasing function such that \(\omega (t)>0\) for \(t\to 0+\), then, for any \(g\in D(A)\), the problem has a unique (generalized) solution \(u(t)=U(t)f\) for some \(f\in E\). If, in addition, \(g\in D(A^2)\), then \(f\in D(A)\), i.e., the solution \(u(t)=U(t)f\) is classical. We give a corollary in the case of an ordered Banach space \(E\).