Linear functional differential equations of the type
\[ (Mx)(t) := x'(t) + (Bx)(t) = f(t) \; (t \in [0, \omega]) \]
and with the additional condition \(lx = c \) are considered, where \(B\) is a (possibly non-local) operator on \(C^0([0, \omega])\) and \( l \) is a continuous linear functional on this space. One main result asserts unique solvability of the general problem, if \( B\) is a Volterra operator (i.e., \((Bx)\) restricted to \([0,a] \) depends only on \(x\) restricted to \([0,a]\)), and both \(-B\) and \(l\) are positive.
Under the assumption that \(B\) is positive, the unique solvability of the same problem is related to sign properties of Green’s and Cauchy functions associated to the homogeneous equation \(Mx = 0\), and to the property that nonzero solutions of this equation have no zeroes on \([0, \infty)\).
These methods are applied to obtain, e.g., eventual nonoscillation of solutions to delay equations like \(x'(t) -a(t) x(g(t)) + b(t) x(h(t)) = f(t) \). The key arguments (e.g., in the proof of Corollary 9.2) make use of one-dimensional linear ODEs and of appropriately chosen assumptions.
In the last section, exponential stability results for
\[ \dot x(t) - (A_1x)(t) - (A_2x)(t) + (Bx)(t) =0 \]
are derived, essentially from positivity of \(B\), boundedness of the delay involved in the operator \(B\), and from conditions which express that \(B\) dominates \(A_1\) and \(A_2\).