In this paper, the authors investigate the boundary value problem (BVP) \[ {\mathcal L}x=f,\;x(a)=\alpha \text{ and } x(b)= \beta, \] where \({\mathcal L}={\mathcal L}_0-T, {\mathcal L}_0:W^2[a,b]\to L[a,b]\) and \(T:C[a,b]\to L[a,b]\) are linear, bounded Volterra-type operators and \((Tx_1)(t)\leq (Tx _2)(t)\) for \(t\in [a,b]\), whenever \(x_1(t)\geq x_2(t)\) on \([a,b]\). Defining the mapping \(H:C[a,b]\to C[a,b],\;(Hx)(t)=\int_a^bG_0(t,s) (Tx) (s)ds\), \(t\in[a,b]\), \(x\in C([a,b])\), where \(G_0\) is the Green function of (BVP), the authors prove that the following statements are equivalent:
(i) There exists \(\nu\in W^2[a,b]\), such that \(\nu(t)\geq 0\), \(({\mathcal L}\nu) (t)\leq 0\) \((a\leq t\leq b)\), and \(\nu(a)+ \nu(b)-\int_a^b({\mathcal L}\nu) (s)ds>0\);
(ii) The spectral radius of operator \(H\) is less than one;
(iii) (BVP) has exactly one solution for any \(f\in L[a, b]\) and any \(\alpha,\beta\), and its Green operator is antitone;
(iv) The fundamental system of \({\mathcal L}x=0\) is not oscillatory.
For the entire collection see [Zbl 0863.00015].