Proc. Steklov Inst. Math. 211, 28-34 (1995); translation from Tr. Mat. Inst. Steklova 211, 32-39 (1995).
In this paper, the authors investigate the boundary value problem (BVP)
where and are linear, bounded Volterra-type operators and for , whenever on . Defining the mapping , , , where is the Green function of (BVP), the authors prove that the following statements are equivalent:
(i) There exists , such that , , and ;
(ii) The spectral radius of operator is less than one;
(iii) (BVP) has exactly one solution for any and any , and its Green operator is antitone;
(iv) The fundamental system of is not oscillatory.