ABSTRACT
Theorem 15.1. Let X = [U - V] or X = [U - V], and suppose that the linear operator K = L + M +N acts in X and is regular. Assume that the estimates (15.12) hold, that the c01Tesponding resolvent kernels satisfy (15.15), and that one of the operators N +LM or N + M L is compact in X. Then the linear partial integral equation (15.1) satisfies the Fredholm alternative in the space X. In particular, (15.1) admits a unique solution x E X for any f E X if and only if the equation x = Kx has only the trivial solution z(t,s) == O. The conditions of Theorem 15.1 are sufficient but not necessary. The crucial points in this theorem are the compactness of N + LM or N +M L and the invertibility of both 1-L(s) (s E S) and 1-M(t) (t E T), Le. the hypothesis (15.20) 1 ¢ (7(L(s» u (7(M(t» (t E T, s E S). Even in the case 'et, s, r) == 'et, r), met, s, CT) == m(", (7), and net, r, ", CT) =0 (which frequently occurs in applications), one cannot drop the fundamental assumption (15.20).
