ABSTRACT
Z(2)(t,S) = h(t)U(8), Z(3){t,8) =g(8)"{t) with h E N{I - L) n E I , U E F2' g E N(I - M) n E2, and "E Ft· All these arguments do not only allow us to give explicit criteria for the solvability of equation (15.32) in the case of degenerate kernels, but they also provide a constructive method for solving this equation. We summarize with the following
Theorem 15.10. Suppose that the kemelsl and m in equation (15.32) are of the form (15.45). Then the following solvability criteria hold true:
(a) If 1 '/. (T(L) + (T{AI), the equation (15.32) has a unique solution for any function f E L2(T x S). (b) If1 E (T(L)+(T(St) but I'/. (T{L)U(T(M), then the equation (15.32) is solvable if and only if the equation (15.58) is solvable; moreover, equation (15.32) has then a finite number of linearly independent solutions. (c) If 1 E (T(L)+(T{AI) an~ one of the following conditions is satisfied: (el) 1 E u(L) but 1 rI. (T(M), (c2) 1 rI. (T(L) but 1 E (T(St), (c3) 1 E (T{L) and 1 E (T(St), then the equation (15.31) is solvable if and only if both (15.58) and (15.59) (in case (c1», (15.58) and (15.60) {in case (c2», or (15.58), (15.59) and (15.61) (in case (c3» are solvable; moreover, the equation (15.32) has then an infinite number of linearly independent solutions.
We remark that a fairly complete description equation (15.32) in the case of degenerate kernels and matrix Ljapunov equations was established in VITOVA [1975, 1976, 1976a, 1977). As we have seen, the study of equation (15.32) may be reduced to the study of the
