On iteration procedures for equations of the first kind, $Ax=y$, and Picard’s criterion for the existence of a solution

Suppose that the (not identically zero) linear operator A, on a real Hilbert space H to itself, is compact, selfadjoint, and positive semidefinite; that y is a vector of H which is perpendicular to the null space of A; and that u is a real number such that 0 0 for all u E H). The "family" of iterative procedures, under consideration in Section 1, is described by Eq. (10) of Section 1 (see subsection (cl) of that section). Given the vector y, Received June 13, 1969, revised February 24, 1970. AMS 1967 subject classifications. Primary 4511, 6575.