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

Abstract:

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 $\mu$ is a real number such that $0 < \mu < 2/||A||$. Then, the "iteration scheme" ${x_{n = + 1}} = {x_n} + \mu (y - A{x_n}),n = 0,1,2, \cdot \cdot \cdot$, yields a strongly convergent sequence of vectors $\{x_n\}_{n = 0}^\infty$ if and only if "Picard’s criterion" for the existence of a solution of $Ax = y$ holds (i.e., if and only if $y$ is perpendicular to the null space of $A$, and $\sum \nolimits _{k = 1}^\infty {{{(y,{u_k})}^2}} /\lambda _k^2 < \infty$, where the ${u_k}$ and the ${\lambda _k}$ are the orthonormalized eigenvectors, and the corresponding eigenvalues, of $A$, respectively). An analogous result holds when $A$ is only required to be compact.