On iteration procedures for equations of the first kind, 𝐴𝑥=𝑦, and Picard’s criterion for the existence of a solution
J. B. Diaz, FREDERIC T. METCALF · Mathematics of Computation · 1970
Suppose that the (not identically zero) linear operator A A , on a real Hilbert space H H to itself, is compact, selfadjoint, and positive semidefinite; that y y is a vector of H H which is perpendicular to the null space of A A ; and that μ \mu is a real number such that 0 > μ > 2 / | | A | | 0 > \mu > 2/||A|| . Then, the "iteration scheme" x n = + 1 = x n + μ ( y − A x n ) , n = 0 , 1 , 2 , ⋅ ⋅ ⋅ {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 ∞ \{x_n\}_{n = 0}^\infty if and only if "Picard’s criterion" for the existence of a solution of