ABSTRACT
Justification of Theorems 2.5.1-2.5.4 on the convergence of the continual and discrete Schwartz method is deduced using their ε-uniform monotonicity. First, we give themonotonicity principle for the continual Schwartz method. Let on the sets D
i ⊂ D (0), for i = 1, . . . , I,with D i = D i(2.91), that form a covering of the set D
(2.90), functions w (k) i (x), x ∈ D
i , where i = 1, . . . , I
and k = 1, 2, 3, . . ., be defined; these functions are 2pi-periodic in xs for s = 2, . . . , n. Let w[0](x), x ∈ D(0), be a prescribed 2pi-periodic function in xs, for s = 2, . . . , n; the functions w[k]i (x), w
[k](x), x ∈ D(0), are defined by the relations
w [k] i (x) = u
( x; {w(k)l (·)}, 1 ≤ l ≤ i, w[k−1](·)
) , i = 1, . . . , I;
w[k](x) = w[k]I (x), x ∈ D (0) , k = 1, 2, 3, . . . .
