By Stanislaw Brzychczy, Roman R. Poznanski

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**Example text**

12) is satisfied with κ = 0, then f i is called parabolic in Szarski’s sense with respect to the function u in D. The uniform parabolicity in Besala’s sense implies the parabolicity in Szarski’s sense. 10). If the operators F i , i ∈ S, are uniformly parabolic in D, then these operators are uniformly parabolic in D with respect to any function u ∈ C S∗ (D). ,m , then the condition m Dq jk f i (t, x, y, p, q, s)ξ j ξk ≥ λ |ξ |2 for all ξ ∈ Rm j,k=1 with a constant λ > 0, implies the uniform parabolicity in Besala’s sense.

2 The convergence of the method of direct iterations may also be derived from the comparison theorem. The comparison theorem allows us to prove that the sequences of direct iterations {u n } and {vn } are Cauchy sequences. 1, as a particular case there follows a theorem on the existence and uniqueness of a solution of the Fourier first initial-boundary value problem for the infinite system of equations. This theorem is a generalization of the well-known Kusano theorem to the case of the infinite system.

1) in D; iii. the inequalities u n (t, x) ≤ vn (t, x) n = 1, 2, . . 6) hold for (t, x) ∈ D; iv. the following estimate is true vni (t, x) − u in (t, x) ≤ N0 [(L 1 + L 2 )t]n i ∈ S, n = 1, 2, . . n! 7) for (t, x) ∈ D, where N0 = v0 − u 0 0 = const < ∞; v. limn→∞ vni (t, x) − u in (t, x) = 0 uniformly in D and i ∈ S; vi. 29) within the sector u 0 , v0 , and z ∈ C S2+α (D). Before going into the proof of the theorem, we will introduce the fundamental operator P , the nonlinear operator, and prove some lemmas.