By Stanislaw Brzychczy, Roman R. Poznanski
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Example text
Thus it is a pseudo-linearization of the nonlinear problem. Applying the above iterative method to the problem here considered under appropriate assumptions on the functions f , the sequence {u˜ n } tends to the searched for exact solution at a rate not lower than that of the successive approximation sequence {u n } given by the iteration F i [u in ](t, x) = f i (t, x, u n−1 , u n−1 ). 47) and for n = 1, 2, . . 6). 6) are well defined in C S2+α (D), the functions u˜ n and v˜n (n = 1, 2, . . 48) and u n (t, x) ≤ u˜ n (t, x) ≤ v˜n (t, x) ≤ vn (t, x) hold for (t, x) ∈ D and n = 1, 2, .
Thus we obtain the system of equations F i [z i ](t, x) + k i (t, x)z i (t, x) = f i (t, x, z(t, x), z) + k i (t, x)z i (t, x) for i ∈ S, whose right-hand sides are increasing with respect to the variable y i for each i, i ∈ S. 39) to which monotone iterative methods, including in particular the method of direct iteration, are applicable. 4 Let assumptions A0 , (Ha ), (Hf ) hold, and conditions (W), ∗ ∗ in the set K. 41) for (t, x) ∈ D and i ∈ S for n = 1, 2, . . 6) where the function ∗ ∗ k i = k i (t, x), i ∈ S, fulfill the assumption (Ha ) and let u 0 = u 0 , v0 = v0 .
6) in D and u 1 ∈ CN2+α (D). 6) in D, u n ∈ CN2+α (D) and (i) is proved by induction. 6). 6). 1 we obtain u 0 (t, x) ≤ u 1 (t, x) for (t, x) ∈ D. 6) in D. 6). 1 we obtain u n−1 (t, x) ≤ u n (t, x) for (t, x) ∈ D. 6) in D. 53). 54) by induction.