Monotone Iterative Methods

CHAPTER 4

Monotone Iterative Methods

4.1 Method of Direct Iterations We consider a cylindrical domain D := (0, T ] × G, where T is an arbitrary positive constant, G is an open and bounded domain in Rm whose boundary ∂G is an (m −1)-dimensional surface of a class C 2+α for some 0 < α < 1 and we use notations: S0 := {(t, x) : t = 0, x ∈ G}, the lateral surface σ := (0, T ] × ∂G, the parabolic boundary of domain D is  := S0 ∪ σ , and D := D ∪ . We assume that f i : D × B (S) × C S (D) → R, (t, x, y, s)  → f i (t, x, y, s), i ∈ S are given functions and we consider an infinite system of equations with a homogeneous initial-boundary condition in D, i.e., the problem  i i F [z ](t, x) = f i (t, x, z(t, x), z) for (t, x) ∈ D, (4.1) z i (t, x) = 0 for (t, x) ∈ , and i ∈ S, where the operators F i , i ∈ S, are diagonal and uniformly parabolic in D. The following theorem holds true. ˜ (V) hold in the Theorem 4.1 Let assumptions A0 , (Ha ), (Hf ) hold, and conditions (L), (W), set K. If we define the successive terms of the approximation sequences {u n } and {vn } as regular solutions in D of the following infinite systems of equations:     (4.2) F i u in (t, x) = f i t, x, u n−1 (t, x), u n−1 for (t, x)∈D and i ∈ S and

    F i vni (t, x) = f i t, x, vn−1 (t, x), vn−1 for (t, x) ∈ D and i∈S

(4.3)

for n = 1, 2, . . . in D with homogeneous initial-boundary condition (2.6), and let an ordered pair of a lower u 0 and an upper solution v0 (given by Assumption A0 ), and such that u 0 ≤ v0 in D, will be the pair of initial iterations in this iterative process, then i. {u n }, {vn } are well defined and u n , vn ∈ C S2+α (D) for n = 1, 2, . . .; Mathematical Neuroscience. http://dx.doi.org/10.1016/B978-0-12-411468-5.00004-1 © 2014 Elsevier Inc. All rights reserved.

49

50 Chapter 4 ii. the inequalities u 0 (t, x) ≤ u n (t, x) ≤ u n+1 (t, x) for n = 1, 2, . . . ,

(4.4)

hold for (t, x) ∈ D and the functions u n for n = 1, 2, . . . , are lower solutions of problem (4.1) in D, and analogously vn+1 (t, x) ≤ vn (t, x) ≤ v0 (t, x) for n = 1, 2, . . . ,

(4.5)

hold for (t, x) ∈ D and vn for n = 1, 2, . . . , are upper solutions of problem (4.1) in D; iii. the inequalities u n (t, x) ≤ vn (t, x) n = 1, 2, . . . , (4.6) 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!

(4.7)

for (t, x)∈ D, where N0 = v  0 − u 0 0 = const < ∞; i i v. limn→∞ vn (t, x) − u n (t, x) = 0 uniformly in D and i ∈ S; vi. the function z = z(t, x) = lim u n (t, x) n→∞

is the unique global (in time) regular solution of problem (2.29) 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. Since the proofs are straightforward and similar for the lower and upper solutions, we present the proof for the upper solution only. Let β = β(t, x) be a sufficiently regular function defined in D. Denote by P the operator P : β  → P [β] = γ

generated by differential-functional problem in this way that the function γ = γ (t, x) is the (supposedly unique) solution of the linear initial-boundary value problem  i i F [γ ](t, x) = f i (t, x, β(t, x), β) for (t, x) ∈ D, (4.8) γ i (t, x) = 0 for (t, x) ∈  and i ∈ S. The operator P is the composition of the nonlinear operator F = {Fi }i∈S generated by the functions f i (t, x, y, s), i ∈ S, and defined for any sufficiently regular function β as follows: F : β  → F[β] = δ,

Monotone Iterative Methods 51 where Fi [β](t, x) := f i (t, x, β(t, x), β) := δ i (t, x) for (t, x) ∈ D and i ∈ S,

(4.9)

and the operator G : δ  → G [δ] = γ ,

where γ = γ (t, x) is the (supposedly unique) solution of the linear initial-boundary value problem  i i F [γ ](t, x) = δ i (t, x) for (t, x) ∈ D, (4.10) γ i (t, x) = 0 for (t, x) ∈  and i ∈ S. Hence P = G ◦ F.

The introduction of the nonlinear operator enables proofs of certain theorems to be simplified significantly and renders the proofs elegant. The examination of properties of the operator P , including its monotonicity or compactness, is the founding idea of the theory of monotone iterative methods. From the form of infinite system (4.8), precisely from the diagonal form of differential operators F i , i ∈ S, it follows that this weakly coupled system has the following fundamental property: the ith equation depends on the ith unknown function only. Roughly speaking, system (4.8) (and systems (4.2) and (4.3)) is a collection of several equations each of which is an equation in one unknown function only. Therefore, its solution is a collection of separate solutions of the scalar problems considered in suitably selected space. Lemma 4.1 If β ∈ C S0+α (D) and the function f = { f i }i∈S , generating the nonlinear operator F, satisfies conditions (Hf ) and (L), then F : C S0+α (D) → C S0+α (D) β  → F[β] = δ. Proof.

Because β ∈ C S0+α (D),    α  α  i  β (t, x) − β i (t , x ) ≤ HαD (β) t − t  2 + x − x 

for (t, x), (t , x ) ∈ D and all i ∈ S, where HαD (β) > 0 is a constant independent of the index i. From assumption (Hf ) and (L) it follows that |δ i (t, x) − δ i (t , x )| = |Fi [β](t, x) − Fi [β](t , x )| = | f i (t, x, β(t, x), β) − f i (t , x , β(t , x ), β)|

52 Chapter 4 ≤ | f i (t, x, β(t, x), β) − f i (t , x , β(t, x), β)| +| f i (t , x , β(t, x), β) − f i (t , x , β(t , x ), β)| 



α  α ≤ HαD ( f ) t − t  2 + x − x  + L β(t, x) − β(t , x ) B(S)   α  α α  α ≤ HαD ( f ) t − t  2 + x − x  + L HαD (β) t − t  2 + x − x   α  α ≤ HαD (δ) t − t  2 + x − x  for all (t, x), (t , x ) ∈ D, i ∈ S, where HαD (δ) = HαD ( f ) + L HαD (β). Therefore, F[β] ∈ C S0+α (D). This completes the proof.



From Lemma 4.1 and Theorem A.1, the next lemma follows directly. Lemma 4.2 If δ ∈ C S0+α (D), ∂G ∈ C 2+α and all the coefficients of the operators Li , i ∈ S, satisfy condition (Ha ), then problem (4.10) has the unique regular solution γ and γ ∈ C S2+α (D). Corollary 4.1 From Lemmas 4.1 and 4.2 it follows that P = G ◦ F : C S0+α (D) → C S2+α (D).

Lemma 4.3 Let all the assumptions of Lemmas 4.1 and 4.2 hold, β be an upper solution and ˜ holds. Then α be a lower solution of problem (4.1) in D, α, β ∈ u 0 , v0 and condition (W) α(t, x) ≤ P [β](t, x) ≤ β(t, x) for (t, x) ∈ D and i ∈ S

(4.11)

and γ = P [β] is an upper solution of problem (4.1) in D, and analogously α(t, x) ≤ P [α](t, x) ≤ β(t, x) for (t, x) ∈ D and i ∈ S and η = P [α] is a lower solution of problem (4.1) in D. Proof.

If β is an upper solution, then by virtue of (2.21) there is F i [β i ](t, x) ≥ f i (t, x, β(t, x), β) for (t, x) ∈ D and i ∈ S.

From the definition (4.8) of the operator P it follows that F i [γ i ](t, x) = f i (t, x, β(t, x), β) for (t, x) ∈ D and i ∈ S.

Therefore F i [β i − γ i ](t, x) ≥ 0 for (t, x) ∈ D and i ∈ S

and β(t, x) − γ (t, x) = 0 for (t, x) ∈ .

(4.12)

Monotone Iterative Methods 53 Hence, by Lemma 3.1 there is β(t, x) − γ (t, x) ≥ 0 for (t, x) ∈ D so γ (t, x) = P [β](t, x) ≤ β(t, x) for (t, x) ∈ D. ˜ we obtain From (4.8) and conditions (W) F i [γ i ](t, x) − f i (t, x, γ (t, x), γ ) = f i (t, x, β(t, x), β) − f i (t, x, γ (t, x), γ ) ≥ 0

for (t, x) ∈ D, i ∈ S, and γ (t, x) = 0 for (t, x) ∈ . From Corollary 4.1 we infer that γ is a regular function and γ ∈ C S2+α (D), so it is an upper solution of problem (4.1) in D and from Remark 2.6 it follows that: α(t, x) ≤ γ (t, x) = P [β](t, x) ≤ β(t, x) for (t, x) ∈ D. 

This completes the proof.

Lemma 4.4 If all assumptions of Lemma 4.3 hold, α, β ∈ u 0 , v0 and α(t, x) ≤ β(t, x) in D, then P [α](t, x) ≤ P [β](t, x) for (t, x) ∈ D (4.13) i.e., the operator P is isotone1 in the sector u 0 , v0 . Proof.

Let η = P [α] and γ = P [β], then by (4.8) F i [ηi ](t, x) = f i (t, x, α(t, x), α) for (t, x) ∈ D and i ∈ S

and F i [γ i ](t, x) = f i (t, x, β(t, x), β) for (t, x) ∈ D and i ∈ S

and η(t, x) = γ (t, x) = 0 for (t, x) ∈ . ˜ there is By (W),   F i ηi − γ i (t, x) = f i (t, x, β(t, x), β) − f i (t, x, α(t, x), α) ≥ 0 for (t, x) ∈ D and i ∈ S 1 Let X and Y be partially ordered sets with the ordering given by cones K and K and denoted by “≤” in each X Y

set. A transformation T : X → Y is called isotone (monotone increasing) if for each u, v ∈ X , u ≤ v implies T [u] ≤ T [v] and strictly isotone if u < v implies T [u] < T [v].

54 Chapter 4 and γ (t, x) − η(t, x) = 0 on . By Lemma 3.1, there is γ (t, x) ≥ η(t, x) in D, i.e., P [α](t, x) ≤ P [β](t, x) in D. This means that the operator P is isotone in the sector u 0 , v0 . This completes the proof.



From Corollary 4.1, Lemmas 4.3 and 4.4, the next corollary follows. Corollary 4.2 If all the assumptions of Lemma 4.3 hold and if α and β are a lower and an upper solution of problem (4.1) in D, respectively, α, β ∈ u 0 , v0 , and α(t, x) ≤ β(t, x) then α(t, x) ≤ P [α](t, x) ≤ P [β](t, x) ≤ β(t, x) for (t, x) ∈ D.

(4.14)

This means that P [ α, β ] ⊂ α, β . Proof of Theorem 4.1. Starting from the lower solution u 0 and the upper solution v0 , we define by induction two sequences of functions {u n } and {vn } as regular solutions of systems (4.2) and (4.3). Therefore we have u n = P [u n−1 ], vn = P [vn−1 ] for n = 1, 2, . . . . From Lemmas 4.1, 4.2, and 4.4 it follows that u n and vn , for n = 1, 2, . . . , are well defined and are the lower and the upper solutions of problem (4.2) and (4.3) in D, respectively. Using mathematical induction, from Lemma 4.3, we obtain u n (t, x) ≤ P [u n ](t, x) = u n+1 (t, x), n = 1, 2, . . . and vn+1 (t, x) = P [vn ](t, x) ≤ vn (t, x) for (t, x) ∈ D and n = 1, 2, . . . . Therefore, inequalities (4.4) and (4.5) hold. Analogously, using mathematical induction, from Lemma 4.4 and Assumption A0 we also obtain inequalities (4.6). Using mathematical induction, we will prove the inequalities 0 ≤ vni (t, x) − u in (t, x) := wni (t, x) ≤ N0 for (t, x) ∈ D and i ∈ S, n = 1, 2, . . . .

[(L 1 + L 2 )t]n , n!

(4.15)

Monotone Iterative Methods 55 It is obvious that inequality (4.15) holds for w0 . Let inequality (4.15) hold for wn . The functions f i (t, x, y, s), i ∈ S, fulfill Lipschitz condition (L) with respect to y and s, and condition (V). Therefore , using the interrelation between conditions (V) and (L) ⇐⇒ (L∗ ) (see Section 2.7), f i fulfills condition (L∗ ). By (4.2), (4.3), and (4.15), there is  i F i wn+1 (t, x) = f i (t, x, vn (t, x), vn ) − f i (t, x, u n (t, x), u n ) ≤ L 1 wn (t, x)B(S) + L 2 wn 0,t . By the definition of the norm ·0,t in the space C S (D) and by inequality (4.15), wn 0,t ≤

[(L 1 + L 2 )t]n . n!

So we finally obtain  (L 1 + L 2 )n+1 t n i for (t, x) ∈ D and i ∈ S F i wn+1 (t, x) ≤ N0 n!

(4.16)

and wn+1 (t, x) = 0 for (t, x) ∈ .

(4.17)

Let us consider the comparison system of equations  (L 1 + L 2 )n+1 t n i for (t, x) ∈ D and i ∈ S F i Mn+1 (t, x) = N0 n!

(4.18)

with the initial-boundary condition Mn+1 (t, x) ≥ 0 for (t, x) ∈ .

(4.19)

It is obvious that the functions i Mn+1 (t, x) = N0

[(L 1 + L 2 )t]n+1 for (t, x) ∈ D and i ∈ S (n + 1)!

are the regular solutions of comparison problem (4.18) and (4.19) in D. Applying the Szarski theorem on weak differential inequalities (Theorem 3.1) to problems (4.16), (4.17) and (4.18), (4.19), we obtain i i wn+1 (t, x) ≤ Mn+1 (t, x) = N0

[(L 1 + L 2 )t]n+1 for (t, x) ∈ D and i ∈ S (n + 1)!

so the induction step is proved and so estimate (4.7) is true.

56 Chapter 4 As a direct conclusion from formula (4.15) we obtain  lim vni (t, x) − u in (t, x) = 0 uniformly in D for all i ∈ S. n→∞

(4.20)

The sequences of functions {u n (t, x)} and {vn (t, x)} are monotonous and bounded, and (4.20) holds, so there exists a continuous function U = U (t, x) in D such that lim u i (t, x) n→∞ n

= U i (t, x) and

lim v i (t, x) n→∞ n

= U i (t, x)

(4.21)

uniformly in D, i ∈ S. ˜ from (4.4) it follows that the Since functions f i (i ∈ S) are monotonous (condition W), i functions f (t, x, u n−1 (t, x), u n−1 ), i ∈ S, are uniformly bounded in D with respect to n. Hence by Theorem A.1 we conclude that all the functions u n ∈ C S2+α (D) for n = 1, 2, . . . , satisfy the Hölder condition with a constant independent of n. Hence U ∈ C S0+α (D). If we now consider the system of equations   F i [z i ](t, x) = f i t, x, U (t, x), U := Fi [U ](t, x) for (t, x) ∈ D i ∈ S

(4.22)

with initial-boundary condition (2.6), then by Lemma 4.1 there is Fi [U ] ∈ C S0+α (D). Therefore, by virtue of Lemma 4.2 this problem has the unique regular solution z, and z ∈ C S2+α (D). Let us now consider systems (4.2) and (4.22) together. Let us apply, to these systems, Szarski’s theorem on the continuous dependence of the solution of the first problem on the initial-boundary values and on the right-hand sides of systems. Since the functions f i , i ∈ S, satisfy the Lipschitz condition (L), by (4.21), there is   lim f i (t, x, u n (t, x), u n ) = f i t, x, U (t, x), U uniformly in D, i ∈ S. n→∞

Hence lim u i (t, x) n→∞ n

= z i (t, x) for i ∈ S.

(4.23)

By virtue of (4.21) and (4.23), z = z(t, x) = U (t, x) for (t, x) ∈ D is the regular solution of problem (4.1) in D and z ∈ C S2+α (D). The uniqueness of the solution follows directly from Szarski’s uniqueness criterion (Theorem 3.6). It also follows directly from inequality (4.15). Because the assumptions of our theorem hold in the set K only, the uniqueness of a solution is ensured only with respect to the given upper and lower solutions, and it does not rule out the existence of other solutions outside the sector u 0 , v0 . Thus the theorem is proved.



Monotone Iterative Methods 57 Remark 4.1 In the case of estimate (4.15) we say that the sequences of successive approximations {u n } and {vn } defined by (4.2) and (4.3) converge to the searched solution z with the power speed. Remark 4.2 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. From Theorem 4.1, 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. Theorem 4.2 Let us consider the Fourier first initial-boundary value problem for the infinite system of equations of the form  i i F [z ](t, x) = f˜i (t, x, z(t, x)) for (t, x) ∈ D, (4.24) z i (t, x) = 0 for (t, x) ∈  and i ∈ S. Let the following assumptions be satisfied: the operators F i , i ∈ S, are uniformly parabolic in D; Assumption A0 holds; all the coefficients of the operators F i , i ∈ S, fulfill condition (Ha ); the functions f˜i (t, x, y), i ∈ S, are defined for (t, x, y) ∈ D × B (S), satisfy assumption (Hf ), Lipschitz condition (L), and condition (W) with respect to y; v. the above assumptions hold locally in the sector u 0 , v0 formed by u 0 and v0 .

i. ii. iii. iv.

Under these assumptions, problem (4.24) possesses the unique global regular solution z within the sector u 0 , v0 , and z ∈ C S2+α (D).

4.2 Chaplygin Method To solve problem (4.1), we now apply another monotone iterative method, namely the Chaplygin method, in which we use the linearization with respect to the nonfunctional argument y only. ˜ and (V) We assume that the functions f i (t, x, y, s), i ∈ S, satisfy conditions (Hf ), (L), (W), with respect to y and s in the set K. We additionally assume that each function f i (t, x, y, s), i ∈ S, has the continuous derivatives i D y i f i := ∂∂ yf i := f yi i (t, x, y, s), i ∈ S, which satisfy the following conditions in the set K:

58 Chapter 4 Assumption H p . Condition (Hf ). Assumption Lp . The Lipschitz condition with respect to y and s. ˜ p . They are increasing with respect to y and s. Assumption W ˜ ), Theorem 4.3 Let assumptions A0 , (Ha ), (Hf ), (Hp ) hold, and conditions (W ˜ (Wp ), (L), (Lp ), (V) hold in the set K. Let us assume that the successive terms of approximation sequences {uˆ n } and {vˆn } are defined as regular solutions in D of the following infinite systems of linear equations:  F i uˆ in (t, x) = f i (t, x, uˆ n−1 (t, x), uˆ n−1 )  + f yi i (t, x, uˆ n−1 (t, x), uˆ n−1 ) · uˆ in (t, x) − uˆ in−1 (t, x) , i ∈ S (4.25) and

 F i vˆni (t, x) = f i (t, x, vˆn−1 (t, x), vˆn−1 ) + f yi i (t, x, vˆn−1 (t, x), vˆn−1 )  i (t, x) for (t, x) ∈ D, i ∈ S · vˆni (t, x) − vˆn−1

(4.26)

for n = 1, 2, . . . , in D with homogeneous initial-boundary condition (2.6). Let us use uˆ 0 for u 0 and vˆ0 for v0 (the initial iteration in this iterative process). Then i. {uˆ n }, {vˆn } are well defined and uˆ n , vˆn ∈ C S2+α (D) for n = 1, 2, . . .; ii. the inequalities u 0 (t, x) ≤ uˆ n (t, x) ≤ uˆ n+1 (t, x) ≤ vˆn+1 (t, x) ≤ vˆn (t, x) ≤ v0 (t, x)

(4.27)

hold for (t, x) ∈ D, n = 1, 2, . . . , and the functions uˆ n and vˆn for n = 1, 2, . . . , are lower and upper solutions of problem (4.1) in D, respectively; iii. the following inequalities: u n (t, x) ≤ uˆ n (t, x) ≤ vˆn (t, x) ≤ vn (t, x)

(4.28)

hold for (t, x) ∈ D and n = 1, 2, . . . , where the sequences {u n } and {vn } are defined by (4.2) and (4.3); iv. the following estimate: vˆni (t, x) − uˆ in (t, x) ≤ N0

[(L 1 + L 2 )t]n , n!

where N0 = v0 − u 0 0 = const < ∞;

holds for (t, x) ∈ D, i ∈ S, n = 1, 2, . . . ,

(4.29)

Monotone Iterative Methods 59 v.

 lim vˆni (t, x) − uˆ in (t, x) = 0 uniformly in D, i ∈ S;

n→∞

vi. the function z = z(t, x) = lim uˆ n (t, x) n→∞

is the unique global regular solution of problem (4.1) within the sector u 0 , v0 , and z ∈ C S2+α (D). Before proving the theorem, we introduce the nonlinear operators and prove some lemmas. Let β ∈ C S (D) be a sufficiently regular function. Let Pˆ denote the operator Pˆ : β  → Pˆ [β] = γ ,

where γ is the (supposedly unique) solution of the following problem: ⎧ i i i i ⎪ ⎨ F [γ ](t, x) + f y i (t, x, β(t, x), β) · γ (t, x) = f i (t, x, β(t, x), β) + f yi i (t, x, β(t, x), β) · β i (t, x) for (t, x) ∈ D, ⎪ ⎩ i γ (t, x) = 0 for (t, x) ∈  and i ∈ S.

(4.30)

Observe that system (4.30) has the following property: the ith equation depends on the ith unknown function only. Therefore, its solution is a collection of separate solutions of these scalar problems for linear equations in a suitable space. It is convenient to define two nonlinear operators related to the functions f i (t, x, y, s) and f yi i (t, x, y, s), i ∈ S, to examine them separately. They are: the operator C = {Ci }i∈S C : β  → C[β] = η, where Ci [β](t, x) := f yi i (t, x, β(t, x), β) = ηi (t, x)

(4.31)

for (t, x) ∈ D, i ∈ S and the operator Fˆ = {Fˆ i }i∈S ˆ Fˆ : β  → F[β] = δ, where Fˆ i [β](t, x) := f i (t, x, β(t, x), β) + f yi i (t, x, β(t, x), β) · β i (t, x) = f i (t, x, β(t, x), β) + Ci [β](t, x) · β i (t, x) := δ i (t, x), for (t, x) ∈ D and i ∈ S.

(4.32)

60 Chapter 4 One may use the notation just introduced to write problem (4.30) in a simpler way. The operator Pˆ is the composition of the nonlinear operators Fˆ and C with the operator Gˆ : δ  → G [δ] = γ ,

where γ is the (supposedly unique) solution of the linear problem  i i F [γ ](t, x) + Ci [β](t, x) · γ i (t, x) = Fˆ i [β](t, x) for (t, x) ∈ D, γ i (t, x) = 0 for (t, x) ∈  and i ∈ S. Hence

(4.33)

ˆ C). Pˆ = Gˆ ◦ (F,

The following lemmas hold, too. Lemma 4.5 If β ∈ C S0+α (D) and the functions f y i = { f yi i } generating the nonlinear operator C satisfy conditions (Hp ) and (Lp ), then C[β] = η ∈ C S0+α (D). If β ∈ C S0+α (D), the functions f = { f i }i∈S and f y i = { f yi i }i∈S generating the nonlinear operator Fˆ satisfy conditions (Hf ), (L), (Hp ), and (Lp ), then ˆ F[β] = δ ∈ C S0+α (D). Proof.

It runs analogously to the proof of Lemma 4.1. This completes the proof.



From Theorem 3.3 and Lemma 4.5, one may derive the following statements. Lemma 4.6 If β ∈ C S0+α (D) and all the coefficients of the operators Li , i ∈ S satisfy assumption (Ha ), then problem (4.33) has exactly one regular solution, γ ∈ C S2+α (D). Corollary 4.3 It follows from Lemmas 4.5 and 4.6 that Pˆ : C S0+α (D) → C S2+α (D).

Lemma 4.7 Let all the assumptions of Lemmas 4.5 and 4.6 hold, β be an upper solution and ˜ (W ˜ p ) hold. Then α be a lower solution of problem (4.1) in D, and conditions (W), Pˆ [β](t, x) ≤ β(t, x) for (t, x) ∈ D

(4.34)

and γ = Pˆ [β] is an upper solution of problem (4.1) for (t, x) ∈ D, and analogously Pˆ [α](t, x) ≥ α(t, x) for (t, x) ∈ D

and η = Pˆ [α] is a lower solution of problem (4.1) in D.

(4.35)

Monotone Iterative Methods 61 Proof.

If β is an upper solution, then due to (2.21) and the notation introduced, there is F i [β i ](t, x) ≥ f i (t, x, β(t, x), β), i ∈ S, for (t, x) ∈ D

thus β satisfies the following system of inequalities:  F i [˜z i ](t, x) ≥ f i (t, x, β(t, x), β) + f yi i (t, x, β(t, x), β) · z˜ i (t, x) − β i (t, x) ,

(4.36)

i ∈ S, for (t, x) ∈ D and initial-boundary condition (2.6). From definition (4.30) of the operator Pˆ it follows that the function γ is a solution of the system of equations     F i z˜˜ i (t, x) = f i (t, x, β(t, x), β) + f yi i (t, x, β(t, x), β) · z˜˜ i (t, x) − β i (t, x) (4.37) i ∈ S, for (t, x) ∈ D, with condition (2.6). Applying Theorem 4.2 to systems (4.36) and (4.37), we obtain z˜˜ i (t, x) ≤ z˜ i (t, x), i ∈ S, for (t, x) ∈ D so,

γ (t, x) = Pˆ [β](t, x) ≤ β(t, x) for (t, x) ∈ D.

˜ p )), then the Since the functions f yi i (t, x, y, s) are increasing in variable y i (condition (W functions f i (t, x, y, s) are convex in y i . ˜ give Now (4.30), (4.34), and condition (W) F i [γ i ](t, x) = f i (t, x, β(t, x), β) + f yi i (t, x, β(t, x), β) · [γ i (t, x) − β i (t, x)]

≥ f i (t, x, γ (t, x), β) ≥ f i (t, x, γ (t, x), γ ), i ∈ S, for (t, x) ∈ D and γ (t, x) = 0 on for (t, x) ∈ . Thus from (2.21) and Corollary 4.2 it follows that the function γ is an upper solution of problem (4.1) for (t, x) ∈ D. This completes the proof. Proof of Theorem 4.3. One may prove statements (i)–(iii) of the theorem using the induction argument. Using Assumption A0 it is easy to see that those statements are immediate consequences of Lemmas 4.5–4.7 and Corollary 4.3. ˆ (4.30), (4.25), and (4.26), we obtain Indeed, due to the definition of P, uˆ n = Pˆ [uˆ n−1 ], vˆn = Pˆ [vˆn−1 ] for n = 1, 2, . . .



62 Chapter 4 which means that the sequences {uˆ n } and {vˆn } are well defined. Statement (iv) follows immediately from the comparison of system (4.25) and (4.26) with system (4.2) and (4.3) defining the sequences {u n } and {vn }. ˜ p ), there is Indeed, due to (4.30) and condition (W F i [uˆ i1 ](t, x) − F i [u i1 ](t, x)

 = f i (t, x, u 0 (t, x), u 0 ) + f yi i (t, x, u 0 (t, x), u 0 ) · uˆ i1 (t, x) − u i0 (t, x) − f i (t, x, u 0 (t, x), u 0 )

 = f yi i (t, x, u 0 (t, x), u 0 ) · uˆ i1 (t, x) − u i0 (t, x) ≥ 0 for (t, x) ∈ D and i ∈ S and uˆ 1 (t, x) − u 1 (t, x) = 0 for (t, x) ∈ . Consequently, by Lemma 3.1 uˆ 1 (t, x) ≥ u 1 (t, x) for (t, x) ∈ D. By mathematical induction, we obtain u n (t, x) ≤ uˆ n (t, x) ≤ vˆn (t, x) ≤ vn (t, x) for (t, x) ∈ D and n = 1, 2, . . . . The other statements, (v) and (vi), follow immediately from inequality (4.29) and Theorem 4.1. Thus the theorem is proved.



4.3 Certain Variants of the Chaplygin Method Now we present the next two monotone iterative methods being certain variants of the Chaplygin method, applicable to problem (4.1), whose right-hand sides of equations are quasi-increasing functions; more precisely, they meet condition (K). The last of these methods consists of adding appropriate linear terms k i (t, x)z i including the unknown functions z i = z i (t, x) to both sides of (2.2), in order to render the new sides monotonous. 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. Setting Fκi := Dt − Li + k i I

(4.38)

Monotone Iterative Methods 63 we obtain a system of the following form: Fki [z i ](t, x) = f i (t, x, z(t, x), z) + k i (t, x)z i (t, x) for (t, x) ∈ D and i ∈ S

(4.39)

to which monotone iterative methods, including in particular the method of direct iteration, are applicable. ˜ (L), (K), (V) hold Theorem 4.4 Let assumptions A0 , (Ha ), (Hf ) hold, and conditions (W), ∗ ∗ in the set K. Let the successive terms of the approximation sequences {u n } and {vn } be defined as regular solutions in D of the following infinite systems of equations:  ∗ ∗ ∗ ∗ Fki u in (t, x) = f i (t, x, u n−1 (t, x), u n−1 ) + k i (t, x)u in−1 (t, x) (4.40) for (t, x) ∈ D and i ∈ S and

  Fki

∗

∗

∗

∗

i vni (t, x) = f i (t, x, vn−1 (t, x), vn−1 ) + k i (t, x)vn−1 (t, x)

(4.41)

for (t, x) ∈ D and i ∈ S for n = 1, 2, . . . with homogeneous initial-boundary condition (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 . Then the statements of Theorem 4.3 hold, thus there exists the unique time-global regular solution z of problem (4.1) within the sector u 0 , v0 , and z ∈ C S2+α (D). Proof. After some small changes in technical details, the proof of this theorem is identical to those of previous theorems, so we may omit it here. If condition (K κ ) holds, then we will define the approximation sequences in the following way. ˜ (L), (V) hold Theorem 4.5 Let assumptions A0 , (Ha ), (Hf ) hold, and conditions (Kκ ), (W), in the set K. Let the successive terms of the approximation sequences {u˚ n } and {v˚n } be defined as regular solutions in D of the following infinite systems of equations:  Fκi u˚ in (t, x) = f i (t, x, u˚ n−1 (t, x), u˚ n−1 ) + κ u˚ in−1 (t, x) (4.42) f or (t, x) ∈ D and i ∈ S and

 i Fκi v˚ni (t, x) = f i (t, x, v˚n−1 (t, x), v˚n−1 ) + κ v˚n−1 (t, x) f or (t, x) ∈ D and i ∈ S

(4.43)

64 Chapter 4 for n = 1, 2, . . . for (t, x) ∈ D with the condition (2.6) and let u˚ 0 = u 0 , v˚0 = v0 . Then there exists the unique global regular solution z of problem (4.1) z = z(t, x) = lim u˚ n (t, x) = lim v˚n (t, x) n→∞

n→∞

within the sector u 0 , v0 , and z ∈ C S2+α (D). Remark 4.3 Let u 0 be a lower and v0 be an upper solution on the initial-boundary value problem for equation  (Dt − L)[z](t, x) = F[z](t, x) := f (t, x, z(t, x)) for (t, x) ∈ D, (4.44) z(t, x) = 0 on  and suppose u 0 ≤ v0 in D. Let α := min u 0 (t, x) and β := max v0 (t, x). D

D

If the function f (t, x, y) fulfills the Lipschitz condition (L) with respect to y in the set K˜ = {(t, x, y) : (t, x) ∈ D, α ≤ y ≤ β},

then for all (t, x) ∈ D and ξ, η with α ≤ ξ < η ≤ β we have f (t, x, η) − f (t, x, ξ ) ≥ −L(η − ξ ). In other words, for every (t, x) ∈ D the function f L (t, x, y) := f (t, x, y) + L y is increasing with respect to y in the interval α, β . The initial-boundary value problem (4.44) is obviously equivalent to the initial-boundary value problem  (Dt − L L )[z](t, x) = F L [z](t, x) := f L (t, x, z(t, x)) for (t, x) ∈ D, (4.45) z(t, x) = 0 for (t, x) ∈ , where L L := L − L I .

Moreover, u 0 (v0 ) is a lower (upper) solution of problem (4.44) if and only if it is a lower (upper) solution of problem (4.45); F and F L have the same continuity properties and L L is a differential operator of the same type as L.

Monotone Iterative Methods 65

4.4 Certain Variants of Monotone Iterative Methods Above, the four monotone iterative methods have been used to examine the existence of a solution for problem (4.1). Here we shall apply two other monotone iterative methods. These methods make it again possible to build sequences of successive approximations that converge to a solution sought for at a rate higher than in the case of the iterative sequences of successive approximations defined by (4.2) and (4.3). In general, it consists of the following: if we consider some nonlinear system of equations whose right-hand sides are functions of the form f = f (t, x, s, s), then the successive approximation sequence {u˜ n } arising from the iteration f (t, x, u˜ n , u˜ n−1 ) is considered. 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 ). We will define the successive terms of approximation sequences {u˜ n } and {v˜n } as regular solutions of the following infinite systems of equations:  F i u˜ in (t, x) = f i (t, x, u˜ n (t, x), u˜ n−1 ) for (t, x) ∈ D and i ∈ S (4.46) and

 F v˜ni (t, x) = f i (t, x, v˜n (t, x), v˜n−1 ) for (t, x) ∈ D and i ∈ S

(4.47)

for n = 1, 2, . . . , satisfying the homogeneous initial-boundary condition (2.6). Theorem 4.6 Under assumptions A0 , (Ha ), (Hf ), and conditions (W), (L), (V) which hold in the set K, if we start the approximation process from the same ordered pair of a lower u 0 and an upper v0 solution of problem (4.1) in D, as the pair of initial iterations then the sequences {u˜ n } and {v˜n } given by (4.46), (4.47), and (2.6) are well defined in C S2+α (D), the functions u˜ n and v˜n (n = 1, 2, . . . ) are a lower and an upper solution of problem (4.1) in D, respectively, and these sequences converge monotonously and uniformly to a solution z of problem (4.1) in D at a rate not lower than that of the iterative sequences {u n } and {vn } defined by (4.2), (4.3), and (2.6) in D, namely the inequalities u 0 (t, x) ≤ u˜ n (t, x) ≤ u˜ n+1 (t, x) ≤ v˜n+1 (t, x) ≤ v˜n (t, x) ≤ v0 (t, x)

(4.48)

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, . . . The function z = z(t, x) = lim u˜ n (t, x) uniformly in D n→∞

is the unique global regular solution of problem (4.1) within the sector u 0 , v0 , and z ∈ C S2+α (D).

(4.49)

66 Chapter 4 Proof. From Theorem 4.2 it follows that the successive terms of approximation sequences u˜ n and v˜n are well defined and u˜ n , v˜n ∈ C S2+α (D) for n = 1, 2, . . . To prove the theorem, it is enough to show inequality (4.49), because the other statements of the theorem are obvious. To this end we use the mathematical induction. Indeed, for n = 1, by (4.48), (4.46), (4.2), and (W) we obtain     F i u˜ i1 (t, x) − F i u i1 (t, x) = f i (t, x, u˜ 1 (t, x), u 0 ) − f i (t, x, u 0 (t, x), u 0 ) ≥ 0, i ∈ S, in D with homogeneous initial-boundary condition (2.6). Hence, by virtue of Lemma 3.1, we obtain u˜ 1 (t, x) ≥ u 1 (t, x) for (t, x) ∈ D and i ∈ S. If now u˜ n−1 (t, x) ≥ u n−1 (t, x) for (t, x) ∈ D and i ∈ S, then by (4.46), (4.48), (4.1), and (W) we come to the inequalities     F i u˜ in (t, x) − F i u in (t, x) = f i (t, x, u˜ n (t, x), u˜ n−1 ) − f i (t, x, u n−1 (t, x), u n−1 ) ≥ f i (t, x, u˜ n−1 (t, x), u˜ n−1 ) − f i (t, x, u n−1 (t, x), u n−1 ) ≥ 0 for (t, x) ∈ D and i ∈ S with condition (2.6). Hence by Lemma 3.1 we obtain u˜ n (t, x) ≥ u n (t, x) for (t, x) ∈ D and i ∈ S. Analogously vn (t, x) ≥ v˜n (t, x) for (t, x) ∈ D and i ∈ S. Therefore, by induction, inequality (4.49) is proved. This completes the proof.



4.5 Another Variant of the Monotone Iterative Method Let us consider the infinite countable system of equations of the form F j [z j ](t, x) = f˜ j (t, x, z) for (t, x) ∈ D and j ∈ N

(4.50)

with homogeneous initial-boundary condition (2.6). We will define the successive terms of the approximation sequences of problem (4.50) and (2.6) as solutions of the following infinite countable systems of equations with Volterra functionals2    F j u in (t, x) = f˜ j t, x, [u n , u n−1 ]i (4.51) for (t, x) ∈ D, j ∈ N 2 For any η, y ∈ B(S) and for every fixed i ∈ S, let [η, y]i denote an element of B(S) with the description

Monotone Iterative Methods 67 and

  j F j v n (t, x) = f˜ j t, x, [v n , v n−1 ] j for (t, x) ∈ D,

j ∈N

(4.52)

for n = 1, 2, . . . in D, with initial-boundary condition (2.6). Theorem 4.7 Let assumptions A0 , (Ha ), (Hf ) hold, and conditions (W), (L), (V) hold in the set K˜ . If we define the successive terms of approximation sequences {u n } and {v n } as regular solutions in D of systems (4.51) and (4.52) with homogeneous initial-boundary condition (2.6) and if u 0 = u 0 , v 0 = v0 , then i. {u n }, {v n } are well defined and u n , v n ∈ CN2+α (D) for n = 1, 2, . . .; ii. the inequalities u 0 (t, x) ≤ u n (t, x) ≤ u n+1 (t, x) ≤ v n+1 (t, x) ≤ v n (t, x) ≤ v0 (t, x)

(4.53)

for (t, x) ∈ D and n = 1, 2, . . . hold; iii. the functions u n and v n for n = 1, 2, . . . are lower and upper solutions of problem (4.1) in D, respectively; iv. if we start two approximation processes from the same pair of a lower solution u 0 and an upper solution v0 of problem (4.50) and (2.6) in D, then the sequences {u n } and {v n } converge monotonously and uniformly to a solution z of problem (4.50) and (2.6) in D at a rate not lower than that of the iterative sequences {u n } and {vn } defined by (4.2), (4.3), and (2.6) in D, i.e., the inequalities u n (t, x) ≤ u n (t, x) ≤ v n (t, x) ≤ vn (t, x)

(4.54)

for (t, x) ∈ D and n = 1, 2, . . . hold; v. the function z = z(t, x) = limn→∞ u n (t, x) uniformly in D is the unique global regular solution of problem (4.50) and (2.6) within the sector u 0 , v0 , and z ∈ CN2+α (D). Proof. Using the auxiliary lemmas we prove the theorem by induction. The proof is simple and similar for the lower and upper solutions, so in both cases we present the proof for lower solutions only. Since u 0 ∈ CN0+α (D), then from Lemma 4.1 and the theorem on the existence and uniqueness of the solution of the Fourier first initial-boundary value problem for the finite system of equations, it follows that there exists the regular unique solution u 1 of problem (4.51) and [η, y]i :=

 j y for all j  = i, ηi for j = i.

j ∈ N,

In the case if S = N then η, y ∈ l ∞ and [η, y]i := (y 1 , y 2 , . . . , y i−1 , ηi , y i+1 , . . . ).

68 Chapter 4 (2.6) in D and u 1 ∈ CN2+α (D). Analogously, if u n−1 ∈ CN2+α (D), then there exists the regular unique solution u n of problem (4.52) and (2.6) in D, u n ∈ CN2+α (D) and (i) is proved by induction. Since u 0 is a lower solution, it satisfies the inequalities   j F j u 0 (t, x) ≤ f˜ j (t, x, u 0 ) = f˜ j t, x, [u 0 , u 0 ] j for (t, x) ∈ D and j ∈ N with condition (2.6). The function u 1 is a solution of the equation   j F j u 1 (t, x) = f˜ j t, x, [u 1 , u 0 ] j for (t, x) ∈ D and j ∈ N with condition (2.6). Therefore, by Lemma 3.1 we obtain u 0 (t, x) ≤ u 1 (t, x) for (t, x) ∈ D.

(4.55)

Moreover, by (4.55) and (W) there is    j F j u 1 (t, x) = f˜ j t, x, [u 1 , u 0 ] j ≤ f˜ j t, x, [u 1 , u 1 ] j for (t, x) ∈ D, and j ∈ N, so the function u 1 is a lower solution of problem (4.50) and (2.6) in D. Analogously, if u n−1 is a lower solution, then by (2.20) we obtain  j  F j u n−1 (t, x) ≤ f˜ j (t, x, u n−1 ) for (t, x) ∈ D and j ∈ N and u n is a solution of system (4.51) with condition (2.6). Therefore, by Lemma 3.1 we obtain u n−1 (t, x) ≤ u n (t, x) for (t, x) ∈ D.

(4.56)

Moreover, by (4.56) and (W) there is    j F j u n (t, x) = f˜ j t, x, [u n , u n−1 ] j ≤ f˜ j t, x, [u n , u n ] j = f˜ j (t, x, u n ), for (t, x) ∈ D and j ∈ N, so the function u n is a lower solution of problem (4.50) and (2.6) in D. From inequalities (4.55) and (4.56) for lower solutions and the analogous inequalities for upper solutions, using mathematical induction, we obtain inequality (4.53). We now prove inequality (4.54) by induction. From (4.2) and (4.56) for n = 1, (4.55) and (W) we obtain    j  j F j u 1 (t, x) − F j u 1 (t, x) = f˜ j t, x, [u 1 , u 0 ] j − f˜ j t, x, [u 0 , u 0 ] j ≥ 0 for (t, x) ∈ D and j ∈ N

Monotone Iterative Methods 69 with condition (2.6). By virtue of Lemma 3.1 we obtain u 1 (t, x) ≥ u 1 (t, x) for (t, x) ∈ D.

(4.57)

u n−1 (t, x) ≥ u n−1 (t, x) for (t, x) ∈ D,

(4.58)

Let now then by (W ), because u n−1 is a lower solution of problem (4.50) and (2.6) in D, we derive     F j u in−1 (t, x) ≤ f˜ j t, x, [u n−1 , u n−1 ]i ≤ f˜ j t, x, [u n−1 , u n−1 ] j for (t, x) ∈ D and j ∈ N with condition (2.6). Therefore, by (4.51) and Lemma 3.1 we have u n (t, x) ≥ u n−1 (t, x) for (t, x) ∈ D.

(4.59)

From (4.2), (4.51), (4.58), and (4.59) we get    j  j F j u n (t, x) − F j u n (t, x) = f˜ j t, x, [u n , u n−1 ] j − f˜ j t, x, [u n , u n−1 ] j   ≥ f˜ j t, x, [u n , u n−1 ] j − f˜ j t, x, [u n , u n−1 ] j ≥ 0 for (t, x) ∈ D and j ∈ N with condition (2.6). By virtue of Lemma 3.1 we obtain u n (t, x) ≥ u n (t, x) in D.

(4.60)

From (4.53), (4.57), and (4.60) we finally obtain inequality (4.54). By (4.54) and (4.15) there is (Lt)n , j ∈ N, n = 1, 2, . . . , n! where N0 := v0 − u 0 0 = const < ∞ for (t, x) ∈ D. j

j

v n (t, x) − u n (t, x) ≤ N0

As a direct consequence, we obtain  j j lim v n (t, x) − u n (t, x) = 0 uniformly in D, n→∞

(4.61)

j ∈ N.

By arguments similar to that used in Section 4.1 we show that z = z(t, x) := lim u n (t, x) uniformly in D n→∞

(4.62)

is the unique global regular solution of problem (4.50) and (2.6) within the sector u 0 , v0 . Obviously, z ∈ CN2+α (D).  Remark 4.4 It is easy to see that the method used in the proof does not need the assumption that the considered system is countable. Therefore, in Theorem 4.7 a countable infinite system may be replaced by an arbitrary infinite system.

70 Chapter 4

4.6 Method of Direct Iterations in Unbounded Domains We consider an infinite system of equations of the form (2.2) with initial-boundary condition (2.5), i.e., the problem F i [z i ](t, x) = f˜i (t, x, z) for (t, x) ∈ and i ∈ S,

(4.63)

z(t, x) = φ(t, x) for (t, x) ∈  ,

(4.64)

where is an arbitrary open domain in the time-space Rm+1 , unbounded with respect to x, and the operators F i , i ∈ S, are uniformly parabolic in , where  is the parabolic boundary of and := ∪  . Now we assume that the open and unbounded (with respect to x) domain, ⊂ Rm−1 , has property P . The boundary ∂ of domain consists of m-dimensional (bounded or unbounded) domains S0 (where S0 is a nonempty set) and ST lying in the hyperplanes t = 0 and t = T , respectively, and a certain surface σ (not necessarily connected) of a class C 2+α , i.e., it consists of a finite number of surfaces of a class C 2+α , not overlapping but having common boundary points lying in the zone 0 < t < T , which is not tangent to any hyperplane t = const. We denote  := S0 ∪ σ . We assume that the number T is sufficiently small. Precisely, T ≤ h 0 = const, where h 0 is a positive constant depending on problem (4.64). We denote by R the part of the domain contained inside the cylindric surface  R  described by the equation nj=1 x 2j = R 2 and  R := ∂ R \ST . Moreover, for an arbitrary fixed τ, 0 < τ ≤ T , we define: τ := ∩ {(t, x) : 0 < t ≤ τ, x ∈ Rm }, σ τ := σ ∩ {(t, x) : 0 < t ≤ τ, x ∈ Rm }, τ

 τ := S0 ∪ σ τ , := τ ∪  τ .

Obviously, T = . Analogously as in Section 2.2, we define the Banach space B (S) and denote by C S ( ) the Banach space of mappings w : → B (S), (t, x)  → w(t, x) and w(t, x) : S → R, i  → wi (t, x), where the functions wi are continuous in , with the finite norm      w0 := sup wi (t, x) : (t, x) ∈ , i ∈ S .

Monotone Iterative Methods 71 In the Banach space C S ( ) the partial order is defined by means of the positive cone C S+ ( ) := {w : w = {wi }i∈S ∈ C S ( ), wi (t, x) ≥ 0 for (t, x) ∈ , i ∈ S}. In the theory of parabolic equations it is well known that initial-value problems  in unbounded  spatial domains are considered with the growth condition |w(t, x)| ≤ M exp K |x|2 ; otherwise, these problems are ill posed. Therefore, we introduce the class E 2 of real functions w = w(t, x) for which there exist positive constants M and K such that the following growth condition is fulfilled   |w(t, x)| ≤ M exp K |x|2 for (t, x) ∈ . We then say that w ∈ E 2 (M, K ; ) or shortly w ∈ E 2 . Denote by C S,E 2 ( ) a Banach space of mappings w ∈ C S ( ) belonging to the class E 2 (M, K ; ) with the finite weighted norm        w0E 2 := sup wi (t, x) exp −K |x|2 : (t, x) ∈ , i ∈ S . For w ∈ C S,E 2 ( ) and for a fixed t, 0 ≤ t ≤ T , we define         t   E2 w0,t ˜ 2 : t˜, x˜ ∈ , i ∈ S . := sup wi t˜, x˜  exp −K |x| A mapping w ∈ C S,E 2 ( ) will be called regular in if the functions wi , i ∈ S, have continuous derivatives Dt wi , Dx j wi , Dx2 j xk wi in for j, k = 1, 2, . . . , m (i.e., reg w ∈ C S,E 2 ( )). k+α We define the Hölder space C k+α ( ) and by C S,E ( ) we denote a Banach space of 2 i mappings w = {w }i∈S with the finite weighted norm     E2 wk+α := sup |wi |k+α exp −K |x|2 : (t, x) ∈ , i ∈ S ,

where K > 0 is a constant. ˜ a . We assume that the coefficients a i = a i (t, x), a i = a i , bi = bi (t, x) Assumption H jk jk jk kj j j ( j, k = 1, 2, . . . , m, i ∈ S) of the operators Li , i ∈ S, are continuous with respect to t and x in , bounded and locally Hölder continuous with exponent α (0 < α < 1) with respect to t and x in and their Hölder norms are uniformly bounded. We assume that the functions f˜i : × C S ( ) → R (t, x, s)  → f˜i (t, x, s), i ∈ S are continuous and satisfy the following assumptions:

72 Chapter 4 ˜ f . The functions f˜i , i ∈ S, are locally Hölder continuous with exponent α Assumption H (0 < α < 1) with respect to t and x in , and their Hölder norms are uniformly bounded. Assumption Ef . f˜i (t, x, 0) ∈ E 2 (M f , K f ; ), i ∈ S. Condition L˜ E2 . The functions f˜i (t, x, s), i ∈ S, fulfill the Lipschitz condition ( L˜ E 2 ) with respect to s, i.e., for arbitrary s, s˜ ∈ C S,E 2 ( ) there is    ˜i  E  f (t, x, s) − f˜i (t, x, s˜ ) ≤ L 1 s − s˜ 0 2 for (t, x) ∈ , where L 1 > 0 is a constant. Condition L˜ ∗ . We say that the functions f˜i (t, x, s), i ∈ S, fulfill the Lipschitz-Volterra  E2  ˜ ∗ with respect to s if for arbitrary s, s˜ ∈ C S,E 2 ( ) there is condition L E2

    ˜i E  f (t, x, s) − f˜i (t, x, s˜ ) ≤ L 2 s − s˜ 0,t2

for (t, x) ∈ ,

where L 2 > 0 is a constant. Moreover, we assume that ˜ φ . φ ∈ C 2+α ( ), where 0 < α < 1. Assumption H S Assumption Eφ . φ ∈ E 2 (Mφ , K φ ;  ). reg

Functions u, v ∈ C S,E 2 ( ) satisfying the infinite systems of inequalities 

F i [u i ](t, x) ≤ f˜i (t, x, u) for (t, x) ∈ , u i (t, x) ≤ φ i (t, x) for (t, x) ∈  and i ∈ S,  i i F [v ](t, x) ≥ f˜i (t, x, v) for (t, x) ∈ , v i (t, x) ≥ φ i (t, x) for (t, x) ∈  and i ∈ S

(4.65) (4.66)

are called, respectively, a lower and an upper solution of problem (4.63), (4.64) in unbounded domain . ˜ We assume that there exists at least one pair u 0 and v0 of a lower and an Assumption A. upper solution of problem (4.63), (4.64) in and u 0 , v0 ∈ C S0+α ( ). A pair of a lower u 0 and an upper solution v0 of problem (4.63), (4.64) in D is called an ordered pair if u 0 ≤ v0 in D. The inequality u 0 ≤ v0 does not follow directly from inequalities (4.65) and (4.66). Therefore, we adopt the following fundamental assumption. ˜ 0 . We assume that there exists at least one ordered pair u 0 and v0 of a lower Assumption A and an upper solution of problem (4.63), (4.64) in , and u 0 , v0 ∈ C S0+α ( ).

Monotone Iterative Methods 73 Let β ∈ C S ( ) be a sufficiently regular function. Denote by P˜ the operator P˜ : β  → P˜ [β] = γ ,

where γ is the (supposedly unique) solution of the initial-boundary value problem  i i F [γ ](t, x) = f˜i (t, x, β) for (t, x) ∈ , γ i (t, x) = φ i (t, x) for (t, x) ∈  and i ∈ S.

(4.67)

  The operator P˜ is the composition of the nonlinear operator F˜ = F˜ i i∈S generated by the functions f˜i (t, x, s), i ∈ S, and defined for any β ∈ C S ( ) as follows: F˜ i : β  → F˜ i [β] = δ i , where

F˜ i [β](t, x) := f˜i (t, x, β) = δ i (t, x) for (t, x) ∈ and i ∈ S,

(4.68)

and the operator G : δ  → G [δ] = γ ,

where γ is the (supposedly unique) solution of the linear problem  i i F [γ ](t, x) = δ i (t, x) for (t, x) ∈ , γ i (t, x) = φ i (t, x) for (t, x) ∈  and i ∈ S. Hence

(4.69)

˜ P˜ = G ◦ F.

0+α ( ) and the function f˜ = { f˜i }i∈S generating the nonlinear Lemma 4.8 If β ∈ C S,E 2 ˜ f ), (Ef ), and (L˜ ∗ ), then operator satisfies conditions (H E2

0+α 0+α ˜ ( ) → C S,E ( ), β  → F[β] = δ. F˜ : C S,E 2 2

Proof. Using the same argument as in the proof of Lemma 4.3 we prove that δ ∈ C S0+α ( ) and by (Ef ) and (L˜ E 2 ) there is δ(t, x) ∈ E 2 (Mδ , K δ ; ). This completes the proof.  ˜ E2 ) hold, then the operator ˜ a ), (H ˜ f ), (Ef ), (H ˜ φ ), (Eφ ), and (L Lemma 4.9 If assumptions (H 0+α P˜ is well defined for β ∈ C S,E 2 ( ), where T ≤ h 0 and 0+α 2+α P˜ : C S,E ( ) → C S,E ( ), β  → P˜ [β] = γ . 2 2

Proof. Observe that system (4.67) has the following property: the ith equation depends on the ith unknown function only. Therefore, its solution is a collection of separate solutions of

74 Chapter 4 these scalar problems for linear equations considered in a suitable space. This fact, Lemma 4.8, and assumptions on the domain imply that the theorem on the existence and uniqueness of a solution for a linear parabolic problem in an unbounded domain holds. Therefore, problem 2+α (4.63), (4.64) has exactly one regular solution, γ ∈ C S,E ( ), provided that T ≤ h 0 .  2 Using the same arguments as in Section 4.1, we may prove the following lemma. Lemma 4.10 Let all the assumptions of Lemmas 4.8 and 4.9 hold, β be an upper solution and α be a lower solution of problem (4.63), (4.64) in , α, β ∈ u 0 , v0 and condition (W) hold. Then α(t, x) ≤ P˜ [β](t, x) ≤ β(t, x) in (4.70) and γ = P˜ [β] is an upper solution of problem (4.63) and (4.64) in . Analogously α(t, x) ≤ P˜ [α](t, x) ≤ β(t, x) in

(4.71)

and η = P˜ [α] is a lower solution of problem (4.63), (4.64) in . Lemma 4.11 If the assumptions of Lemma 4.9 and condition (W) hold, then the operator P˜ is monotone increasing (isotone). Proof. Of course, P˜ is monotone increasing, because F˜ is monotone increasing by assumption (W) and G is monotone increasing operator by the maximum principle. This completes the proof.



˜ 0 , (H ˜ a ), (H ˜ f ), (Ef ), (H ˜ φ ), (Eφ ) hold, and conditions Theorem 4.8 Let assumptions A ˜ ˜ (LE2 ), (W), (L ∗ ) hold in the set E2

K˜ := {(t, x, s) : (t, x) ∈ , s ∈ u 0 , v0 },

where

  u 0 , v0 := w ∈ C S,E 2 ( ) : u 0 (t, x) ≤ w(t, x) ≤ v0 (t, x) for (t, x) ∈ .

Consider the following infinite systems of linear equations:   F i u in (t, x) = f˜i (t, x, u n−1 ) for (t, x) ∈ , and i ∈ S and

  F i vni (t, x) = f˜i (t, x, vn−1 ) for (t, x) ∈ , and i ∈ S

(4.72) (4.73)

for n = 1, 2, . . . with initial-boundary condition u n (t, x) = φ(t, x) for (t, x) ∈  and vn (t, x) = φ(t, x) for (t, x) ∈  ,

respectively, and let N˜ 0 = v0 − u 0 0E 2 < ∞, where an ordered pair of a lower u 0 and an upper v0 solution will be the pair of initial iteration in this iterative process.

Monotone Iterative Methods 75 Then i. there exist unique solutions u n and vn for n = 1, 2, . . . of systems (4.72) and (4.73) with 2+α suitable initial-boundary conditions in and u n , vn ∈ C S,E ( ); 2 ii. the inequalities u 0 (t, x) ≤ u n (t, x) ≤ u n+1 (t, x) ≤ vn+1 (t, x) ≤ vn (t, x) ≤ v0 (t, x)

(4.74)

for (t, x) ∈ and n = 1, 2, . . . hold; iii. the functions u n and vn for n = 1, 2, . . . are a lower and an upper solution of problem (4.63), (4.64) in , respectively; iv. the following estimate: (Lt) wni (t, x) ≤ N˜ 0 n!

n

for (t, x) ∈ , and n = 1, 2, . . . , i ∈ S,

(4.75)

holds, where wni (t, x) := vni (t, x) − u in (t, x) ≥ 0 for (t, x) ∈ ; and i ∈ S,   v. limn→∞ vni (t, x) − u in (t, x) = 0 almost uniformly in , i ∈ S; vi. the function z = z(t, x) = lim u n (t, x) n→∞

is the unique regular solution of problem (4.63), (4.64) within the sector u 0 , v0 , and 2+α z ∈ C S,E ( ). 2 Proof. Starting from the lower solution u 0 and the upper solution v0 of problem (4.63), (4.64), we define by induction the successive terms of the iteration sequences {u n }, {vn } as solutions of systems (4.72) and (4.73) of linear equations with initial-boundary condition (4.64) in , or shortly u n = P˜ [u n−1 ], vn = P˜ [vn−1 ] for n = 1, 2, . . . . 2+α From Lemmas 4.8–4.10 it follows that u n and vn , for n = 1, 2, . . . exist, u n , vn ∈ C S,E ( ) 2 and they are the lower and the upper solution of problem (4.63), (4.64) in , respectively.

By induction, from Lemma 4.10, we derive u n−1 (t, x) ≤ P˜ [u n−1 ](t, x) = u n (t, x), vn (t, x) = P˜ [vn−1 ](t, x) ≤ vn−1 (t, x), for (t, x) ∈ and n = 1, 2, . . . .

(4.76) (4.77)

76 Chapter 4 Therefore, inequalities (4.74) hold. ˜ 0 it follows that u n , vn ∈ E 2 (M0 , N0 ; ) for n = 1, 2, . . .. From (4.74) and Assumption A We now show by induction that (4.75) holds. It is obvious that (4.75) holds for w0 . Suppose it ˜ ∗ ), by holds for wn . Since the functions f˜i , i ∈ S, satisfy the Lipschitz-Volterra condition (L E2 (4.72)–(4.75), we obtain  i  E2 (t, x) = f˜i (t, x, vn ) − f˜i (t, x, u n ) ≤ L wn 0,t F i wn+1 . E2 By the definition of the norm ·0,t and by (4.75) we obtain

(Lt) E2 wn 0,t , ≤ N˜ 0 n! n

so we finally obtain  i  L n+1 t n for (t, x) ∈ , and i ∈ S (t, x) ≤ N˜ 0 F i wn+1 n!

(4.78)

i wn+1 (t, x) = 0 for (t, x) ∈  .

(4.79)

and

Consider the comparison system  i  L n+1 t n for (t, x) ∈ , and i ∈ S (t, x) = N˜ 0 F i Mn+1 n!

(4.80)

with the initial-boundary condition i Mn+1 (t, x) ≥ 0 for (t, x) ∈  .

(4.81)

It is obvious that the functions (Lt) i for i ∈ S Mn+1 (t, x) = N˜ 0 (n + 1)! n+1

are regular solutions of (4.80) and (4.81) in . Applying the theorem on differential inequalities of parabolic type in an unbounded domain to systems (4.78) and (4.80) we get (Lt) i i wn+1 for (t, x) ∈ , and i ∈ S (t, x) ≤ Mn+1 (t, x) = N˜ 0 (n + 1)! n+1

so the induction step is proved by inequality (4.75).

(4.82)

Monotone Iterative Methods 77 As a direct consequence of (4.75) we obtain  lim vni (t, x) − u in (t, x) = 0 almost uniformly in , i ∈ S. n→∞

(4.83)

The iteration sequences {u n } and {vn } are monotone and bounded, and (4.83) holds, so there is a continuous function U = U (t, x) in such that lim u i (t, x) n→∞ n

= U i (t, x),

lim v i (t, x) n→∞ n

= U i (t, x)

(4.84)

almost uniformly in , i ∈ S, and this function satisfies initial-boundary condition (4.64). To prove that the function U (t, x) defined by (4.84) is the regular solution of system (4.63), (4.64) in , it is enough to show that it fulfills (4.63), (4.64) in any compact set contained in . Consequently, because of the definition of R , we only need to prove that it is a regular solution in R for any R > 0. Since the functions f˜i , i ∈ S, are monotone (condition (W)) and (4.74) holds, it follows that the functions f˜i (t, x, u n−1 ) i ∈ S are uniformly bounded in with respect to n. On the basis of results concerning the properties of weak singular integrals, by means of which the solution of the linear system of equations is expressed   F i u in (t, x) = f˜i (t, x, u n−1 ) for (t, x) ∈ R , and i ∈ S, (4.85) we conclude that the function u n (t, x) locally satisfies the Lipschitz condition with respect to x, with a constant independent of n. Hence by (4.84), we conclude that the boundary function U (t, x) locally satisfies the Lipschitz condition with respect to variable x. If we now take the system of equations F i [z i ](t, x) = f˜i (t, x, U ) for (t, x) ∈ R , and i ∈ S

(4.86)

with the initial-boundary condition z(t, x) = U (t, x) for (t, x) ∈  R

(4.87)

then the last property of U (t, x) together with conditions (H f ) and (L) implies that the right-hand sides of system (4.86) are continuous with respect to t, x in R and locally Hölder continuous with respect to x. Hence, in Lemma 4.9, there exists the unique regular solution z of problem (4.86) and (4.87) 2+α in R and z ∈ C S,E ( R ). 2

78 Chapter 4 On the other hand, using (4.84), we conclude that the right-hand sides of (4.85) converge uniformly in R to the right-hand sides of (4.86), lim f i (t, x, u n ) = f i (t, x, U ) uniformly in R i ∈ S.

n→∞

(4.88)

Moreover, the boundary values of u n (t, x) converge uniformly on  R to the respective values of U (t, x). Hence, using the theorem on the continuous dependence of the solution on the right-hand sides of the system and on the initial-boundary conditions to systems (4.85) and (4.86), we obtain lim u in (t, x) = z i (t, x) uniformly in R i ∈ S. (4.89) n→∞

By (4.84) and (4.89), there is z = z(t, x) = U (t, x) for (t, x) ∈ , for an arbitrary R, which means that z = z(t, x) = U (t, x) for (t, x) ∈ i.e., z is the regular solution of problem (4.63), (4.64) within the sector u 0 , v0 , and 2+α z ∈ C S,E ( ). 2 The uniqueness of the solution of this problem follows directly from the uniqueness criterion of Szarski (Theorem 3.6), which ends the proof.