Accelerated finite difference schemes for second order degenerate elliptic and parabolic problems in the whole space

We give sufficient conditions under which the convergence of finite difference approximations in the space variable of possibly degenerate second order parabolic and elliptic equations can be accelerated to any given order of convergence by Richardson's method.


Introduction
This is the third article of a series studying a class of finite difference equations, related to finite difference approximations in the space variable of second order parabolic and elliptic PDEs in R d . These PDEs are given on the whole R d in the space variable, and may degenerate and become first order PDEs. Denote by u h the solutions of the finite difference equations corresponding to a given grid with mesh-size h. By shifting the grid so that x becomes a grid point we define u h for all x ∈ R d rather than only at the points of the original grid. In [5] and [6], the first and second articles of the series, we focus on the smoothness in x of u h , rather than their convergence for h → 0. The main results in [5] and [6] give estimates, independent of h, for the first order derivatives Du h and for derivatives D k u h in x of any order k, respectively.
In the present paper one of our main concerns is the smoothness of the approximations u h in (x, h). In particular, we are interested in the convergence of u h , and their derivatives in x, in the supremum norm, as h → 0. We give conditions ensuring that for any given integer k ≥ 0 the approximations u h admit power series expansions up to order k + 1 in h near 0 like formula for u h in h. We obtain it by proving first Theorems 2.1 and 2.2 below on the solvability of the PDE that is being approximated, and of a system of degenerate parabolic PDEs, respectively, for the coefficients u (j) , j = 0, . . . , k. Of course, u (0) is the true solution of the corresponding PDE. The remainder term r h satisfies a finite difference equation, with the same difference operator appearing in the equation for u h , and we estimate r h by making use of the maximum principle enjoyed by this operator. This is a standard approach to get power series expansions for finite difference approximations in general, and it works well in many situations, when suitable results regarding the equations for the coefficients u (j) are available. In our situation it requires some facts either from the theory of diffusion processes or from the theory of degenerate parabolic equations. However, we do not use any facts from these theories. We prove Theorem 2.1, and hence Theorem 2.2, relying on results on finite difference schemes, obtained in [6] by elementary techniques. It is worth saying that since long ago finite difference equations were already used to prove the solvability of partial differential equations (see, for instance, [8] and [9]). Our contribution lies in considering degenerate equations. After establishing the expansions of u h in h not only we obtain the possibility to prove the convergence of u h to the true solution in the sup norm as h → 0 but also the possibility to accelerate it to any order under appropriate assumptions. We prove the latter by taking linear combinations of finite difference approximations corresponding to different mesh-sizes. This method is especially effective when many of the coefficients in the expansion of u h are zero. These results are given by Theorem 2.21 and Corollary 2.8. Their counterparts in the elliptic case are presented by Corollary 3.7.
The idea of accelerating the convergence of finite difference approximations in the above way is well-known in numerical analysis. It is due to L.F. Richardson, who showed that it works in some cases and demonstrated its usefulness in [15] and [16]. This method is often called Richardson's method or extrapolation to the limit, and is applied to various types of approximations. The reader is referred to the survey papers [2] and [4] for a review on the history of the method and on the scope of its applicability and to the textbooks (for instance, [10] and [11]) concerning finite difference methods and their accelerations.
We are interested in approximating in the sup norm not only the true solution but also its derivatives. Note that even if the coefficients u (j) are bounded smooth functions of (t, x), the derivatives D k u h of u h in x may not admit similar expansions, since the derivatives of r h may not be bounded in h near 0. Note also that the bounds on the sups of u (j) and r h generally depend on T , and may grow exponentially in T . This becomes a big obstacle on the way of extending our results to the elliptic case.
Our next result on power series expansions, Theorem 2.7, improves the previous theorem in two directions. It gives sufficient conditions such that for any given integer k ≥ 0 (a) D k u h admits an expansion similar to (1.1), (b) the bounds on the coefficients are independent of T . Having (a) we can approximate the k-th derivatives of the true solution by D k u h with rate of order h and accelerate the rate under appropriate assumptions. We can also approximate the k-th derivatives of the true solution with finite difference operators in place of D k applied to u h , which is more convenient in applications because it does not require computing the derivatives of u h .
We ensure (a) and (b) by relying heavily on derivative estimates, independent of T , obtained in [5] and [6] for solutions of finite difference equations. Property (b) of the expansions allows us to extend Theorem 2.7 to the elliptic case. This extension is Theorem 3.5.
As a consequence of the derivative estimates proved in [6] we obtain also, see Theorem 2.9 below, estimates, independent of h and T , for the derivatives of u h in x and h. Clearly, Theorem 2.9 immediately implies Taylor's formula for u h in h, up to appropriate order, with bounded coefficients. It is interesting to notice that the converse implication does not hold: If for k ≥ 1 the function u h admits a power series expansion up to order k + 1 in h near 0 with bounded coefficients, it does not imply, in general, that the derivative of u h in h up to order k + 1 are bounded functions. That is why Theorem 2.7 does not imply Theorem 2.9, and the latter implies the former only if condition (i) in Theorem 2.7 is satisfied. Additional information on the behaviour of the derivatives of u h in x and h when h is near 0 is given by Theorem 2.11. The corresponding result in the elliptic case is Theorem 3.4.
In this article we are working with equations in the whole space having in mind considering equations in bounded smooth domains in a subsequent article. Still it may be worth noting that the results of this article are applicable to the one dimensional ODE The point is that one need not prescribe any boundary value of u at the points ±1 and if one considers this equation on all of R, the values of its coefficients and f outside (−1, 1) do not affect the values of u(x) for |x| < 1.

Formulation of the main results for parabolic equations
We fix some numbers h 0 , T ∈ (0, ∞) and for each number h ∈ (0, h 0 ] we consider the integral equation for u, where g(x) and f (s, x) are given real-valued Borel functions of x ∈ R d and (s, x) ∈ H T = [0, T ] × R d , respectively, and L h is a linear operator defined by x), and c(t, x) are given real-valued Borel functions of (t, As usual, we denote For smooth ϕ and integers k ≥ 0 we introduce D k ϕ as the collection of partial derivatives of ϕ of order k, and define is continuous in R d and for all multi-indices α with |α| ≤ m the generalized functions D α ψ(t, x) are bounded on H T . In this case we use the notation This notation will be also used for functions ψ independent of t. Let m ≥ 0 be a fixed integer. We make the following assumptions.
Assumption 2.1. For any λ ∈ Λ 1 , we have p λ , q λ , c, f, g ∈ B m and, for k = 0, ..., m and some constants M k we have  [5] it is required that the derivatives of the data up to order m be continuous in H T , but its proof can be easily adjusted to include our case (see Remark 2.6 below).
Naturally, we view (2.1) as a finite difference schemes for the problem where By a solution of (2.5)-(2.6) we mean a bounded continuous function u(t, x) on H T , such that it belongs to B 2 and satisfies (2.8) in H T in the sense of generalized functions, that is, for any t ∈ [0, T ] and Observe that if u ∈ B 2 , then (2.9) implies that (2.8) holds almost everywhere with respect to x and if u ∈ B 3 then the second derivatives of u in x are continuous in x and (2.8) holds everywhere.
The reader can find in [7] a discussion showing that in all practically interesting cases of parabolic equations like (2.8) the operator L can be represented as in (2.7), so that considering operators L 0 h in form (2.3) is rather realistic.
The following theorem on existence and uniqueness of solutions is a classical result (see, for instance, [12], [13], [14]) which we are going to obtain by using finite-difference approximations. Observe that this result is rather sharp in what concerns the smoothness of solutions, which is seen if all the coefficients of L are identically zero and f is independent of t in which case the solution is tf (x) + g(x).
The existence part in Theorem 2.1 is proved in Section 6 and uniqueness in Section 4.
In Section 6 a repeated application of this theorem allows us to prove a result on the solvability of (2.13) below. First introduce where is the derivative of ϕ in the direction of λ. Consider the system of equations Remark 2.2. Quite often in the article we use the following symmetry condition: (S) Λ 1 = −Λ 1 and q λ = q −λ for all λ ∈ Λ 1 . Notice that, if condition (S) holds, then for j = 1, . . . , k.
(ii) If the symmetry condition (S) holds and Assumption 2.1 is satisfied with m ≥ 2k + 2, then (2.13) has a unique solution {u (j) } k j=1 , such that for j = 1 . . . , k. In addition, if  We prove this theorem in Section 7. The following corollary is one of the results of [3] proved there by using the theory of diffusion processes. We obtain it immediately from case (iii) with k = 1. Of course, the result is well known for uniformly nondegenerate equations but we do not assume any nondegeneracy of L, which becomes just a zero operator at those points where q λ = p λ = c = 0. Actually, in [3] a full discretization in time and space is considered for parabolic equations, so that, formally, Corollary 2.4 does not yield the corresponding result of [3]. On the other hand, a similar corollary can be derived from Theorem 3.5 below which treats elliptic equations and it does imply the corresponding result of [3]. It also generalizes it because in [3] one of the assumptions, unavoidable for the methods used there, is that q λ = r 2 λ with functions r λ that have four bounded derivatives in x, which may easily be not the case under the assumptions of Theorem 3.5.
To formulate our main result about acceleration for parabolic equations we fix an integer k ≥ 0 and set where, naturally, u 2 −j h are the solutions to (2.1), with 2 −j h in place of h, and V −1 is the inverse of the Vandermonde matrix with entries The following result is a simple corollary of Theorem 2.3.
Theorem 2.5. In each situation when Theorem 2.3 is applicable we have that the estimate Proof. By Theorem 2.3 and the theorem is proved.
Sometimes it suffices to combine fewer terms u 2 −j h to get accuracy of order k + 1. To consider such a case for odd integers k ≥ 1 definẽ andṼ −1 is the inverse of the Vandermonde matrix with entries The above results show that if the data in equation (2.8) are sufficiently smooth, then the order of accuracy in approximating the solution u (0) can be as high as we wish if we use suitable mixtures of finite difference approximations calculated along nested grids with different mesh-sizes. Assume now that we need to approximate not only u (0) but its derivative D α u (0) for some multi-index α as well. What accuracy can we achieve? The answer is closely related to the question whether the expansion holds for all (t, x) ∈ H T and h ∈ (0, h 0 ], such that The result concerning this expansion and the following series of results appeared after the authors tried to extend the above theorems from the parabolic to the elliptic case. The main and rather hard obstacle is that the constants in our estimates depend on T and, actually, may grow exponentially in T . By the way, this obstacle is caused by possible degeneration of our equations and exists even if we consider equations in bounded smooth domain. To be able to give some conditions under which this does not happen, we introduce new notation and investigate smoothness properties of u h with respect to x. As a simple byproduct of this investigation we also obtain smoothness of u h with respect to h, which, by the way, cannot be derived from (2.18).
Take a function τ λ defined on Λ 1 taking values in [0, ∞) and for λ ∈ Λ 1 introduce the operators For uniformity of notation we also introduce Λ 2 as the set of fixed distinct vectors ℓ 1 , ..., ℓ d none of which is in Λ 1 and definē Below B(R d ) is the set of bounded Borel functions on R d and K is the set of bounded operators K h = K h (t) mapping B(R d ) into itself preserving the cone of nonnegative functions and satisfying K h 1 ≤ 1.
Assumption 2.6. We have m ≥ 2 and, for any h ∈ (0, h 0 ] and n = 1, ..., m, there exists an operator K h = K h,n ∈ K, such that on H T for all smooth functions ϕ. Obviously Assumptions 2.5 and 2.6 are satisfied if q λ and p λ are independent of x. In the general case, as it is discussed in [5], the above assumptions impose not only analytical conditions, but they are related also to some structural conditions, which can somewhat easier be analized under the symmetry condition (S). (2.30) In the main case of applications we will require the last sum to be identically zero as in Assumption 2.3.
Remark 2.4. Assumptions 2.5 and 2.6 are discussed at length and in many details in [5] and [6], and sufficient conditions, without involving test functions ϕ are given for these assumptions to be satisfied. In particular, it is shown in [6] that if condition (S) holds, m ≥ 2, τ λ = 1, Assumptions 2.1 and 2.2 are satisfied, and q λ ≥ κ for a constant κ > 0, then both Assumptions 2.5 and 2.6 are satisfied for any c 0 > 0 and δ ∈ (0, 1), if h 0 is sufficiently small and τ 0 , K, and K are chosen appropriately. Moreover, the condition κ > 0 can be dropped, provided, additionally, that c 0 is large enough (this time we need not assume that h is small). Remember, that by Remark 2.3 the condition that c 0 be large is, actually, harmless as long as we are concerned with equations on a finite time interval. Mixed situations, when c is large at those points where some of q λ can vanish are also considered in [6].
In [5] we have seen that Assumption 2.5 imposes certain nontrivial structural conditions on q λ which cannot be guaranteed by the size of c 0 if q λ is only once continuously differentiable. In contrast, even without condition (S), given that Assumptions 2.1, 2.5, 2.7 are satisfied and m ≥ 2, as is shown in [6], Assumption 2.6 is also satisfied if c 0 is large enough. We prove this theorem in Section 7. Remember the definition ofū h andũ h in (2.21) and (2.24). The following is an obvious consequence of Theorem 2.7.
and if condition (iii) is met then where N depends only on on d, m, δ, K, τ 0 , c 0 , if m ≥ 3 + |α| and Assumption 2.1 through 2.6 hold. In addition one can and e i is the ith basis vector in R d . This follows easily from the mean value theorem and Theorem 2.9 below. The reader understands that similar assertion is true in case of Corollary 2.8 with the only difference that one needs larger m and better finite-difference approximations of D α .
Next we investigate the smoothness of u h in x and h. Recall that for functions ϕ depending on h we use the notation D r h ϕ for the r-th derivative of ϕ in h. As usual, D 0 h ϕ := ϕ. Remark 2.6. Suppose that Assumption 2.1 is satisfied. Take an h 1 ∈ (0, h 0 ), consider equation (2.1) as an equation about a function u h (t, x) as function of (h, t, x) ∈ [h 1 , h 0 ]×H T and look for solutions in the space It is obvious that the integrand in (2.1) can be considered as the result of application of an operator, which is bounded in B m (h 1 ), to u h (s, x). Therefore, a standard abstract theorem on solvability of ODEs in Banach spaces shows that there exists a solution of (2.1) in B m (h 1 ). Since just bounded solutions are uniquely defined by (2.1), we conclude that our u h belongs to B m (h 1 ) for any h 1 ∈ (0, h 0 ). Obviously, if the derivatives of the data are continuous in x, the same will hold for u h .
The above argument, actually, works if we replace |α| + 3r ≤ m with |α| + r ≤ m in (2.32). We talk about (2.32) in the above form because we will show that under our future assumptions the quantity (2.32) is bounded independently of h 1 . We prove this theorem in Section 5, and in Section 6 we show that the following fact, used when we come to the elliptic case, is a simple corollary of it. Additional information on the behavior of D r D k h u h for small h is provided by the following result which we prove in Section 5. Then, for any integer r ≥ 0 such that

Main results for elliptic equations
Here we assume that p λ , q λ , c, and f are independent of t and turn now our attention to the equations 2) Naturally by a solution of (3.2) we mean a function v on R d such that it belongs to B 2 and (3.2) holds almost everywhere. Clearly, if a solution v belongs to B 3 and q λ , p λ , c, and f are continuous functions on R d , then (3.2) holds everywhere.
First we prove the existence and uniqueness of the solutions of equations (3.1) and (3.2).
It is seen that the existence and uniqueness of bounded solution of (3.1) follows by contraction principle. Using smooth successive iterations yields that v h ∈ B m .
where the last equality is obtained by integration by parts. Consequently, v is a solution of (3.4) and it satisfies estimate (3.3).

5)
where N is a constant depending only on m, δ, c 0 , τ 0 , K, Proof. To prove (3.5), take a constant ν > 0 as in the proof of Theorem 3.2, define u(t, x) := v h (x)e νt , and observe that u is the unique bounded solution of By Theorem 2.9 for any T > 0  Proof. This theorem can be deduced from Theorem 2.11 in the same way as Theorem 3.3 is obtained from Theorem 2.9.
Now we want to establish an expansion for v h , i.e., to show for an integer k ≥ 0 the existence of some functions v (0) ,...,v (k) on R d , and a function R h on R d for each h ∈ (0, h 0 ] such that for all x ∈ R d and h ∈ (0, with a constant N . Proof. Take a small constant ν > 0, as in the proof of Theorem 3.2, let u be a function defined on H ∞ such that for each T > 0 its restriction onto H T is the unique solution Remark 3.1. We can show similarly that v (i) , i = 1, ..., k, is the unique solution of the system

By Theorem 2.3 in each of the cases (i)-(iii) we have
in an appropriate class of functions (cf. Theorem 2.2).
The following result can be obtained easily from Theorem 2.7 by inspecting the proof of the previous theorem.

Proof of uniqueness in Theorem 2.1 and a stipulation
We will see later that the proof of Theorem 2.3 only uses the existence of sufficiently smooth solutions of (2.8) and (2.13). Therefore, if m ≥ 3, uniqueness of u (0) follows from expansion (2.18). If m = 2, one can use simple ideas based on integrating by parts. We briefly outline these ideas referring for details to [12], [13], [14].
First, one may assume that g = f = 0 and let u (0) be the corresponding solution. Then, by introducing a new function v = u (0) (cosh |x|) −1 one reduces the issue to uniqueness of v, which satisfies an equation similar to (2.5) with g = f = 0 and different coefficients which we denote byq λ ,p λ , andĉ = c, and, moreover, v, Dv, D 2 v ∈ L 2 (H T ). After that one multiplies the equation for v by v and integrates over H T . One uses integration by parts, and the fact that due to the assumption q λ ≥ 0 we have |Dq λ | 2 ≤ 4q λ sup |D 2q λ |. One also uses Young's inequality implying that

Then one quickly arrives at a relation like
where N is a constant independent of c. If c is large enough, the above inequality is only possible if v = 0, which proves uniqueness if c is large enough. In the general case it only remains to observe that the usual change of the unknown function taking v(t, x)e λt in place of v for an appropriate λ will lead to as large c as we like.
Remark 4.1. Notice that apart from uniqueness in Theorems 2.1 and 2.2 all our other assertions and assumptions are stable under applying mollifications of the data with respect to x. For instance, take a nonnegative ζ ∈ C ∞ 0 (R d ) with unit integral, for ε > 0 define ζ ε (x) = ε −d ζ(x/ε) and for locally summable ψ(x) use the notation λ , c (ε) , f (ε) , and g (ε) will satisfy the same assumptions with the same constants as the original ones and will be infinitely differentiable in x.
It is not hard to see that if our assertions are true for the mollified data, then they are also true for the original ones. For instance, let v ε be the solution of (2.5) with the new data. The uniform in ε estimates of the derivatives in x and the equation itself, guaranteeing that the first derivatives in time are bounded, show that v ε are uniformly continuous in [0, T ]×{|x| ≤ R} for any R. Then there is a sequence ε n ↓ 0 such that v εn converges uniformly in [0, T ]×{|x| ≤ R} for any R to a bounded continuous function v.
This along with uniform boundedness of |D α v ε |, |α| ≤ m, lead to the fact that the generalized derivatives |D α v|, |α| ≤ m, are bounded and admit the same estimates as those of v ε . Also since D α v εn → D α v in the sense of distributions and all of them are uniformly bounded, we conclude that this convergence is true in the weak sense in any L 2 ([0, T ] × {|x| ≤ R}). Now it is easy to pass to the limit in equation (2.9) written for modified coefficients and v ε in place of u concluding that since the derivatives converge weakly and q (ε) λ → q λ ,..., f (ε) → f uniformly on H T , v satisfies (2.9). Similar argument takes care of Theorem 2.2 (in which uniqueness will be derived from uniqueness in Theorem 2.1).
Our claim about stability of other results is almost obvious and from this moment on we will assume that the data are as smooth in x as we like.

5.
Proof of Theorems 2.9 and 2.11 In [5] (see there Theorems 2.3 and 2.1 and Corollary 3.2 if m = 0) and [6] we obtained the following result on the smoothness in x of the solution u h to equation (2.1).
To proceed further we need a few formulas. on H T , where ∂ λ ϕ is introduced in (2.12).
(ii) Assume that the derivatives of ϕ in x up to order n + 2 are continuous functions in x, and that Assumption 2.3 holds. Then Proof. By Taylor's formula applied to ϕ(t, x+ hθλ) as a function of θ ∈ [0, 1] Multiplying the first equality by p λ and summing up in λ over Λ 1 we obtain (5.2) for n = 0. Multiplying the second equality by q λ , summing up in λ over Λ 1 we obtain (5.3) for n = 0 since Now we are ready to prove Theorems 2.9 and 2.11.
Proof of Theorem 2.9. If m = 2 or k = 0, our assertion follow directly from Theorem 5.1. Therefore, in the rest of the proof we assume that m ≥ 3 and k ≥ 1.
We will be using (5.4). Observe that if 1 ≤ i ≤ k, then Thus by Remark 2.6 we know that D i+2+r u (k−i) h are bounded and continuous on H T . It follows that R k h ∈ B r . By Theorem 5.1 with r in place of m we obtain I kr := sup It is not hard to see that Hence, Here on the right the first index of I kr is reduced by at least 1 and the sum of indices increased by 2. Therefore, after k iterations we will come to the inequality I kr ≤ N I 0,k+2k+r .
It only remains to observe that I 0,3k+r ≤ I 0,m and the latter quantity is estimated in Theorem 5.1. The theorem is proved.
Proof of Theorem 2.11. First of all observe that the symmetry assumption and (2.16) imply that for any smooth function ϕ(x), odd i ≥ 0, and any multi-index α, such that |α| ≤ m, we have If k = 1 and an integer n ≤ r, then owing to (5.5) where the last two estimates follow from the fact that r + 4 = r + 3k + 1 ≤ m and from Theorem 2.9, respectively. Similarly,

Hence, sup
and applying Theorem 5.1 to (5.4) yields (2.34). Now we proceed by induction on k. Assume that for an odd number j estimate (2.34) holds whenever 3k + r ≤ m − 1 and odd k ≤ j. This hypothesis is justified by the above for j = 1 and to prove the theorem it suffices to show that the hypothesis also holds with j + 2 in place of j. Take an odd k and an integer r such that k ≤ j + 2, 3k + r ≤ m − 1 and again use (5.4). As above, to obtain (2.34) it suffices to prove that Take an integer n ≤ r. Observe that if 1 ≤ i ≤ k and i is even, then k − i is odd and k − i ≤ j + 2 − i ≤ j and If 1 ≤ i ≤ k and i is odd, then i + 2 is odd too and as in the beginning of the proof D n where the last sup is majorated by N J owing to Theorem 2.9 since In both situations we have (5.7). Similarly, if 1 ≤ i ≤ k and i is odd, then i + 1 is even and where the last sup is majorated by N J again owing to Theorem 2.9 since which is now shown to hold in both subcases. By combining this with (5.7) we come to (5.6) and the theorem is proved.
If λ ∈ ±(Λ 1 \Λ s 1 ) we setq λ = (1/2)q ±λ . ThenΛ 1 andq λ satisfy the symmetry condition (S) and can be used to represent the first term on the right in (2.7) in place of the original ones. Next, we redefine and extend p λ introducingp λ onΛ 1 , so thatp λ = M 0 + p λ on Λ s 1 , for λ ∈ Λ 1 \ Λ s 1 we setp ±λ = M 0 ± (1/2)p λ , and for −λ ∈ Λ 1 \ Λ s with N independent of h, we conclude from the equation for u h that their first derivatives in t are bounded uniformly in h. Therefore, there exists a sequence h(n) ↓ 0 such that u h(n) converges uniformly on [0, T ] × {x : |x| ≤ R} for any R to a continuous function v.
with the same N as in (6.1). If we take τ λ ≡ 1, then Remark 6.4 of [5] and Remark 4.3 of [6] imply that both N 's can be chosen to depend only on d, m, inf c, |Λ 1 |, and M 0 , ..., M m . Next, the modified equation (2.1) yields that for any φ ∈ C ∞ 0 (R d ) and We pass to the limit in this equation and find that v satisfies an integral equation, integrating by parts in which proves that v is a solution of (2.8).
Finally, we notice that the case that c is not large is reduced to the above one by usual change of the unknown function taking v(t, x)e λt in place of v for an appropriate λ, which leads to subtracting λv from the right-hand side of (2.5). For the new equation we then find a solution admitting estimate (6.2) with N independent of T but coming back to the solution of the original equation will bring an exponential factor depending on T .
This and uniqueness proved in Section 4 finish proving the theorem.

Proof of Theorem 2.2.
Notice that for each j = 1, . . . , k equation (2.13) does not involve the unknown functions u (l) with indices l > j. Therefore we can solve (2.13) and prove the statements (i) and (ii) recursively on j.
First we prove that there is at most one solution (u (1) , . . . , u (k) ) in the space B 2 × · · · × B 2 . Denote We may assume that u (0) = 0. Then clearly S 1 = 0 and by Theorem 2.1 we have u (1) = 0. If for a j ∈ {2, . . . k} we have u (1) = u (2) = · · · = u (j−1) = 0, then clearly S j = 0 which by Theorem 2.1 yields u (j) = 0. Hence the statements on uniqueness follow because for every j = 1, 2, . . . , k we obviously have B m−3j ⊂ B 2 when m ≥ 3k + 2 and B m−2j ⊂ B 2 when m ≥ 2k + 2. While dealing with the existence of a solution first take j = 1. Observe that by Theorem 2.1 we have u (0) ∈ B m with m ≥ 5 in case (i) and with m ≥ 4 in case (ii). Thus in case (i) we have S 1 ∈ B m−3 ⊂ B 2 and by Theorem 2.1 it follows that there exists u (1) ∈ B m−3 satisfying (2.13) and admitting the estimate Taking the estimate of the last term again from Theorem 2.1 we obtain (2.14) for j = 1. In case (ii) we have actually better smoothness of S 1 , because the first sum in (2.11) is zero for i = 1 and, for that matter, for all odd i. It follows that S 1 ∈ B m−2 and this leads to (2.15) for j = 1 as above. By adding that under the conditions (S) and (2.16) we have L (1) = 0, S 1 = 0, and u (1) = 0, we obtain (2.17) for j = 1.
Passing to higher j we assume that k ≥ 2. Suppose that, for a j ∈ {2, ..., k} we have found u (1) ,...,u (j−1) with the asserted properties. Then in the case (i) we have Hence S j ∈ B m−3j and therefore by Theorem 2.1 there exists u (j) ∈ B m−3j satisfying (2.13) and admitting the estimate where the last inequality follows by the induction hypothesis.
In case (ii) we take into account that due to condition (S) we have and due to condition (2.16) we have for odd numbers i and sufficiently smooth functions ϕ. It follows that in case (ii) for i = 1, ..., j we have Hence S j ∈ B m−2j and therefore by Theorem 2.1 there exists u (j) ∈ B m−2j satisfying (2.13) and admitting the estimate and by using the induction hypothesis we come to (2.15). Furthermore, in case (ii) if (2.16) is satisfied, our induction hypothesis says that u (l) = 0 for all odd l ≤ j − 1. If j is even, then, obviously, u (l) = 0 for all odd l ≤ j as well. If j is odd then to carry the induction forward it only remains to prove that u (j) = 0. However, for odd i we have due to (6.3)-(6.4). This equality also holds if i ≥ 2 and i is even, since then j − i is odd and u (j−i) = 0 by assumption. Thus, S j = 0 and u (j) = 0.
Remark 6.2. The above proof is based on Theorem 2.1 and leads to estimates (2.14) and (2.15) with N depending only on the same parameters as in Theorem 2.1. Therefore, according to Remark 6.1 if Assumptions 2.1 through 2.6 are satisfied and the restrictions on m and k from Theorem 2.2 are met, then the constants N in estimates (2.14) and (2.15) depend only on m, δ, c 0 , τ 0 , K, M 0 , ..., M m , |Λ 1 |, and Λ 1 . This proves the part of assertions of Theorem 2.10 concerning Theorem 2.2. The proof of its remaining assertions can be obtained in the same way and is left to the reader.

Proof of Theorem 2.3 and 2.7
We need some lemmas. The first one is a simple lemma from undergraduate calculus on Taylor's expansion.
To formulate our next lemma we recall the operators L h , L and L (i) , defined in (2.2), (2.7), and (2.11), respectively, and for each h ∈ (0, h 0 ] and integer j ≥ 0 introduce the operator Thus applying Lemma 7.1 to F (h) := L h ϕ(t, x) for fixed (t, x) and using Lemma 5.2, we have Now estimate (7.1) follows easily.
The next lemma is a version of the maximum principle for ∂/∂t − L h . It is a special case of Corollary 3.2 in [5].
Proof of Theorem 2.7. Coming back to the above proof of Theorem 2.3 we see that function (7.5) satisfies (7.4) with F given by (7.6). We notice that k − j + 3 + l ≤ m − 3j for j = 0, 1, . . . , k in case (i),