ON WELL-POSEDNESS OF THE NONLOCAL BOUNDARY VALUE PROBLEM FOR PARABOLIC DIFFERENCE EQUATIONS

We consider the nonlocal boundary value problem for difference equations (uk −uk−1)/τ +Auk = φk, 1 ≤ k ≤ N , Nτ = 1, and u0 = u[λ/τ] + φ, 0 < λ ≤ 1, in an arbitrary Banach space E with the strongly positive operatorA. The well-posedness of this nonlocal boundary value problem for difference equations in various Banach spaces is studied. In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference scheme of the mixed-type boundary value problems for the parabolic equations are obtained. Some results of numerical experiments are given.


Introduction
In [6], the coercive stability estimates in Hölder norms for the solution of the nonlocal boundary value problem in arbitrary Banach space E with the strongly positive operator A were proved.The exact Shauder's estimates in Hölder norms of solution of the boundary value problem on the range {0 ≤ t ≤ 1, x ∈ R n } for 2m-order multidimensional parabolic equations were obtained.In [5], the convergence estimates for the solution of first-order accuracy implicit Rothe difference scheme for approximate solution of this boundary value problem were obtained.We are interested in studying the well-posedness of this difference nonlocal boundary value problem (1.2) in various Banach spaces.
Applying the method of [9,11,14], the well-posedness of difference problem (1.2) in various Banach spaces is studied.In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference schemes of the mixed-type boundary value problems for the parabolic equations are obtained.Some results of numerical experiments are given.Finally, notice that the well-posedness of differential and difference equations of the parabolic type has been developed extensively, see, for instance, [4,10,12].

The estimate of stability
Assume that A is the strongly positive operator, that is, −A is the generator of the analytic semigroup exp{−tA} (t ≥ 0) of the linear bounded operators with exponentially decreasing norm.
Let F τ (E) be the linear space of mesh functions ϕ τ = {ϕ k } N 1 with values in the Banach space E. Next, on F τ (E) we denote C τ (E)-and C α τ (E)-Banach spaces with the norms (2.1) The nonlocal boundary value problem (1.2) is said to be stable in F τ (E) if we have the inequality where M is independent not only of ϕ, ϕ τ but also of τ.
Theorem 2.1.Let τ be a sufficiently small number.Then the boundary value problem (1.2) is stable in C τ (E) and C α τ (E).Proof.The stability of the boundary value problem (1.2) in C τ (E) was obtained in [5].The proof of this result is based on the stability in C τ (E) of the initial value problem for difference equations u 0 is given and also the estimate for solutions of the boundary value problem (2.3), when τ is a sufficiently small number.
If τ is a sufficiently small number, then from the last estimate for solutions of the boundary value problem (2.3) it follows that where M is independent not only of ϕ, ϕ τ but also of τ.
Since the nonlocal boundary value problem (1.1) in the space C(E) of continuous functions defined on [0,1] and with values in E is not well posed for the general positive operator A and space E, then the well-posedness of the difference boundary value problem (1.2) in C τ (E)-norm does not take place uniformly with respect to τ > 0. This means that the coercive norm tends to ∞ as τ → 0+.The investigation of the difference problem (1.2) permits us to establish the order of growth of this norm to ∞. Theorem 2.2.Let τ be a sufficiently small number.Then, for solution of the difference problem (1.2), obey the almost coercive inequality Proof.The almost coercive stability for the solution of the initial value problem for difference (2.3) was obtained in [14].Therefore, the proof of Theorem 2.2 is based on the estimate for the solution of the boundary value problem for difference (1.2), when τ is a sufficiently small number.From the strong positivity A, it follows that 276 On well-posedness of the nonlocal boundary value problem for the solution of the boundary value problem for difference (2.3).Here, (2.12) Let τ be a sufficiently small number.Then, using the estimates and the identity we obtain the estimate Now, using formula (2.11) and the estimate (2.15), we obtain the estimate (2.16) Finally, using the estimate (see [14]) we obtain the estimate (2.10).Theorem 2.2 is proved.
We denote E α = E α (A,E) as the fractional spaces consisting of all v ∈ E for which the following norm is finite: (2.18) A. Ashyralyev et al. 277 Theorem 2.3.Let τ be a sufficiently small number.Then the coercivity inequality holds: where M does not depend on ϕ, ϕ k , 1 ≤ k ≤ N, α, and τ.
Proof.The proof of this theorem is based on the abstract theorem on the well-posedness C τ (E α ) of the initial value difference problem (2.3) of [9] and on the estimate for the solution of problem (1.2), when τ is a sufficiently small number.By formula (2.11), We estimate the norm of each term.Using the estimate (2.15), we obtain To estimate Av 0 in the norm of E α , we use the following Cauchy-Riesz representation formula for the operator A(λ + A) −1 R [λ/τ]− j+1 (see, e.g., [11]): where

.24)
Since z = ρe iψ , with |ψ| < π/2, from the strong positivity of A, it follows that (2.25) 278 On well-posedness of the nonlocal boundary value problem Hence, (2.26) Summing the geometric progression and using the estimate (2.15), we get  [14] and on the estimate for the solution of problem (1.2), when τ is a sufficiently small number.By formula (2.21), we have where We estimate the norm of each term.Using the estimate (2.15), we obtain (2.31) A. Ashyralyev et al. 279 Using the estimates (2.13) and (2.15), we obtain (2.32) Finally, using the triangle inequality, the last estimate, and (2.30), we obtain the estimate (2.28).Theorem 2.4 is proved.
Remark 2.5.By passing to the limit for τ → 0, we obtain the well-posedness of the boundary value problem (1.1) in the spaces of smooth functions.
Remark 2.6.Applying the method of the present paper and of [3], we can obtain similar results for solutions of the difference scheme of the second-order accuracy for approximate solution of problem (1.1).
Remark 2.7.Using this approach, we can obtain the same results for solutions of firstorder accuracy implicit Rothe difference scheme for approximate solutions of the general boundary value problem has a bounded inverse in E. Finally, we consider the applications of these results to the parabolic equations.
280 On well-posedness of the nonlocal boundary value problem

Applications
First, we consider the nonlocal boundary value problem for heat equations where a(x), f (t,x) are given sufficiently smooth functions, δ = const > 0, and a(x) > 0.
We associate with the nonlocal boundary value problem (3.1) the corresponding difference problem We introduce the Banach spaces It is known (see [7,8]) that the difference operator acting on the space of grid functions ϕ h (x) = {ϕ n } M 0 satisfying the conditions ϕ 0 = ϕ M , ϕ 1 − ϕ 0 = ϕ M − ϕ M−1 is a positive operator.Therefore, we can replace the difference problem (3.2) by the abstract boundary value problem (1.2).Using the results of [7,8] and of Theorems 2.1, 2.2, 2.3, and 2.4, we obtain the following result.Theorem 3.1.Let τ be a sufficiently small number.Then, for the solution of the difference problem (3.2), the following inequalities are valid: where M 1 is independent of (ϕ h ) τ , ϕ h , α, and β.
A. Ashyralyev et al. 283 The strong positivity of this elliptic difference operator A x h in the C β 01 (Ω h )-, C(Ω h )-, L p (Ω h )-norms was established in [1,2,14].Using (1.2), we have the nonlocal difference problem The abstract result of this paper and [1,2,11,14,15] permits us to obtain the following result.
Theorem 3.2.Let τ be a sufficiently small number.Then, for the solution of the difference problem (3.14), the following stability inequalities are valid: where M 1 is independent of {ϕ h k (x)} N 1 , ϕ h (x), α, β, p, and M(α,β) is independent of . Third, we consider the boundary value problem on the range {0 ≤ t ≤ 1, x ∈ R n } for 2m-order multidimensional differential equations of parabolic type: 284 On well-posedness of the nonlocal boundary value problem where a r (x), f (t,x) are given sufficiently smooth functions and δ > 0 is a sufficiently large number.We will assume that the symbol of the differential operator of the form acting on functions defined on the space R n satisfies the inequalities for ξ = 0. Problem (3.16) has a unique smooth solution.This allows us to reduce the boundary value problem (3.16) to the boundary value problem (1.1) in Banach space E with a strongly positive operator A x = B x + δI defined by (3.18).We define the grid space R n h (0 ≤ h ≤ h 0 ) as the set of all points of the space R n whose coordinates are given by ) of all grid functions u h (x), defined by the norms By replacing the differential operator A x with the difference operator A x h that acts on the Banach spaces ) and is uniformly strongly positive in h for 0 ≤ h ≤ h 0 , and using (1.2), we have the nonlocal difference problem The abstract result of this paper and of [11,13] permits us to obtain the stability in For approximate solutions of the nonlocal boundary value problem (4.1), we will use Rothe's scheme and a second-order accuracy difference scheme with τ = 1/30, h = 1/30.We have the second-order or fourth-order difference equations with matrix coefficients.
To solve these difference equations, we have applied a procedure of modified Gauss elimination method for difference equations with matrix coefficients.The exact and numerical solutions are given in Table 4.1 Thus, the second-order accuracy difference scheme was more accurate compared with Rothe's difference scheme.
.5) A. Ashyralyev et al. 275 Finally, from the last estimate for solutions of the boundary value problem (2.3) and the stability in C α τ (E) of the initial value problem (2.3), the stability in C α τ (E) of the boundary value problem (1.2) follows.Theorem 2.1 is proved.
Proof.The proof of this theorem is based on the abstract theorem on the well-posedness in C α .27) Finally, using the triangle inequality, the last estimate, and (2.22), we obtain the estimate (2.20).Theorem 2.3 is proved.Theorem 2.4.Let τ be a sufficiently small number.Then the boundary value problem (1.2) is coercive stable in C α τ (E).τ (E) of the initial value difference problem (2.3) of