Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in Hölder Spaces

and Applied Analysis 3 Let Fτ H F a, b τ ,H be the linear space of mesh functions φ τ {φk} Na defined on a, b τ {tk kh,Na ≤ k ≤ Nb,Naτ a,Nbτ b} with values in the Hilbert space H. Next, C a, b τ ,H , C α −1, 1 τ ,H , Cα/2 −1, 0 τ ,H , and C 0, 1 τ ,H 0 < α < 1 denote Banach spaces on Fτ H with norms: ∥ ∥φ ∥ ∥ C a,b τ ,H max Na≤k≤Nb ∥ ∥φk ∥ ∥ H, ∥ ∥φ ∥ ∥ Cα −1,1 τ ,H ∥ ∥φ ∥ ∥ C −1,1 τ ,H sup −N≤k<k r≤0 ∥ ∥φk r − φk ∥ ∥ Hr −α/2 sup 1≤k<k r≤N−1 ∥ ∥φk r − φk ∥ ∥ Hr −α, ∥ ∥φ ∥ ∥ Cα/2 −1,0 τ ,H ∥ ∥φ ∥ ∥ C −1,0 τ ,H sup −N≤k<k r≤0 ∥ ∥φk r − φk ∥ ∥ Hr −α/2, ∥ ∥φ ∥ ∥ Cα 0,1 τ ,H ∥ ∥φ ∥ ∥ C 0,1 τ ,H sup 1≤k<k r≤N−1 ∥ ∥φk r − φk ∥ ∥ Hr −α. 2.2 With the help of the self-adjoint positive definite operator B in a Hilbert space H, the Banach space Eα Eα B,H 0 < α < 1 consists of those v ∈ H for which the norm see 22, 23 : ‖v‖Eα sup z>0 z ∥∥∥B z B −1v ∥∥∥ H ‖v‖H, 2.3 is finite. By the definition of Eα B,H , D B ⊂ Eα B,H ⊂ Eβ B,H ⊂ H, 2.4 for all β < α. Lemma 2.2. For 0 < α < 1, the norms of the spaces Eα B,H and Eα/2 A,H are equivalent (see [24]). Theorem 2.3. Suppose μ ∈ D A , Aμ ∈ Eα B,H , f0 g0 ∈ Eα/2 A,H , f−N gN ∈ Eα B,H , g t ∈ C 0, 1 τ ,H , and f t ∈ Cα/2 −1, 0 τ ,H , 0 < α < 1. Boundary value problem 1.2 is wellposed in Hölder space C −1, 1 τ ,H and the following coercivity inequality holds: ∥∥∥ { τ−2 uk 1 − 2uk uk−1 }N−1 1 ∥∥∥ Cα 0,1 τ ,H ∥∥∥{Auk}N−1 −N ∥∥∥ Cα −1,1 τ ,H ∥∥∥ { τ−1 uk − uk−1 }0 −N 1 ∥∥∥ Cα/2 −1,0 τ ,H ≤M [∥∥Aμ ∥ Eα B,H 1 α 1 − α ∥fτ ∥ Cα/2 −1,0 τ ,H ∥gτ ∥ Cα 0,1 τ ,H ] ∥ I τB ( f0 g0 )∥ Eα/2 A,H ∥ I τB ( f−N gN )∥ Eα B,H ] , 2.5 whereM is independent of not only f , g , and μ but also of τ and α. 4 Abstract and Applied Analysis Proof. First of all, let us get the formulae for solution of problem 1.2 . By 21, 25 , uk ( I − R2N )−1 {[ R − R2N−k ] ξ [ RN−k − R k ] ψ − [ RN−k − R k ] I τB 2I τB −1B−1 N−1 ∑ s 1 [ RN−s − R s ] gsτ } I τB 2I τB −1B−1 N−1 ∑ s 1 [ R|k−s| − R s ] gsτ, 1 ≤ k ≤N 2.6 is the solution of boundary value difference problem: −τ−2 uk 1 − 2uk uk−1 Auk gk gk g tk , tk kτ, 1 ≤ k ≤N − 1, u0 ξ, uN ψ, 2.7


Introduction
Nonlocal boundary value problems for partial differential equations have been applied by various researchers in order to model numerous processes in different fields of applied sciences when they are unable to determine the boundary values of the unknown function see, e.g., 1-15 and the references therein .
Well-posedness of difference schemes of elliptic-parabolic equations with nonlocal boundary conditions in H ölder spaces with a weight was studied in 16-19 .In paper 20 in H ölder spaces without a weight was established.The coercivity inequalities for solutions of the boundary value problem for elliptic-parabolic equations were obtained.
In the present paper, the first order of accuracy difference scheme for the approximate solution of problem 1.1 is considered.The well-posedness of difference scheme 1.2 in H ölder spaces without a weight is established.As an application, coercivity inequalities for solutions of difference scheme for elliptic-parabolic equations are obtained.Throughout the paper, H denotes a Hilbert space and A is a self-adjoint positive definite operator with A ≥ δI for some δ > δ 0 > 0.Then, it is wellknown that B 1/2 τA A 4 τ 2 A is a self-adjoint positive definite operator and B ≥ δ 1/2 I. Furthermore, R I τB −1 and P P τA I τA −1 which are defined on the whole space H, are bounded operators, where I is the identity operator.

Well-Posedness of 1.2
First of all, let us start with some auxiliary lemmas that are used throughout the paper.

Abstract and Applied Analysis 3
Let F τ H F a, b τ , H be the linear space of mesh functions ϕ τ {ϕ k } N b N a defined on a, b τ {t k kh, N a ≤ k ≤ N b , N a τ a, N b τ b} with values in the Hilbert space H. Next, C a, b τ , H , C α −1, 1 τ , H , C α/2 −1, 0 τ , H , and C α 0, 1 τ , H 0 < α < 1 denote Banach spaces on F τ H with norms:

2.2
With the help of the self-adjoint positive definite operator B in a Hilbert space H, the Banach space E α E α B, H 0 < α < 1 consists of those v ∈ H for which the norm see 22, 23 : for all β < α.
2 is wellposed in Hölder space C α −1, 1 τ , H and the following coercivity inequality holds: where M is independent of not only f τ , g τ , and μ but also of τ and α.
Proof.First of all, let us get the formulae for solution of problem 1.2 .By 21, 25 , is the solution of boundary value difference problem: is the solution of inverse Cauchy problem:

2.11
Operator equation follows from formulas 2.10 , 2.11 , and the condition u 1 − u 0 u 0 − u −1 .As the operator has an inverse for the solution of operator equation 2.12 .Hence, we have formulas 2.10 , 2.11 , and 2.15 for the solution of difference problem 1.2 .
Using formulae 2.10 and 2.15 , we can get 16

2.17
Abstract and Applied Analysis 7 Finally, we will get coercivity estimate 2.5 .It is based on estimates for the solution of boundary value difference problem 2.7 , for the solution of inverse Cauchy difference problem 2.9 , and for the solution of problem 1.2 .Estimates 2.18 and 2.19 were established in 21, 25 , respectively.
Estimates 2.20 are derived from the formulas 2.16 and 2.17 for the solution of problem 1.2 , estimates 2.1 and following estimates which were established in 26 .This finalizes the proof of Theorem 2.3.

An Application
In this section, an application of the abstract Theorem 2.3 is considered.First, let Ω be the unit open cube in the n-dimensional Euclidean space R n 0 < x k < 1, 1 ≤ k ≤ n with boundary S, Ω Ω ∪ S. In −1, 1 × Ω, the mixed boundary value problem for multidimensional mixed equation: Here, a r x x ∈ Ω , μ x μ x 0, x ∈ S , g t, x t ∈ 0, 1 , x ∈ Ω , and f t, x t ∈ −1, 0 , x ∈ Ω are given smooth functions and a r x ≥ a > 0.
The discretization of problem 3.1 is carried out in two steps.In the first step, the grid sets are defined.To the differential operator A generated by problem 3.1 , the difference operator A x h is assigned by formula: acting in the space of grid functions u h x , satisfying the conditions u h x 0 for all x ∈ S h .With the help of A x h , we arrive at the nonlocal boundary-value problem for an infinite system of ordinary differential equations.
In the second step, problem 3.4 is replaced by difference scheme 1.2 see 21 :

10 Abstract and Applied Analysis
To formulate the result, we introduce the Hilbert spaces L 2h L 2 Ω h , W n be sufficiently small numbers.Then, the solutions of difference scheme 3.5 satisfy the following coercivity stability estimate: where M is not dependent on τ, h, μ h x , g h k x , 1 ≤ k ≤ N − 1, and The proof of Theorem 3.1 is based on Theorem 2.3, the symmetry properties of the difference operator A x h defined by formula 3.3 , and along with the following theorem on the coercivity inequality for the solution of elliptic difference equation in L 2h .

Theorem 3.2. For the solution of elliptic difference problem:
A x h u h x ω h x , x ∈ Ω h , u h x 0, x ∈ S h ,

3.9
Here, M depends neither on h nor w h x .
, the well-posedness of abstract nonlocal boundary value problem Ω h of the grid functions ϕ h x {ϕ h 1 m 1 , . . ., h n m n } defined on Ω h , equipped with the norms: