We use, respectively, the third and fourth order accuracy approximate formulas
(2.1)
(2.2)
for . Here, , is a notation for the greatest integer function. Applying formulas (2.1) and (2.2) to , we get, respectively,
(2.3)
the third order of accuracy difference problem and
(2.4)
the fourth order of accuracy difference problem for inverse problem (1.1).
For solving of problems (2.3) and (2.4), we use the algorithm [14], which includes three stages. For finding a solution of difference problems (2.3) and (2.4) we apply the substitution
(2.5)
In the first stage, applying approximation (2.5), we get a nonlocal boundary value difference problem for obtaining . In the second stage, we put and find . Then, using the formula
we define an element p. In the third stage, by using approximation (2.5), we can obtain the solution of difference problems (2.3) and (2.4). In the framework of the above mentioned algorithm for , we get the following auxiliary nonlocal boundary value difference scheme:
(2.7)
for the third order of accuracy difference problem (2.3) and
(2.8)
for the fourth order of accuracy difference problem (2.4).
For a self-adjoint positive definite operator A, it follows that [22] is a self-adjoint positive definite operator, where , , I is the identity operator. Moreover, the bounded operator D is defined on the whole space H.
Now we give some lemmas that will be needed below.
Lemma 2.1 The following estimates hold [23]:
Lemma 2.2 The following estimate holds [23]:
where
Lemma 2.3 For , the operator
has an inverse such that
and the estimate
(2.9)
is valid.
Proof We have
(2.10)
where
(2.11)
Applying estimates of Lemma 2.1, we have
(2.12)
By using the triangle inequality, formula (2.10), estimates (2.9), (2.12), and Lemma 2.2 of paper [15], we obtain
for any small positive parameter τ. From that follows estimate (2.9). Lemma 2.3 is proved. □
Lemma 2.4 For , the operator
has an inverse
and the estimate
(2.13)
is satisfied.
Proof We can get
(2.14)
where G is defined by formula (2.11) and
Applying estimates of Lemma 2.1, we have
(2.15)
Using the triangle inequality, formula (2.14), estimates (2.13), (2.15), and Lemma 2.3 of paper [15], we get
for any small positive parameter τ. From that follows estimate (2.13). Lemma 2.4 is proved. □
Let and be the spaces of all H-valued grid functions in the corresponding norms,
Theorem 2.1 Assume that A is a self-adjoint positive definite operator, and (). Then, the solution of difference problem (2.3) obeys the following stability estimates:
(2.16)
(2.17)
(2.18)
Proof We will obtain the representation formula for the solution of problem (2.7). Applying the formula [23], we get
(2.19)
By using formula (2.19) and nonlocal boundary conditions
we get the system of equations
(2.20)
Solving system (2.20), we obtain
(2.21)
Therefore, difference problem (2.7) has a unique solution which is defined by formulas (2.19), (2.21), and (2.22). Applying formulas (2.19), (2.21), (2.22), and the method of the monograph [23], we get
(2.23)
The proofs of estimates (2.17), (2.18) are based on formula (2.5) and estimate (2.23). Using formula (2.5) and estimates (2.23), (2.17), we obtain inequality (2.16). Theorem 2.1 is proved. □
Theorem 2.2 Suppose that A is a self-adjoint positive definite operator, and (). Then, the solution of difference problem (2.4) obeys the stability estimates (2.16), (2.17), and (2.18).
Proof By using the representation formula (2.19) for the solution of (2.8), formula (2.19), and the nonlocal boundary conditions
we obtain the system of equations
(2.24)
Solving system (2.24), we have
(2.25)
So, the difference problem (2.8) has a unique solution , which is defined by formulas (2.19), (2.25), and (2.26). By using formulas (2.19), (2.25), (2.26), and the method of the monograph [23], we can get the stability estimate (2.23) for the solution of difference problem (2.8). The proofs of estimates (2.17), (2.18) are based on (2.5) and (2.23). Applying formula (2.5) and estimates (2.23), (2.17), we get estimate (2.16). Theorem 2.2 is proved. □
Theorem 2.3 Assume that A is a self-adjoint positive definite operator, and . Then, the solutions of difference problems (2.3) and (2.4) obey the following almost coercive inequality:
(2.27)
Theorem 2.4 Assume that A is a self-adjoint positive definite operator, and (). Then, the solutions of difference problems (2.3) and (2.4) obey the following coercive inequality:
(2.28)
The proofs of Theorems 2.3 and 2.4 are based on formulas (2.5), (2.19), (2.21), (2.22), (2.25), (2.26), Lemmas 2.1 and 2.2.