Let us associate the nonlocal boundary-value problem (1) with the corresponding difference problem,
(2)
We will study the problem (2) under the following assumption:
(3)
It is well known [28] that for a self-adjoint positive definite operator A it follows that is self-adjoint positive definite and , which is defined on the whole space H is a bounded operator. Here, I is the unit operator. Furthermore, we have
(4)
In this paper, positive constants, which can differ in time (hence they are not a subject of precision considerations) will be indicated with M. On the other hand is used to focus on the fact that the constant depends only on .
Lemma 1
The operator
has an inverse
and the following estimate is satisfied:
(5)
where M does not depend on τ.
The proof of the estimate (5) is based on the estimate
(6)
Here
The estimate (6) follows from the spectral representation of A and the Cauchy inequality.
Theorem 2 For any , , the solution of the problem (2) exists and the following formula holds:
(7)
for ,
for .
Proof
(8)
has a solution and the following formula holds [29]:
(9)
Applying formula (9) and the nonlocal boundary condition
we obtain
Since the operator
has an inverse , it follows that
Theorem 2 is proved. □
Let be the linear space of the mesh functions with values in the Hilbert space H. We denote by and , , Banach spaces with the norms
The nonlocal boundary-value problem (2) is said to be stable in if we have the inequality
Theorem 3 The solutions of the difference scheme (2) under the assumption (3) satisfy the stability estimate
(10)
Proof By [29],
(11)
is proved for the solution of difference scheme (8). Then the proof of (10) is based on (11) and on the estimate
Using the formula (7) and the estimates (4), (5), we get
Theorem 3 is proved. □
Theorem 4 The solutions of the difference problem (2) in under the assumption (3) obey the almost coercive inequality
Proof By [29],
is proved for the solution of the boundary-value problem (8). Using the estimates (4), (5) and the formula (7), we obtain
(12)
for the solution of difference scheme (2). Applying formula (7) and , we get
where
(13)
(14)
To this end it suffices to show that
(15)
and
(16)
The estimate (15) follows from formula (13) and the estimates (4), (5). Using formula (14) and the estimates (4), (5), we obtain
From the last estimate and the estimate (15) follows the estimate (12). Theorem 4 is proved. □
Theorem 5 The difference problem (2) is well posed in the Hölder spaces under the assumption (3) and the following coercivity inequality holds:
(17)
Proof By [29],
(18)
is proved for the solution of difference scheme (8). Then the proof of (17) is based on (18) and on the estimate
Applying the triangle inequality, formula (7) and the estimate (15), we get
To this end it suffices to show that
(19)
Applying formula (14), we get
(20)
where
Second, let us estimate for any separately. We start with . Using estimates (4), (5) and the definition of the norm of the space , we get
From (3) it follows that
Now, let us estimate . Using estimates (4), (5) and the definition of the norm of the space , we obtain
The sum
is the lower Darboux integral sum for the integral
It follows that
By the lower Darboux integral sum for the integral it concludes that
For , applying (4), (5) and the definition of the norm of the space , we get
The sum
is the lower Darboux integral sum for the integral
Since
it follows that
By the lower Darboux integral sum for the integral it follows that
Combining and , we get
From (3) it follows that
Next, let us estimate . Using the estimates (4), (5), and the definition of the norm space , we obtain
The sum
is the lower Darboux integral sum for the integral
Since
it follows that
By the lower Darboux integral sum for the integral it follows that
Finally, let us estimate . Using the estimates (4), (5), and the definition of the norm space , we get
The sum
is the lower Darboux integral sum for the integral
Thus, we show that
By the lower Darboux integral sum for the integral it follows that
Combining and , we get
From (3) it follows that
Combining estimates for , we get the estimate (19). Theorem 5 is proved. □