The structure of fractional spaces generated by the two-dimensional difference operator on the half plane
We consider a difference operator approximation $A^x_h$ of the differential operator $A^xu(x) = a_{11}(x)u_{x_1 x_1}(x) - a_{22}(x)u_{x_2x_2} (x) + \sigma u(x),\; x = (x_1, x_2)$ defined in the region $R^{+} \times R$ with the boundary condition $u(0, x_2) = 0,\; x_2 \in R$. Here, the coefficients...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507431427112960 |
|---|---|
| author | Akturk, S. Ashyralyev, A. Актюрк, С. Аширалієв, A. |
| author_facet | Akturk, S. Ashyralyev, A. Актюрк, С. Аширалієв, A. |
| author_sort | Akturk, S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:21:04Z |
| description | We consider a difference operator approximation $A^x_h$ of the differential operator $A^xu(x) = a_{11}(x)u_{x_1 x_1}(x) - a_{22}(x)u_{x_2x_2} (x) + \sigma u(x),\; x = (x_1, x_2)$ defined in the region $R^{+} \times R$ with the boundary condition $u(0, x_2) = 0,\; x_2 \in R$. Here, the coefficients $a_{ii}(x), i = 1, 2$, are continuously differentiable, satisfy the uniform ellipticity condition
$a^2_{11}(x) + a^2_{22}(x) \geq \delta > 0$. We investigate the structure of the fractional spaces generated by the
analyzed difference operator. Theorems on well-posedness in a Holder space of difference elliptic problems are obtained
as applications. |
| first_indexed | 2026-03-24T02:09:12Z |
| format | Article |
| fulltext |
UDC 517.9
A. Ashyralyev (Near East Univ., Nicosia, Mersin, Turkey, Inst. Math. and Math. Modeling, Almaty, Kazakhstan),
S. Akturk (Yakuplu the Neighborhood Street Kubilay, Istanbul, Turkey)
THE STRUCTURE OF FRACTIONAL SPACES
GENERATED BY THE TWO-DIMENSIONAL DIFFERENCE OPERATOR
ON THE HALF PLANE*
СТРУКТУРА ДРОБОВИХ ПРОСТОРIВ, ПОРОДЖЕНИХ ДВОВИМIРНИМ
ДИФЕРЕНЦIАЛЬНИМ ОПЕРАТОРОМ НА ПIВПЛОЩИНI
We consider a difference operator approximation Ax
h of the differential operator Axu(x) = - a11(x)ux1x1(x) -
- a22(x)ux2x2(x) + \sigma u(x), x = (x1, x2) defined in the region \BbbR + \times \BbbR with the boundary condition u(0, x2) = 0,
x2 \in \BbbR . Here, the coefficients aii(x), i = 1, 2, are continuously differentiable, satisfy the uniform ellipticity condi-
tion a2
11(x) + a2
22(x) \geq \delta > 0, and \sigma > 0. We investigate the structure of the fractional spaces generated by the
analyzed difference operator. Theorems on well-posedness in a Hölder space of difference elliptic problems are obtained
as applications.
Розглянуто апроксимацiю рiзницевими операторами Ax
h диференцiального оператора Axu(x) = - a11(x)ux1x1(x) -
- a22(x)ux2x2(x) + \sigma u(x), x = (x1, x2), що визначений у областi \BbbR + \times \BbbR , з граничною умовою u(0, x2) = 0,
x2 \in \BbbR . У даному випадку коефiцiєнти aii(x), i = 1, 2, є неперервно диференцiйовними та задовольняють
рiвномiрну умову елiптичностi a2
11(x) + a2
22(x) \geq \delta > 0 i, крiм того, \sigma > 0. Теореми про коректнiсть рiзницевих
елiптичних задач у просторах Гьольдера одержанi як застосування.
1. Introduction. The importance of the positivity property of the differential operators in a Banach
space in the study of various properties for partial differential equations is well-known (see, for
example, [1 – 8] and the references therein). The positivity of wider class of differential and difference
operators in Banach spaces have been investigated by many scientists (see [9 – 14] and the references
therein).
The structure of fractional spaces generated by positive multidimensional differential and dif-
ference operators on space \BbbR n and one dimensional differential and difference operators in Banach
spaces has been well investigated (see [16 – 21] and the references therein). Note that the structure
of fractional spaces generated by positive multidimensional differential and difference operators with
local and nonlocal conditions on \Omega \subset \BbbR n in Banach spaces C (\Omega ) has not been well studied.
It is well-known that (see, for example, [16]) the operator A is said to be positive in E if its
spectrum \sigma (A) lies inside of the sector S of the angle \phi , 0 < 2\phi < 2\pi , symmetric with respect to
the real axis, and the estimate \bigm\| \bigm\| (A - \lambda ) - 1
\bigm\| \bigm\|
E\rightarrow E
\leq M(\phi )
1 + | \lambda |
holds on the edges S1(\phi ) =
\bigl\{
\rho ei\phi : 0 \leq \rho <\infty
\bigr\}
, S2(\phi ) =
\bigl\{
\rho e - i\phi : 0 \leq \rho <\infty
\bigr\}
of S , and outside
of the sector S. The infimum of all such angles \phi is called the spectral angle of the positive operator
A and is denoted by \phi (A,E).
Throughout the article, M indicates positive constants which may differ from time to time and
we are not interested to precise. If the constant depends only on \alpha , \beta , . . . , then we will write
M(\alpha , \beta , . . .).
* This paper was supported by the “RUDN University Program 5-100”.
c\bigcirc A. ASHYRALYEV, S. AKTURK, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8 1019
1020 A. ASHYRALYEV, S. AKTURK
With the help of the positive operator A we introduce the fractional space E\beta = E\beta (E,A),
0 < \beta < 1, consisting of all elements v \in E for which the norm
\| v\| E\beta
= \mathrm{s}\mathrm{u}\mathrm{p}
\lambda >0
\lambda \beta
\bigm\| \bigm\| A(\lambda +A) - 1v
\bigm\| \bigm\|
E
+ \| v\| E
is finite.
In the paper [21] the structure of fractional spaces generated by the two-dimensional differential
operator Ax defined by
Axu(x) = - a11(x)ux1x1(x) - a22(x)ux2x2(x) + \sigma u(x) (1)
over the region \BbbR 2
+ = \BbbR + \times \BbbR with the boundary condition u(0, x2) = 0, x2 \in \BbbR . Here, the
coefficients aii(t, x), i = 1, 2, are continuously differentiable and satisfy the uniform ellipticity
a211(x) + a222(x) \geq \delta > 0 (2)
and \sigma > 0.
In the present paper, we will study the structure of fractional spaces generated by the two-
dimensional difference operator Ax
h defined by formula
Ax
hu
h(x) = - b11(x)\Delta x1
1+\Delta
x1
1 - u
h(x)h - 2 - b22(x)\Delta
x2
1+\Delta
x2
1 - u
h(x)h - 2 + \sigma uh(x), x = (x1, x2),
(3)
approximates of the differential operator Ax defined by (1). Here
\Delta x1
1\pm u(x) = \pm
\bigl(
uh (x1 \pm h, x2) - uh(x1, x2)
\bigr)
,
x1, x1 + h \in \BbbR +
h , x2 \in \BbbR h, \Delta x2
1\pm u(x) = \pm
\bigl(
uh (x1, x2 \pm h) - uh(x1, x2)
\bigr)
,
x1 \in \BbbR +
h , x2, x2 + h \in \BbbR h, \BbbR h =
\bigl\{
(x2)k = kh, k = 0,\pm 1,\pm 2, . . .
\bigr\}
,
\BbbR +
h =
\bigl\{
(x1)m = mh, m = 0, 1, 2, . . .
\bigr\}
.
Assume that bii(x) satisfy the uniform ellipticity condition
b211(x) + b222(x) \geq \delta > 0.
Next, to formulate our result we need to introduce the Hölder space C2\alpha
h = C2\alpha
\bigl(
\BbbR 2
h,+
\bigr)
of all
bounded grid functions fh defined on \BbbR 2
h,+ = \BbbR +
h \times \BbbR h satisfying a Hölder condition with the
indicator \alpha \in (0, 1/2) with the norm\bigm\| \bigm\| fh\bigm\| \bigm\|
C2\alpha
h
=
\bigm\| \bigm\| fh\bigm\| \bigm\|
C
\bigl(
\BbbR 2
h,+
\bigr) + \mathrm{s}\mathrm{u}\mathrm{p}
(x1,x2),(x\prime
1,x
\prime
2)\in \BbbR 2
h,+
(x1,x2)\not =(x\prime
1,x
\prime
2)
\bigm| \bigm| fh(x1, x2) - fh(x\prime 1, x
\prime
2)
\bigm| \bigm| \biggl( \sqrt{}
| x1 - x\prime 1|
2 + | x2 - x\prime 2| 2
\biggr) 2\alpha .
Here Ch = C
\bigl(
\BbbR 2
h,+
\bigr)
denotes the Banach space of all bounded grid functions fh defined on \BbbR 2
h,+
with the norm \bigm\| \bigm\| fh\bigm\| \bigm\|
Ch
= \mathrm{s}\mathrm{u}\mathrm{p}
(x1,x2)\in \BbbR 2
h,+
\bigm| \bigm| fh(x1, x2)\bigm| \bigm| .
Our goal in this paper is to study the structure of the fractional spaces E\alpha
\bigl(
Ax
h, C
\bigl(
\BbbR 2
h,+
\bigr) \bigr)
. Namelly,
the following main theorem is proved.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
THE STRUCTURE OF FRACTIONAL SPACES GENERATED . . . 1021
Theorem 1. Suppose 0 < \alpha < 1/2. Then the norms of the spaces E\alpha
\bigl(
Ax
h, C
\bigl(
\BbbR 2
h,+
\bigr) \bigr)
and
C2\alpha
\bigl(
\BbbR 2
h,+
\bigr)
are equivalent uniformly in h, 0 < h \leq h0.
The paper is organized as follows. In Section 2, auxiliary results are given. In Section 3, the
proof of main Theorem 1 is presented. In Section 4, theorems on well-posedness in a Hölder space
of difference elliptic problems are established. Finally, Section 5 is conclusion.
2. Auxiliary results. In this section, we give some auxiliary lemmas which will be useful in the
sequel.
Lemma 1. For all n,m \in Z+ and h1 > 0, b > 0 the following estimate holds:
e - b(m+n)h1
\Bigl(
1 + \mathrm{l}\mathrm{n}
\Bigl\{
1 + ((m+ n)h1 + h1)
- 1
\Bigr\} \Bigr)
(m+ n)2\alpha h2\alpha 1 \leq M(\alpha ), n,m \in Z+. (4)
Lemma 2 [15]. Let p and q be mutually conjugate exponents, that is,
1
p
+
1
q
= 1, p > 1,
and let (am)m\in \BbbN and (bn)n\in \BbbN be any two sequences of nonnegative real numbers such that 0 <
<
\sum \infty
m=1
apm <\infty and 0 <
\sum \infty
n=1
bqn <\infty . Then, the following Hilbert’s inequality holds:
\infty \sum
m=1
\infty \sum
n=1
ambn
m+ n
< \pi \mathrm{c}\mathrm{s}\mathrm{c}
\biggl(
\pi
p
\biggr) \Biggl[ \infty \sum
m=1
apm
\Biggr] 1/p \Biggl[ \infty \sum
n=1
bqn
\Biggr] 1/q
.
S. I. Danelich in [12] considered the positivity of a difference analog Ax
h of the 2mth order
multidimensional elliptic operator Ax with dependent coefficients on semispaces \BbbR + \times \BbbR n - 1. Fol-
lowing the paper [12], in the special case m = 1 and n = 2, we consider the problem of finding the
resolvent equation of the operator - Ax
Ax
hu
h(x) + \lambda uh(x) = fh(x), x \in \BbbR 2
h,+. (5)
Let Gh(x1, x2, p, s, \lambda ) be a Green function of the resolvent equation (5) for the difference operator
Ax
h . There exists the inverse operator (Ax
h + \lambda ) - 1 for all \lambda \geq 0 and the formula
(Ax
h + \lambda ) - 1fh(x1, x2) =
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
Gh (x1, x2, p, s, \lambda ) f
h(p, s)h2 (6)
holds. Moreover, we have the following lemma.
Lemma 3 [12]. The estimates for 0 \leq \lambda \leq Lh - 2 (L > 0)\bigm| \bigm| Gh(x1, x2, p, s, \lambda )
\bigm| \bigm| \leq
\leq C \mathrm{e}\mathrm{x}\mathrm{p}
\Bigl\{
- a(\lambda + 1)1/2
\bigl(
| x1 - p| + | x2 - s|
\bigr) \Bigr\}
\times
\times
\biggl(
1 + \mathrm{l}\mathrm{n}
\biggl\{
1 +
\Bigl(
(\lambda + 1)1/2
\bigl(
| x1 - p| + | x2 - s|
\bigr) \Bigr) - 1
\biggr\} \biggr)
, (7)
\bigm| \bigm| Ghx1
(x1, x2, p, s, \lambda )
\bigm| \bigm| , \bigm| \bigm| Ghx2
(x1, x2, p, s, \lambda )
\bigm| \bigm| \leq
\leq C \mathrm{e}\mathrm{x}\mathrm{p}
\Bigl\{
- a(\lambda + 1)1/2
\bigl(
| x1 - p| + | x2 - s|
\bigr) \Bigr\} \bigl(
| x1 - p| + | x2 - s| + h
\bigr) - 1
, (8)
and for \lambda \geq Lh - 2 (L > 0)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1022 A. ASHYRALYEV, S. AKTURK
| Gh(x1, x2, p, s, \lambda )| \leq C \mathrm{e}\mathrm{x}\mathrm{p}
\bigl\{
- ah - 1 (| x1 - p| + | x2 - s| )
\bigr\} \bigm| \bigm| \bigm| \bigl( (\lambda + 1)h2 + 1
\bigr) - 1
\bigm| \bigm| \bigm| , (9)\bigm| \bigm| Ghx1
(x1, x2, p, s, \lambda )
\bigm| \bigm| , \bigm| \bigm| Ghx2
(x1, x2, p, s, \lambda )
\bigm| \bigm| \leq
\leq C \mathrm{e}\mathrm{x}\mathrm{p}
\bigl\{
- ah - 1
\bigl(
| x1 - p| + | x2 - s|
\bigr) \bigr\} \bigl(
(\lambda + 1)h2 + 1
\bigr) - 1
h - 1 (10)
hold. Here a = a (\sigma ) .
Lemma 4 [12]. Let \lambda \geq Lh - 2, where L > 0 is large enough. Then
| Gh(x1, x2, p, s, \lambda )| \leq
\leq Ch \mathrm{e}\mathrm{x}\mathrm{p}
\bigl\{
- ah - 1 (| x1 - p| + | x2 - s| )
\bigr\}
((\lambda + 1)h2) - 2 (| x1 - p| + | x2 - s| ) - 1 . (11)
From (9) and (11) we obtain the estimate (0 < \alpha < 1/2)
| Gh(x1, x2, p, s, \lambda )| \leq
\leq Ch2\alpha \mathrm{e}\mathrm{x}\mathrm{p}
\bigl\{
- ah - 1
\bigl(
| x1 - p| + | x2 - s|
\bigr) \bigr\} \bigl(
(\lambda + 1)h2
\bigr) - 2
(| x1 - p| + | x2 - s| ) - 2\alpha . (12)
Clearly, from estimates (7) – (10) it follows that Ax
h is a positive operator in C
\bigl(
\BbbR 2
h,+
\bigr)
. Moreover,
from that and the commutativity of Ax
h and its resolvent
\bigl(
Ax
h + \lambda
\bigr) - 1
it follows that Ax
h is a positive
operator in E\alpha
\bigl(
A,C
\bigl(
R2
h,+
\bigr) \bigr)
.
3. Proof of Theorem 1. In this section, we will prove the main Theorem 1 on structure of
fractional spaces E\alpha
\bigl(
Ax
h, C
\bigl(
\BbbR 2
h,+
\bigr) \bigr)
. Namely, let (x1, x2) \in \BbbR 2
h,+ and \lambda > 0 be fixed. From
formula (6) it follows that
Ax
h
\bigl(
\lambda +Ax
h
\bigr) - 1
fh(x1, x2) =
=
\lambda \alpha
1 + \lambda
fh(x1, x2) + \lambda \alpha +1
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
Gh(x1, x2, p, s, \lambda )
\bigl(
fh(x1, x2) - fh(p, s)
\bigr)
h2. (13)
By using equation (13), the triangle inequality, and the definition of C2\alpha
h -norm, we obtain\bigm| \bigm| \lambda \alpha Ax
h
\bigl(
\lambda +Ax
h
\bigr) - 1
fh(x1, x2)
\bigm| \bigm| \leq
\leq \lambda \alpha
1 + \lambda
\bigm| \bigm| \bigm| fh(x1, x2)\bigm| \bigm| \bigm| + \lambda \alpha +1
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| \bigm| Gh (x1, x2, p, s, \lambda )
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| fh(x1, x2) - fh(p, s)
\bigm| \bigm| \bigm| h2 \leq
\leq
\bigm\| \bigm\| fh\bigm\| \bigm\|
C2\alpha
h
\left[ \lambda \alpha
1 + \lambda
+ \lambda \alpha +1
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| \bigm| Gh
\bigl(
x1, x2, p, s, \lambda
\bigr) \bigm| \bigm| \bigm| \bigl( | x1 - p| 2 + | x2 - s| 2
\bigr) \alpha
h2
\right] \leq
\leq
\bigm\| \bigm\| fh\bigm\| \bigm\|
C2\alpha
h
\bigl[
I1 + I2
\bigr]
.
Here
I1 =
\lambda \alpha
1 + \lambda
, I2 = \lambda \alpha +1
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| Gh(x1, x2, p, s, \lambda )
\bigm| \bigm| \bigl( | x1 - p| 2 + | x2 - s| 2
\bigr) \alpha
h2.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
THE STRUCTURE OF FRACTIONAL SPACES GENERATED . . . 1023
It is clear that for any \lambda > 0
I1 \leq 1. (14)
Therefore, we will estimate I2. Consider two cases: | \lambda | \leq Lh - 2 and | \lambda | \geq Lh - 2, respectively.
First, assume that | \lambda | \leq Lh - 2. Applying estimate (7), we have
I2 \leq \lambda \alpha +1M
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
e - a(1+\lambda )1/2(| x1 - p| +| x2 - s| )\times
\times
\Bigl(
1 + \mathrm{l}\mathrm{n}
\Bigl\{
1 +
\bigl(
(1 + \lambda )1/2
\bigl(
| x1 - p| + | x2 - s| + h
\bigr) \bigr) - 1
\Bigr\} \Bigr) \bigl(
| x1 - p| 2 + | x2 - s| 2
\bigr) \alpha
h2.
The change of variables x1 - p = mh and x2 - s = nh yields
I2 \leq \lambda \alpha +1M
\sum
m\in Z+
\sum
n\in Z+
e - a(1+\lambda )1/2(m+n)h\times
\times
\Bigl(
1 + \mathrm{l}\mathrm{n}
\Bigl\{
1 +
\bigl(
(1 + \lambda )1/2((m+ n)h+ h)
\bigr) - 1
\Bigr\} \Bigr)
(m+ n)2\alpha h2\alpha h2.
Letting (1 + \lambda )1/2h = h1, we get
I2 \leq
\lambda \alpha +1
(1 + \lambda )\alpha +1
M
\sum
m\in Z+
\sum
n\in Z+
e - a(m+n)h1\times
\times
\Bigl(
1 + \mathrm{l}\mathrm{n}
\Bigl\{
1 + ((m+ n)h1 + h1)
- 1
\Bigr\} \Bigr)
(m+ n)2\alpha h2\alpha 1 h21.
Applying estimate (4) for b =
a
2
, we obtain
I2 \leq M1
\sum
m\in Z+
\sum
n\in Z+
e -
a
2
(m+n)h1h21 \leq M2
\infty \int
0
\infty \int
0
e -
a
2
(q+y) dq dy =M3. (15)
Then from estimates (14), (15) it follows
\mathrm{s}\mathrm{u}\mathrm{p}
(x1,x2)\in \BbbR 2
h,+
\bigm| \bigm| \lambda \alpha Ax
h (A
x
h + \lambda ) - 1 fh(x1, x2)
\bigm| \bigm| \leq M
\bigm\| \bigm\| fh\bigm\| \bigm\|
C2\alpha
h
. (16)
Second, assume that | \lambda | \geq Lh - 2. From estimates (9) and (12) it follows
I2 \leq M\lambda \alpha +1
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
e - ah - 1(| x1 - p| +| x2 - s)
((1 + \lambda ))h2)1+2\alpha \times
\times
\bigl(
| x1 - p| + | x2 - s|
\bigr) - 2\alpha \bigl( | x1 - p| 2 + | x2 - s| 2
\bigr) \alpha
h2\alpha h2.
The change of variables x1 - p = mh and x2 - s = nh m, n = 0, 1, . . . yields
I2 \leq M1\lambda
\alpha +1
\sum
m\in Z+
\sum
n\in Z+
e - a(m+n) h2\alpha +2
((1 + \lambda )h2)1+2\alpha \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1024 A. ASHYRALYEV, S. AKTURK
\leq M2
\bigl(
\lambda h2
\bigr) 1+\alpha
((1 + \lambda ))h2)1+2\alpha \leq M2
1
(\lambda h2)\alpha
\leq M2
1
L\alpha
\leq M3. (17)
Thus, from estimates (14), (17) it follows that
\mathrm{s}\mathrm{u}\mathrm{p}
(x1,x2)\in \BbbR 2
h,+
\bigm| \bigm| \lambda \alpha Ax
h (A
x
h + \lambda ) - 1 fh(x1, x2)
\bigm| \bigm| \leq M
\bigm\| \bigm\| fh\bigm\| \bigm\|
C2\alpha
h
. (18)
Combining estimates (16) and (18), we obtain estimate\bigm\| \bigm\| fh\bigm\| \bigm\|
E\alpha (Ax
h,C(\BbbR 2
h,+)) \leq M
\bigm\| \bigm\| fh\bigm\| \bigm\|
C2\alpha
h
.
Now, we will prove opposite estimate. Applying the triangle inequality and the definition of E\alpha -
norm, we get \bigm| \bigm| \bigm| \bigm| fh(x1 + \tau , x2 + k) - fh(x1, x2)
(\tau 2 + k2)\alpha
\bigm| \bigm| \bigm| \bigm| \leq 2
\bigm\| \bigm\| fh\bigm\| \bigm\|
E\alpha (Ax
h,C(\BbbR 2
h,+)) (19)
for any (x1, x2) \in \BbbR 2
h,+ and \tau 2 + k2 \geq 1. Now, we will prove the estimate\bigm| \bigm| \bigm| \bigm| fh(x1 + \tau , x2 + k) - fh(x1, x2)
(\tau 2 + k2)\alpha
\bigm| \bigm| \bigm| \bigm| \leq M
\alpha (1 - \alpha )
\bigm\| \bigm\| fh\bigm\| \bigm\|
E\alpha (Ax
h,C(\BbbR 2
h,+)) (20)
for any (x1, x2) \in \BbbR 2
h,+ and \tau 2 + k2 < 1. For any positive operator A we can write
v =
\infty \int
0
A(\lambda +A) - 2v d\lambda .
Noted that Ax
h is a positive operator in the Banach space E\alpha
\bigl(
Ax
h, C
\bigl(
\BbbR 2
h,+
\bigr) \bigr)
. Hence, from this
identity and formula (6) it follows that
fh(x1, x2) =
\infty \int
0
Ax
h
\bigl(
\lambda +Ax
h
\bigr) - 2
fh(x1, x2) d\lambda =
=
\infty \int
0
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
Gh(x1, x2, p, s, \lambda )A
x
h
\bigl(
\lambda +Ax
h
\bigr) - 1
fh(p, s)h2 d\lambda .
Consequently,
fh(x1 + \tau , x2 + k) - fh(x1, x2) =
=
\infty \int
0
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigl[
Gh (x1 + \tau , x2 + k, p, s, \lambda ) - Gh(x1, x2, p, s, \lambda )
\bigr]
\times
\times \lambda \alpha Ah (Ah + \lambda ) - 1 fh(p, s)h2 d\lambda ,
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THE STRUCTURE OF FRACTIONAL SPACES GENERATED . . . 1025
whence \bigm| \bigm| \bigm| \bigm| fh(x1 + \tau , x2 + k) - fh(x1, x2)
(\tau 2 + k2)\alpha
\bigm| \bigm| \bigm| \bigm| \leq
\leq
\bigm\| \bigm\| fh\bigm\| \bigm\|
E\alpha (Ax
h,C(\BbbR 2
h,+))
\infty \int
0
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| Gh (x1 + \tau , x2 + k, p, s, \lambda ) - Gh (x1, x2, p, s, \lambda )
\bigm| \bigm|
(\tau 2 + k2)\alpha
h2d\lambda .
Let
P =
\infty \int
0
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| Gh (x1 + \tau , x2 + k, p, s, \lambda ) - Gh (x1, x2, p, s, \lambda )
\bigm| \bigm|
(\tau 2 + k2)\alpha
h2 d\lambda .
Then \bigm| \bigm| \bigm| \bigm| fh(x1 + \tau , x2 + k) - fh(x1, x2)
(\tau 2 + k2)\alpha
\bigm| \bigm| \bigm| \bigm| \leq P
\bigm\| \bigm\| fh\bigm\| \bigm\|
E\alpha (Ax
h,C(\BbbR 2
h,+)). (21)
To estimate P, we will consider two cases: | \lambda | \leq Lh - 2 and | \lambda | \geq Lh - 2, respectively. We denote
that P = L1 + L2, where
L1 =
1/(\tau 2+k2)\int
0
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| Gh (x1 + \tau , x2 + k, p, s, \lambda ) - Gh (x1, x2, p, s, \lambda )
\bigm| \bigm|
(\tau 2 + k2)\alpha
h2 d\lambda ,
L2 =
\infty \int
1/(\tau 2+k2)
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| Gh (x1 + \tau , x2 + k, p, s, \lambda ) - Gh(x1, x2, p, s, \lambda )
\bigm| \bigm|
(\tau 2 + k2)\alpha
h2 d\lambda .
We will estimate L1 and L2. Let us first assume that | \lambda | \leq Lh - 2. By Lemma 2 for p = q = 2 and
using the triangle inequality, estimates (7), (8), we have, for some x\ast 1 between x1, x1 + \tau , and x\ast 2
between x2, x2 + k,
L1 \leq M
1
(\tau 2 + k2)\alpha
\times
\times
1/(\tau 2+k2)\int
0
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| \bigm| \tau Gh
x1
(x\ast 1, x
\ast
2, p, s, \lambda ) + kGh
x2
(x\ast 1, x
\ast
2, p, s, \lambda )
\bigm| \bigm| \bigm| h2 d\lambda \leq
\leq M
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| \bigm| Gh
x1
(x\ast 1, x
\ast
2, p, s, \lambda )
\bigm| \bigm| \bigm| h2 d\lambda \leq
\leq M
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
e - a(1+\lambda )1/2(| x\ast
1 - p| +| x\ast - s| )
h+ | x\ast 1 - p| + | x\ast 2 - s|
h2 d\lambda \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1026 A. ASHYRALYEV, S. AKTURK
\leq M
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\sum
m\in \BbbR +
h
\sum
n\in \BbbR +
h
\lambda - \alpha e
- a(1+\lambda )1/2(mh+nh)
mh+ nh
h2 d\lambda \leq
\leq M
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha
\left( \sum
m\in Z+
e - 2a(1+\lambda )1/2mhh
\right) 1/2\left( \sum
n\in Z+
e - 2a(1+\lambda )
1/2
nhh
\right) 1/2
d\lambda \leq
\leq M1
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha
\sum
m\in Z+
e - 2a(1+\lambda )1/2mhh d\lambda \leq
\leq M2
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha
\infty \int
0
e - 2a(1+\lambda )1/2xdx d\lambda \leq
\leq M3
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
1
\lambda \alpha
1
(1 + \lambda )1/2
d\lambda \leq M4
1 - 2\alpha
,
(22)
L2 \leq
M
(\tau 2 + k2)\alpha
\infty \int
1/(\tau 2+k2)
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\lambda - \alpha e - a(1+\lambda )1/2(| x1 - p| +| x2 - s| )\times
\times
\biggl(
1 + \mathrm{l}\mathrm{n}
\biggl\{
1 +
\Bigl(
(1 + \lambda )1/2
\bigl(
| x1 - p| + | x2 - s| + h
\bigr) \Bigr) - 1
\biggr\} \biggr)
h2d\lambda +
+
M
(\tau 2 + k2)\alpha
\infty \int
1/(\tau 2+k2)
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\lambda - \alpha e - a(1+\lambda ))1/2(| x1+\tau - p| +| x2+k - s| )\times
\times
\biggl(
1 + \mathrm{l}\mathrm{n}
\biggl\{
1 +
\Bigl(
(1 + \lambda )1/2
\bigl(
| x1 + \tau - p| + | x2 + k - s| + h
\bigr) \Bigr) - 1
\biggr\} \biggr)
h2d\lambda .
The change of variables x1 - p = mh and x2 - s = nh, m, n = 0, 1, . . . , yields
L2 \leq
M
(\tau 2 + k2)\alpha
\infty \int
1/(\tau 2+k2)
\sum
m\in Z+
\sum
n\in Z+
\lambda - \alpha e - a(1+\lambda )1/2(m+n)h\times
\times
\biggl(
1 +
\Bigl(
(1 + \lambda )1/2 (mh+ nh+ h)
\Bigr) - 1
\biggr)
h2d\lambda \leq
\leq M1
\infty \int
1/(\tau 2+k2)
\infty \int
0
\infty \int
0
\lambda - \alpha e - a(1+\lambda )1/2(q+y)
\biggl(
1 +
\Bigl(
(1 + \lambda )1/2 (q + y)
\Bigr) - 1
\biggr)
dq dy d\lambda \leq
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THE STRUCTURE OF FRACTIONAL SPACES GENERATED . . . 1027
\leq M2
(\tau 2 + k2)\alpha
\left[ \infty \int
1/(\tau 2+k2)
1
\lambda \alpha
1
1 + \lambda
d\lambda +
\infty \int
1/(\tau 2+k2)
1
\lambda \alpha
1
(1 + \lambda ))1/2
\left( \infty \int
0
e - 2a(1+\lambda ))1/2qdq
\right) d\lambda
\right] \leq
\leq M3
\alpha (1 - \alpha )
. (23)
From estimates (22) and (23) it follows that
P \leq M5
\alpha (1 - \alpha )
. (24)
Second, let us | \lambda | \geq Lh - 2. By using the triangle inequality, estimates (9), (10), we have, for some
x\ast 1 between x1, x1 + \tau , and x\ast 2 between x2, x2 + k,
L1 \leq M
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| \bigm| Gh
x1
(x\ast 1, x
\ast
2, p, s, \lambda )
\bigm| \bigm| \bigm| h2d\lambda \leq
\leq M
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
e - ah - 1(| x\ast
1 - p| +| x\ast
2 - s| )h - 1
\bigl(
1 + (1 + \lambda )h2
\bigr) - 1
h2 d\lambda \leq
\leq M
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha
\sum
m\in Z+
\sum
n\in Z+
e - a(m+n)
(1 + \lambda )h2
\lambda - 1/2
\lambda - 1/2
h - 1h2 d\lambda \leq
\leq M
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha +1/2
1 + \lambda
\sum
m\in Z+
\sum
n\in Z+
e - a(m+n) d\lambda \leq
\leq M
(\tau + k)
(\tau 2 + k2)\alpha
1/(\tau 2+k2)\int
0
\lambda - \alpha - 1/2d\lambda \leq M1
1 - 2\alpha
, (25)
L2 \leq
1
(\tau 2 + k2)\alpha
\infty \int
1/(\tau 2+k2)
\lambda - \alpha
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
\bigm| \bigm| Gh (x1 + \tau , x2 + k, p, s, \lambda ) - Gh (x1, x2, p, s, \lambda )
\bigm| \bigm|
(\tau 2 + k2)\alpha
h2 d\lambda \leq
\leq M
(\tau 2 + k2)\alpha
\infty \int
1/(\tau 2+k2)
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
e - ah - 1
\bigl(
| x1 - p| +| x2 - s|
\bigr)
h2(1 + \lambda )
h2 d\lambda +
+
M
(\tau 2 + k2)\alpha
\infty \int
1/(\tau 2+k2)
\sum
p\in \BbbR +
h
\sum
s\in \BbbR h
e - ah - 1(| x1+\tau - p| +| x2+k - s| )
h2(1 + \lambda )
h2 d\lambda \leq
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1028 A. ASHYRALYEV, S. AKTURK
\leq M1
(\tau 2 + k2)\alpha
\infty \int
1/(\tau 2+k2)
\sum
m\in Z+
\sum
n\in Z+
\lambda - \alpha
(1 + \lambda )
e - a(m+n) d\lambda \leq
\leq M2
(\tau 2 + k2)\alpha
\infty \int
1/(\tau 2+k2)
\lambda - \alpha
1 + \lambda
d\lambda \leq M1
\alpha (1 - \alpha )
. (26)
By using estimates (25) and (26), we have
P \leq M
\alpha (1 - 2\alpha )
. (27)
From estimates (21), (24) and (27) it follows estimate (20). Combining estimates (19) and (20), we
get
\mathrm{s}\mathrm{u}\mathrm{p}
(x1+\tau ,x2+k),(x1,x2)\in \BbbR 2
+
(\tau ,h)\not =(0,0)
\bigm| \bigm| \bigm| \bigm| fh(x1 + \tau , x2 + k) - fh(x1, x2)
(\tau 2 + k2)\alpha
\bigm| \bigm| \bigm| \bigm| \leq
\leq M
\alpha (1 - 2\alpha )
\bigm\| \bigm\| fh\bigm\| \bigm\|
E\alpha (Ax
h,C(\BbbR 2
h,+)).
This means that the following estimate holds:\bigm\| \bigm\| fh\bigm\| \bigm\|
C2\alpha (\BbbR 2
h,+)
\leq M
\alpha (1 - 2\alpha )
\bigm\| \bigm\| fh\bigm\| \bigm\|
E\alpha (Ax
h,C(\BbbR 2
h,+)).
Theorem 1 is proved.
4. Applications of Theorem 1. First, the boundary-value problem
- \partial 2u(y, x)
\partial y2
- a11(x)
\partial 2u(y, x)
\partial x21
- a22(x)
\partial 2u(y, x)
\partial x22
+ \sigma u(y, x) = f(y, x),
0 < y < T, x \in \BbbR 2
+,
u(0, x) = \varphi (x), u(T, x) = \psi (x), x \in \BbbR 2
+; u(y, 0, x2) = 0, 0 \leq y \leq T, x2 \in \BbbR ,
(28)
for the elliptic equation is considered. Here a11(x), a22(x), \varphi (x), \psi (x), and f(y, x) are sufficiently
smooth functions, and satisfy the uniform ellipticity (2) and \sigma > 0.
The discretization of problem (28) is carried out in two steps. In the first step, let us use the
discretization in x. To the differential operator Ax
h generated by the problem (28), we assign the
difference operator Ax
h defined by formula (3).
Theorem 2. For the solution of elliptic problem
Ax
hu
h(x) = \Psi h(x), x \in \BbbR 2
h,+,
uh(0, x2) = 0, x2 \in \BbbR h,
(29)
the following coercive inequality holds:\bigm\| \bigm\| \bigm\| \Delta x1
1+\Delta
x1
1 - u
hh - 2
\bigm\| \bigm\| \bigm\|
C2\alpha (\BbbR 2
h,+)
+
\bigm\| \bigm\| \bigm\| \Delta x
1+\Delta
x
1 - u
hh - 2
\bigm\| \bigm\| \bigm\|
C2\alpha (\BbbR 2
h,+)
\leq M(\alpha )\| \Psi h\| C2\alpha (\BbbR 2
h,+),
where M(\alpha ), 0 < \alpha < 1/2, does not depend on \Psi h.
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THE STRUCTURE OF FRACTIONAL SPACES GENERATED . . . 1029
The proof of Theorem 2 uses the techniques introduced in [22] (Chapter 5) and it is based on
estimates (7) – (10).
With the help of Ax
h, we arrive at the boundary-value problem
- d2uh(y, x)
dy2
+Ax
hu
h(y, x) = fh(y, x), 0 < y < T, x \in \BbbR 2
h,+,
uh(0, x) = \varphi (x), uh(T, x) = \psi (x), x \in \BbbR 2
h,+,
(30)
for the system of ordinary differential equations. In the second step, problem (30) is replaced by the
second order of accuracy difference scheme in y, we get the following difference problem:
- 1
\tau 2
\Bigl(
uhk+1(x) - 2uhk(x) + uhk - 1(x)
\Bigr)
+Ax
hu
h
k(x) = \varphi h
k(x),
\varphi h
k(x) = fh(yk, x), yk = k\tau , 1 \leq k \leq N - 1, N\tau = T, x \in \BbbR 2
h,+,
uh0(x) = \varphi h(x), uhN (x) = \psi h(x), x \in \BbbR 2
h,+,
(31)
for the approximate solution of problem (30).
Theorem 3. For the solution of the difference problem (31) the following inequalities are valid:
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq N - 1
\bigm\| \bigm\| \bigm\| \tau - 2
\bigl(
uhk+1 - 2uhk + uhk - 1
\bigr) \bigm\| \bigm\| \bigm\|
C2\alpha (\BbbR 2
h,+)
+ \mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq N - 1
\bigm\| \bigm\| uhk\bigm\| \bigm\| C2\alpha +2(\BbbR 2
h,+)
\leq
\leq M(\alpha )
\biggl[
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq N - 1
\bigm\| \bigm\| \varphi h
k
\bigm\| \bigm\|
C2\alpha (\BbbR 2
h,+)
+
\bigm\| \bigm\| \varphi h
\bigm\| \bigm\|
C2\alpha +2(\BbbR 2
h,+)
+
\bigm\| \bigm\| \psi h
\bigm\| \bigm\|
C2\alpha +2(\BbbR 2
h,+)
\biggr]
,
where M(\alpha ) does not depend on
\bigl\{
\varphi h
k(x)
\bigr\} N - 1
1
, \varphi h(x), \psi h(x), h and \tau .
The proof of Theorem 3 is based on the Theorem 1 on the structure of the fractional spaces
E\alpha (A
x
h, Ch) and on the positivity of the operator Ax
h and on Theorem 2 coercive inequality on for
the difference elliptic problem (29) and on the theorems on the structure of the fractional spaces
E\prime
\alpha = E\alpha
\bigl(
(Ax
h)
1/2, Ch
\bigr)
of paper [22] and on the coercive inequalities in C\tau (E\alpha ) for the solution of
the second order of accuracy difference scheme
- 1
\tau 2
(uk+1 - 2uk + uk - 1) +Auk = fk, fk = f(tk), tk = k\tau ,
1 \leq k \leq N - 1, N\tau = T, u0 = \varphi , uN = \psi ,
(32)
for the approximate solution of the boundary-value problem
- u\prime \prime (t) +Au(t) = f(t), 0 < t < T,
u(0) = \varphi , u(T ) = \psi ,
(33)
in a Banach space E with positive operator A.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1030 A. ASHYRALYEV, S. AKTURK
Second, the nonlocal boundary-value problem
\partial 2u(y, x)
\partial y2
- a11(x)
\partial 2u(y, x)
\partial x21
- a22(x)
\partial 2u(y, x)
\partial x22
+ \sigma u(y, x) = f(y, x),
0 < y < T, x \in \BbbR 2
+,
u(0, x) = u(T, x), uy(0, x) = uy(T, x), x \in \BbbR 2
+;
u(y, 0, x2) = 0, 0 \leq y \leq T, x2 \in \BbbR ,
(34)
for the elliptic equation is considered.
The discretization of problem (34) is carried out in two steps. In the first step, let us use the
discretization in (x1, x2). To the differential operator Ax
h generated by the problem (34), we assign
the difference operator Ax
h defined by formula (3), too. With the help of Ax
h, we arrive at the nonlocal
boundary-value problem
- d2uh(y, x)
dy2
+Ax
hu
h(y, x) = fh(y, x), 0 < y < T, x \in \BbbR 2
+,
uh(0, x) = uh(T, x), uhy(0, x) = uhy(T, x), x \in \BbbR 2
+,
(35)
for the system of ordinary differential equations. In the second step, problem (35) is replaced by the
second order of accuracy difference scheme in y , we get the following difference problem:
- 1
\tau 2
\bigl(
uhk+1(x) - 2uhk(x) + uhk - 1(x)
\bigr)
+Ax
hu
h
k(x) = \varphi h
k(x), x \in \BbbR 2
h,+,
\varphi h
k(x) = fh(yk, x), yk = k\tau , 1 \leq k \leq N - 1, N\tau = T, x \in \BbbR 2
h,+, (36)
uh0(x) = uhN (x), - uh2(x) + 4uh1(x) - 3uh0(x) = uhN - 2(x) - 4uhN - 1(x) + 3uhN (x), x \in \BbbR 2
h,+,
for the approximate solution of problem (35).
Theorem 4. For the solution of the difference problem (36) the following inequalities are valid:
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq N - 1
\bigm\| \bigm\| \tau - 2
\bigl(
uhk+1 - 2uhk + uhk - 1
\bigr) \bigm\| \bigm\|
C2\alpha (\BbbR 2
h,+)
+ \mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq N - 1
\bigm\| \bigm\| uhk\bigm\| \bigm\| C2\alpha +2(\BbbR 2
h,+)
\leq
\leq M(\alpha ) \mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq N - 1
\bigm\| \bigm\| \varphi h
k
\bigm\| \bigm\|
C2\alpha (\BbbR 2
h,+)
,
where M(\alpha ) does not depend on
\bigl\{
\varphi h
k(x)
\bigr\} N - 1
1
, h and \tau .
The proof of Theorem 4 is based on the Theorem 1 on the structure of the fractional spaces
E\alpha (A
x
h, Ch) and on the positivity of the operator Ax
h and on the theorem on the structure of the
fractional spaces E\prime
\alpha = E\alpha
\bigl(
(Ax
h)
1/2, Ch
\bigr)
of paper [22] and on Theorem 2 coercive inequality on for
the difference elliptic problem (29) and on the theorem on coercive inequalities in C\tau (E\alpha ) (see [22])
for the solution of the second order of accuracy difference scheme
- 1
\tau 2
(uk+1 - 2uk + uk - 1) +Auk = fk, fk = f(tk), tk = k\tau ,
1 \leq k \leq N - 1, N\tau = T, u0 = uN , - u2+4u1 - 3u0 = uN - 2 - 4uN - 1 + 3uN ,
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THE STRUCTURE OF FRACTIONAL SPACES GENERATED . . . 1031
for the approximate solution of the nonlocal boundary-value problem
- u\prime \prime (t) +Au(t) = f(t), 0 < t < T,
u(0) = u(T ), u\prime (0) = u\prime (T ),
in a Banach space E with positive operator A.
5. Conclusion. Banach fixed-point theorem and methods of the present paper and [24] enable
us to establish the existence and uniqueness results which hold under the some sufficient conditions
on nonlinear term for the solution of second order of approximation difference schemes for the
approximate solution of the following mixed problem:
- \partial 2u(y, x)
\partial y2
- a11(y, x)
\partial 2u(y, x)
\partial x21
- a22(y, x)
\partial 2u(y, x)
\partial x22
+ \sigma u(y, x) = f(y, x, u, ux1 , ux2 , uy),
0 < y < T, x \in \BbbR 2
+,
u(0, x) = \varphi (x), u(T, x) = \psi (x), x \in \BbbR 2
+; u(y, 0, x2) = 0, 0 < y < T, x2 \in \BbbR .
6. Acknowledgements. Some of the results of the present article were announced in the confer-
ence proceeding [25] as extended abstract without proofs. The authors would like to thank Prof. Pavel
Sobolevskii (Universidade Federal do Ceará, Brasil) for helpful suggestions to the improvement of
our paper. The second author would also like to thank Scientific and Technological Research Council
of Turkey (TUBİTAK) for the financial support. The paper has been published under target program
BR05236656 of the Science Committee of the Ministry of Education and Science of the Republic of
Kazakhstan.
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Received 18.05.15,
after revision — 02.02.18
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
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| id | umjimathkievua-article-1614 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:12Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/8a/4407088da1844ee25b51e98ca33d5c8a.pdf |
| spelling | umjimathkievua-article-16142019-12-05T09:21:04Z The structure of fractional spaces generated by the two-dimensional difference operator on the half plane Структура дробових просторiв, породжених двовимiрним диференцiальним оператором на пiвплощинi Akturk, S. Ashyralyev, A. Актюрк, С. Аширалієв, A. We consider a difference operator approximation $A^x_h$ of the differential operator $A^xu(x) = a_{11}(x)u_{x_1 x_1}(x) - a_{22}(x)u_{x_2x_2} (x) + \sigma u(x),\; x = (x_1, x_2)$ defined in the region $R^{+} \times R$ with the boundary condition $u(0, x_2) = 0,\; x_2 \in R$. Here, the coefficients $a_{ii}(x), i = 1, 2$, are continuously differentiable, satisfy the uniform ellipticity condition $a^2_{11}(x) + a^2_{22}(x) \geq \delta > 0$. We investigate the structure of the fractional spaces generated by the analyzed difference operator. Theorems on well-posedness in a Holder space of difference elliptic problems are obtained as applications. Розглянуто апроксимацiю рiзницевими операторами $A^x_h$ диференцiального оператора $A^xu(x) = a_{11}(x)u_{x_1 x_1}(x) - a_{22}(x)u_{x_2x_2} (x) + \sigma u(x),\; x = (x_1, x_2)$, що визначений у областi $R^{+} \times R$, з граничною умовою $u(0, x_2) = 0,\; x_2 \in R$. У даному випадку коефiцiєнти $a_{ii}(x), i = 1, 2$, є неперервно диференцiйовними та задовольняють рiвномiрну умову елiптичностi $a^2_{11}(x) + a^2_{22}(x) \geq \delta > 0$ i, крiм того, $\sigma > 0$. Теореми про коректнiсть рiзницевих елiптичних задач у просторах Гьольдера одержанi як застосування. Institute of Mathematics, NAS of Ukraine 2018-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1614 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 8 (2018); 1019-1032 Український математичний журнал; Том 70 № 8 (2018); 1019-1032 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1614/596 Copyright (c) 2018 Akturk S.; Ashyralyev A. |
| spellingShingle | Akturk, S. Ashyralyev, A. Актюрк, С. Аширалієв, A. The structure of fractional spaces generated by the two-dimensional difference operator on the half plane |
| title | The structure of fractional spaces generated by the two-dimensional
difference operator on the half plane |
| title_alt | Структура дробових просторiв, породжених двовимiрним
диференцiальним оператором на пiвплощинi |
| title_full | The structure of fractional spaces generated by the two-dimensional
difference operator on the half plane |
| title_fullStr | The structure of fractional spaces generated by the two-dimensional
difference operator on the half plane |
| title_full_unstemmed | The structure of fractional spaces generated by the two-dimensional
difference operator on the half plane |
| title_short | The structure of fractional spaces generated by the two-dimensional
difference operator on the half plane |
| title_sort | structure of fractional spaces generated by the two-dimensional
difference operator on the half plane |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1614 |
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