The structure of fractional spaces generated by a two-dimensional elliptic differential operator and its applications

Abstract

We consider the two-dimensional differential operator [InlineEquation not available: see fulltext.] defined on functions on the half-plane [InlineEquation not available: see fulltext.] with the boundary conditions [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], where [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.], are continuously differentiable and satisfy the uniform ellipticity condition [InlineEquation not available: see fulltext.], [InlineEquation not available: see fulltext.]. The structure of the fractional spaces [InlineEquation not available: see fulltext.] generated by the operator A is investigated. The positivity of A in Hölder spaces is established. In applications, theorems on well-posedness in a Hölder space of elliptic problems are obtained.

MSC: 35J25, 47E05, 34B27.

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R E S E A R C H Open Access

The structure of fractional spaces generated by a two-dimensional elliptic differential operator and its applications

Allaberen Ashyralyev1,2, Sema Akturk1 and Yasar Sozen1,3*

*Correspondence: mailto:[email protected]

Web End [email protected] ; mailto:[email protected]

Web End [email protected]

1Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey

3Present address: Hacettepe Universitesi, Fen Fakultesi, Matematik Bolumu, 06800 Beytepe, Ankara, TurkeyFull list of author information is available at the end of the article

Abstract

We consider the two-dimensional dierential operatorAu(x1, x2) = a11(x1, x2)ux1x1(x1, x2) a22(x1, x2)ux2x2(x1, x2) + u(x1, x2) dened on functions on the half-plane = R+

R with the boundary conditions u(0, x2) = 0,

R, where aii(x1, x2), i = 1, 2, are continuously dierentiable and satisfy the uniform ellipticity condition a211(x1, x2) + a222(x1, x2) > 0, > 0. The structure of the

fractional spaces E(A, C( )) generated by the operator A is investigated. The positivity of A in Hlder spaces is established. In applications, theorems on well-posedness in a Hlder space of elliptic problems are obtained.

MSC: 35J25; 47E05; 34B27

Keywords: positive operator; fractional spaces; Greens function; Hlder spaces

1 Introduction

It is well known that (see, for example, [] and the references therein) various classical and non-classical boundary value problems for partial dierential equations can be considered as an abstract boundary value problem for an ordinary dierential equation in a Banach space with a densely dened unbounded operator. The importance of the positivity property of the dierential operators in a Banach space in the study of various properties for partial dierential equations is well known (see, for example, [] and the references therein). Several authors have investigated the positivity of a wider class of differential and dierence operators in Banach spaces (see [] and the references therein).

Let us give the denition of positive operators and introduce the fractional spaces and preliminary facts that will be needed in the sequel.

The operator A is said to be positive in E if its spectrum (A) lies inside of the sector S of the angle , < < , symmetric with respect to the real axis, and the following estimate (see, for example, [, ])

(A

)

x2

EE

M() + ||

holds on the edges S() = {ei : < }, S() = {ei : < } of S, and outside of the sector S. The inmum of all such angles is called the spectral angle of the positive operator A and is denoted by (A, E). We say that A is a strongly positive operator in E if (A, E) < /.

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Throughout the article, M indicates positive constants which may dier from time to time, and we are not interested to precise. If the constant depends only on , , . . . , then we will write M(, , . . .).

With the help of the positive operator A, we introduce the fractional space E = E(E, A) ( < < ), consisting of all elements v E for which the norm

v E = sup

>

A(

+ A)v

E + v E

is nite.

Theorem [] Let p + q = , p > , and let f , g : R+

R be any two nonnegative inte-

grable functions such that <

f p(x) dx < and <

gq(y) dy < . Then the following

Hilberts inequality holds:

f (x)g(y)x + y dx dy <

p

csc

p

f p(x) dx

gq(y) dy

q

.

Danelich in [] considered the positivity of a dierence analog Axh of the mth-order multi-dimensional elliptic operator Ax with dependent coecients on semi-spaces R+

Rn.

The structure of fractional spaces generated by positive multi-dimensional dierential and dierence operators on the space Rn in Banach spaces has been well investigated (see [] and the references therein).

In papers [, ] the structure of fractional spaces generated by positive one-dimensional dierential and dierence operators in Banach spaces was studied. Note that the structure of fractional spaces generated by positive multi-dimensional dierential and difference operators with local and nonlocal conditions on

Rn in Banach spaces C( )

has not been well studied.

In the present paper, we study the structure of fractional spaces generated by the two-dimensional dierential operator

Au(x, x) = a(x, x)uxx(x, x) a(x, x)uxx(x, x) + u(x, x), ()

dened over the region R+ = R+

R with the boundary condition u(, x) = , x

R.

Here, the coecients aii(x, x), i = , , are continuously dierentiable and satisfy the uniform ellipticity

a(x, x) + a(x, x) > ,

and > .

Following the paper [], passing limit when h in the special case m = and n = , we get that there exists the inverse operator (A+) for all and the following formula

(A + )f (x, x) =

G(x, x, p, s, )f (p, s) ds dp ()

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holds, where G(x, x, p, s, ) is the Green function of dierential operator (). Moreover, the following estimates

G(x

, x, p, s, )

C exp a(

+ )/

|x p| + |x s|

+

ln

+

( + )/

|x p| + |x s| ,

()

and

G

x (x, x, p, s, )

, G

x (x, x, p, s, )

C exp a(

+ )/

|x p| + |x s| |x p| + |x s|

()

hold. Here a = a().Next, to formulate our result, we need to introduce the Hlder space C = C(R+) of

all continuous bounded functions dened on R+ satisfying a Hlder condition with the indicator [, ] with the norm

f C

(R+) = f C(R

+) + sup(x,x),(x ,x )

|f (x, x) f (x , x )| (

|x x | + |x x |).

Here, C(R+) denotes the Banach space of all continuous bounded functions dened on R+ with the norm

C(R+) = sup

(x,x)

R+

R+

(x,x) =(x ,x )

(x

, x)

.

Clearly, from estimates () and () it follows that A is a positive operator in C(R+). Namely, we have the following.

Theorem Let . Then the following estimate

(A

+ )

C(R+)C(

R+)

M +

is valid.

Here, the structure of fractional spaces generated by the operator A is investigated. The positivity of A in Hlder spaces is studied. The organization of the present paper is as follows. In Section , the positivity of A in Hlder spaces is established. In Section , the main theorem on the structure of fractional spaces E(A, C(R+)) generated by A is investigated. In Section , applications on theorems on well-posedness in a Hlder space of parabolic and elliptic problems are presented. Finally, the conclusion is given.

2 Positivity of A in Hlder spaces C(R2+)

Theorem Let (, ). For , we have the following estimate:

(A

+ )

C (R+)C

(R+)

M(a)

+ .

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Proof Applying formula (), the triangle inequality, the denition of C-norm, estimate (), and Hilberts inequality, we get

(A

+ )f (x, x)

G(x

, x, p, s, )

f

(p, s)

ds dp

M f C

ea(+)/(|xp|+|xs|)

+

ln

+

( + )/

|x p| + |x s| ds dp

M f C

ea(+)/(|xp|+|xs|) ds dp

+

ea(+)/(|xp|+|xs|)

( + )/

|x p| + |x s| ds dp

M(a)

+ f C

+ M(a)( + )/ f C

ea(+)/(p+s)

p + s ds dp

M(a)

+ f C

for (x, x)

R+.

Then from that it follows

sup

(x,x)

R+

(A

+ )f (x, x)

M(a)

. ()

Without loss of generality, we can put , h > . Using formula () and the triangle inequality, we get

(A

+ )

+ f C

f (x + , x + h) f (x, x)

( + h)/

( + h)/

G(x

+ , x + h, p, s; ) G(x, x, p, s, )

f

(p, s)

ds dp ()

R+. Now, we will estimate the right-hand side of inequality ().

Let us consider two cases + h < and + h separately. First, we consider the case

+ h < . Using the triangle inequality, estimate (), the denition of C-norm, Hilberts inequality, and the Lagrange theorem, it follows that for some x between x, x + , and x between x, x + h,

(A

+ )

for (x, x), (x + , x + h)

f (x + , x + h) f (x, x)

( + h)/

M f C

( + h)/

G

x

x, x, p, s;

ds dp

+ h

( + h)/

G

x

x, x, p, s;

ds dp

M f C

Gx

x, x, p, s;

ds dp

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+

G

x

x, x, p, s;

ds dp

. ()

Second, we consider the case + h . Using formula (), the triangle inequality, estimate (), the denition of C-norm, and estimate (), we get

(A

+ )

ea(+)/(|xp|+|x

s|)

M(a)

M f C

|x p| + |x s|

ds dp

+ f C

f (x + , x + h) f (x, x)( + h)/

M f C

ea(+)/(|x+p|+|x+hs|)

+

ln

+

( + )/

|x + p| + |x + h s| ds dp

+

ea(+)/(|xp|+|xs|)

+

ln

+

( + )/

|x p| + |x s| ds dp

M(a)

. ()

Estimates () and () yield that

sup

(x+,x+h),(x,x)

R+

+ f C

+ )

f (x + , x + h) f (x, x)

( + h)/

M(a)

+ f C

. ()

(,h) =(,)

(A

Combining estimates () and (), we obtain

(A

+ )

C (R+)C

(R+)

M(a)

+ .

This nishes the proof of Theorem .

Note that from the commutativity of A and its resolvent (A + ), and Theorem , we have the following theorem.

Theorem Let . Then the following estimate holds:

(A

+ )

E(A,C (R+))E(A,C

(R+))

M(a)

+ .

3 The structure of fractional spaces E(A, C(R2+))

Suppose , + (, ). Consider the fractional space E(A, C(

R+)) and the Hlder space C+(R+). In this section, we prove the following structure theorem.

Theorem The norms of the spaces E(A, C(R+)) and C+(R+) are equivalent.

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Proof Assume that f C+(

R+). Let (x, x)

R+ and > be xed. From formula () it

follows that

A(A + )f (x, x)

=

+ f (x, x) +

f (x, x) f (p, s) ds dp. ()

Using equation (), the triangle inequality, the following inequalities

(a + b)p p ap

+ bp

G(x, x, p, s, )

, < p < , a, b , ()

(at)eat M, a > , [, ], ()

estimates (), (), (), and the denition of C+-norm, we obtain

A(A

+ )f (x, x)

+

f

(x, x)

+ +

G(x

, x, p, s, )

f

(x, x) f (p, s)

ds dp

M f C

+ +

+

+

G(x

, x, p, s, )

|x

p| + |x s| +/

ds dp

M f C

+ +

+

+

ea(+)/(|xp|+|xs|)

+

ln

+

( + )/

|x p| + |x s| |x p| + |x s| +/ds dp

M(a) f C

+ +

+( + )++/

M(a) f C

+

+

R+. ()

Thus, it follows from estimate () that

sup

>

sup

(x,x)

for (x, x)

+ )f (x, x)

M(a) f C

+ . ()

R+

A(A

Let > and (x, x)

R+ be xed. Using equation (), we can write

A(A + )

f (x + , x + h) f (x, x)

( + h)/

=

+

f (x + , x + h) f (x, x)

( + h)/

+ +

G(x + , x + h, p, s; )

f (x + , x + h) f (p, s)

( + h)/

ds dp

+

G(x, x, p, s; )

f (x, x) f (p, s)

( + h)/

ds dp. ()

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Now, we will estimate the right-hand side of equation (). We consider two cases + h < and + h , respectively. Let us rst assume that + h < . Furthermore, this situation will be considered in two cases: ( + )( + h) and ( + )( + h) . Let ( + )( + h) . From equation (), the triangle inequality, the denition of C+-norm, the assumptions + h < and ( + )( + h) , it follows that

A(A + )

f (x + , x + h) f (x, x)( + h)/

M f C

+

( + )( + h)/

+

+ ( + h)/

G(x

+ , x + h, p, s; )

|x

+ p| + |x + h s| +/

ds dp

+

+ ( + h)/

G(x

, x, p, s; )

|x

p| + |x s| +/

ds dp

+ [J + J + J]. ()

We will estimate Ji, i = , , , separately.

First, let us estimate J. Clearly, by the assumption ( + )( + h) , we have

J . ()

From estimates (), (), (), and the assumption ( + )( + h) , it follows that

J M

+ ( + h)/

= M f C

ea(+)/(|x+p|+|x+hs|)

+

ln

+

( + )/

|x + p| + |x + h s|

|x

+ p| + |x + h s| +/

ds dp

+( + )++/( + h)/ M(a). ()

Estimates (), (), (), and the assumption ( + )( + h) yield that

J M

+ ( + h)/

M(a)

ea(+)/(|xp|+|xs|)

+

ln

+

( + )/

|x p| + |x s| |x p| + |x s| +/ds dp

M(a)+( + )++/( + h)/ M(a). ()

Combining estimates ()-(), we get

A(A + )

f (x + , x + h) f (x, x)

( + h)/

M(a) f C

+ . ()

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Now, let us consider the case ( + )( + h) . Then, using equation (), we can write

A(A + )

f (x + , x + h) f (x, x)( + h)/

=

+

f (x + , x + h) f (x, x)( + h)/

+ +

G(x + , x + h, p, s; )

f (x + , x + h) f (x, x)( + h)/

ds dp

+ +

G(x + , x + h, p, s; ) G(x, x, p, s; )

ds dp. ()

From equation (), the triangle inequality, the Lagrange theorem, the denition of C+-norm, and the assumption + h < , it follows that for some x between x, x + , and x between x, x + h,

A(A + )

f (x, x) f (p, s) ( + h)/

f (x + , x + h) f (x, x)( + h)/

M f C

+

+

+ h

+ +

+ h

G(x

+ , x + h, p, s; )

ds dp

+

+ ( + h)/

G

x

x, x, p, s;

|x

p| + |x s| +/

ds dp

+

+h ( + h)/

G

x

x, x, p, s;

|x

p| + |x s| +/

ds dp

+ [L + L + L + L].

We will estimate Li, i = , . . . , , separately. Let us start with L. Clearly, we have

L . ()

Using estimates (), (), (), (), we obtain

L M

= M f C

ea(+)/(|x+p|+|x+hs|)

+

|x + p| + |x + h s| ds dp M. ()

Note that by the triangle inequality and the fact that x between x, x + , we get

x

p

ln

+

( + )/

|x p| x

x

|x p| , ()

x

s

|x s| x

x

|x s| h. ()

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By inequalities (), (), estimates (), (), (), and Hilberts inequality, we have

L M

+ ( + h)/

|x p| + |x s| |x

p| + |x s| +

ea(+)/(|xp|+|x

s|)

ds dp

M

+ ( + h)/

ea(+)/(|xp|+|x

s|)

x

p

x

+ s

+

ds dp

+ M

+ ( + h)/

ea(+)/(|xp|+|x

s|)

x

p

x

+ s

(

+ h)+ ds dp

M(a)

+( + )++/( + h)/ +

+( + h)+ ( + )( + h)/

M(a). ()

Similarly, we get

L M(a). ()

Combining estimates (), (), (), (), we obtain that for + h < , ( + )/( + h) ,

A(A + )

f (x + , x + h) f (x, x)( + h)/

+ . ()

It follows from estimates () and () that for + h < , > ,

A(A + )

M(a) f C

f (x + , x + h) f (x, x)

( + h)/

M(a) f C

+ . ()

Next, let us assume that + h . By equation (), we have

A(A + )

f (x + , x + h) f (x, x)

( + h)/

=

+

f (x + , x + h) f (x, x)

( + h)/

+ +

G(x + , x + h, p, s; )

f (x + , x + h) f (p, s)

( + h)/

ds dp

ds dp. ()

Equation (), estimates (), (), Hilberts inequality, the triangle inequality, the denition of C+-norm, and the assumption + h yield that for > ,

A(A + )

+

G(x, x, p, s; )

f (x, x) f (p, s)

( + h)/

f (x + , x + h) f (x, x)

( + h)/

M f C

+ [ + +

G(x

+ , x + h, p, s; )

|

x + p| + |x + h s| +/

ds dp

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+ +

G(x

, x, p, s; )

|x

p| + |x s| +/

ds dp

M(a) f C

+

++

( + )++/ +

+( + )++/

+ . ()

From estimates () and () it follows that

sup

>

sup

(x+,x+h),(x,x)

M(a) f C

A(A + )

f (x + , x + h) f (x, x)( + h)/

R+

(,h) =(,)

+ . ()

Combining estimates () and (), we obtain

C+ R+

E

M(a) f C

A, C

R+

.

Now, we will prove that E = E(A, C(R+)) C+(

R+). By Theorem , A is a positive operator in the Banach space E. Hence, for V E, we have

V =

A(

+ A)V d. ()

Let f E. It follows from formula () and equation () that

f (x, x) =

A(

+ A)f (x, x) d

=

G(x, x, p, s; )A( + A)f (p, s) ds dp d. ()

Using the triangle inequality, equation (), estimate (), and the denition of E-norm, we obtain

f

(x, x)

M f E

ea(+)/(|xp|+|xs|)

+

ln

+

( + )/

|x p| + |x s| ds dp d

M(a) f E

( + ) d

M(a)

( ) f E.

Thus,

sup

(x,x)

(x, x)

M(a)

( ) f E. ()

R+

f

R+ be xed. From equation () it follows that

f (x + , x + h) f (x, x) ( + h)+/

=

Let (x, x)

G(x + , x + h, p, s; ) ( + h)+/

A(A + )

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f

(p, s) f (x, x)

ds dp d

G(x, x, p, s; ) ( + h)+/

A(A + )

ds dp d. ()

Now, we will estimate the right-hand side of equation (). We consider two cases +h < and +h . Let us rst assume that +h < . By equation (), the triangle inequality, the Lagrange theorem, the denition of E-norm, and the assumption + h < , we

have, for some x between x, x + , and x between x, x + h, that

f (p, s) f (x, x)

f (x + , x + h) f (x, x) ( + h)+/

M f E

|G(x + , x + h, p, s; ) G(x, x, p, s; )|( + h)+/

|x

p| + |x s| /

ds dp d

|G(x + , x + h, p, s; ) G(x, x, p, s; )|( + h)+/

/(+h)

|x

+ p| + |x s| /

ds dp d

+

/(+h)

|G(x + , x + h, p, s; ) G(x, x, p, s; )|( + h)+/

|x

p| + |x s| /

ds dp d

= M f E[I + I + I].

We will estimate Ii, i = , , , separately. Using the triangle inequality, estimates (), (),

(), (), (), the Lagrange theorem, and the assumption + h < , we obtain

I M

|Gx(x, x, p, s; )| + h|Gx(x, x, p, s; )|

( + h)+/

|x

p| + |x s| /

ds dp d

ea(+)/(|xp|+|x

s|)

(|x p| + |x s|) |

x p| + |x s| /

ds dp d

M

ea(+)/(|xp|+|x

s|)(|x p| + |x s|) (|x p| + |x s|)

ds dp d

ea(+)/(|xp|+|x

s|)( + h)

(|x p| + |x s|)

+

ds dp d

M(a)

+/+/ d +

+/ d

M(a).

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From the triangle inequality, estimates (), (), (), (), (), the Lagrange theorem, and the assumption + h < , it follows that

I M

/(+h)

|Gx(x, x, p, s; )| + h|Gx(x, x, p, s; )|( + h)+/

|x

p| + |x s| /

ds dp d

M

/(+h)

( + h)ea(+)/(|xp|+|x

s|)(|x p| + |x s|) ( + h)+/(|x p| + |x s|)

ds dp d

M

/(+h)

( + h)ea(+)/(|xp|+|x

s|)(|x p| + |x s|) ( + h)+/(|x p| + |x s|)

ds dp d

/(+h)

+

( + h)ea(+)/(|xp|+|x

s|)( + h) ( + h)+/(|x p| + |x s|)

ds dp d

M(a)

/(+h)

( + h)/( + h)+/+/+/ d

+

/(+h)

( + h)/+/( + h)+/+/ d

M(a).

Using the triangle inequality, estimates (), (), (), (), (), the assumption +h < , and the following estimate

|x p| = |x p + | |x p + | + ,

|x s| = |x s + h h| |x s + h| + h,

we obtain

I M

/(+h)

ea(+)/(|x+p|+|x+hs|)

( + h)+/

+

ln

+

( + )/

|x + p| + |x + h s|

|x

p + | + |x s + h| + + h

ds dp d

+

/(+h)

ea(+)/(|xp|+|xs|)

+

ln

+

( + )/

|x p| + |x s| / + h

+/

|x

p| + |x s| /

ds dp d

M(a)

/(+h)

( + h)+/+/+ d

+

/(+h)

+ h( + h)+/+ d

M(a).

Thus, for + h < , we have

f (x + , x + h) f (x, x) ( + h)+/

M(a) f E. ()

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Next, let us assume that + h . Using formula (), estimate (), the triangle inequality, Hilberts inequality, and the assumption + h , we get

f (x + , x + h) f (x, x) ( + h)+/

|G(x + , x + h, p, s; )|

A(A

+ )f (p, s)

ds dp d

|G(x, x, p, s; )|

A(A

+ )f (p, s)

ds dp d

M f E

ea(+)/(|x+p|+|x+hs|)

+

ln

+

( + )/

+ |x + p| + |x + h s| ds dp d

+

ea(+)/(|xp|+|xs|)

+

ln

+

( + )/

|x p| + |x s| ds dp d

M(a) f E

( + ) d

M(a)

( ) f E.

Hence, for + h , we obtain

f (x + , x + h) f (t, x) ( + h)+/

M(a)

( ) f E. ()

Combining estimates () and (), we get

sup

>

sup

(x+,x+h),(x,x)

f (x + , x + h) f (x, x) ( + h)+/

M(a)

( ) f E. ()

R+

(,h) =(,)

Estimates () and () yield that

E = E

A, C

R+

C+

R+

.

This is the end of the proof of Theorem .

4 Applications

In this section, we consider some applications of Theorem . First, we consider the boundary value problem for the elliptic equation

u(y,x,x)

y a(x, x)

u(y,x,x)

x a(x, x)

u(y,x,x)

x + u(y, x, x)

= f (y, x, x), < y < T, x

R+, x

R,

()

u(, x, x) = (x, x), u(T, x, x) = (x, x), x

R+, x

R,

u(y, , x) = , y T, x

R.

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Here, a(x, x), a(x, x), (x, x), (x, x), and f (y, x, x) are given smooth functions and they satisfy every compatibility condition and

a(x, x) + a(x, x) > , ()

and > , which guarantees that problem () has a smooth solution u(y, x, x).

Theorem For the solution of boundary value problem (), we have the following estimate:

uyy C(C

+ (R+)) + uxx C(C

+ (R+)) + uxx C(C

+ (R+))

M(, )

xx C

+ (R+) + xx C

+ (R+) + xx C

+ (R+)

+ xx C

+ (R+) + f C(C

+ (R+))

,

where M(, ) is independent of , , and f .

Proof We introduce the Banach space C([, T], E) of all continuous abstract functions u(y) dened on [, T] with values in E, equipped with the norm

u C([,T],E) = max

yT

u(y)

E.

Note that problem () can be written in the form of the abstract boundary value problem

du(y)dy + Au(y) = f (y), < y < T, u() =

, u(T) = ()

in a Banach space E = C(R+) with a positive operator A dened by (). Here f (y) = f (y, x, x) is the given abstract function dened on [, T] with values in E, = (x, x), = (x, x) are elements of D(A). Therefore, the proof of Theorem is based on Theorem on the structure of the fractional spaces E(A, C(R+)), Theorem on the positivity of the operator A, on the following theorems on coercive stability of elliptic problems, nonlocal boundary value for the abstract elliptic equation and on the structure of the fractional space E = E(A/, E). This is the end of the proof of Theorem .

Theorem Under assumption () for the solution of elliptic problem

a(x, x)

u(x,x)

x a(x, x)

u(x,x)

x + u(x, x) = g(x, x),

x

R+, x

R,

u(, x) = , x

R,

the following coercive inequality holds:

u

x

C(R+)

+

u

x

C(R+)

M() g C

(R+),

where M() ( < < ) does not depend on g.

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The proof of Theorem uses the techniques introduced in [, Chapter ] and it is based on estimates () and ().

Theorem ([, Theorem ..]) The spaces E(A, E) and E (A/, E) coincide for any < < , and their norms are equivalent.

Theorem ([, Theorem .]) Let A be a positive operator in a Banach space E and f C([, T], E ) ( < < ). Then, for the solution of nonlocal boundary value problem (), the coercive inequality

u

C([,T],E

) + Au C([,T],E

)

M A E

+ A E

+ M

( ) f C([,T],E

)

holds, where M does not depend on , , , and f .

Second, we consider the nonlocal boundary value problem for the elliptic equation under assumption ()

u(y,x,x)

y a(x, x)

u(y,x,x)

x a(x, x)

u(y,x,x)

x + u(y, x, x)

= f (y, x, x), < y < T, x

R+, x

R,

()

Here, a(x, x), a(x, x), and f (y, x, x) are given smooth functions and they satisfy every compatibility condition and (), which guarantees that problem () has a smooth solution u(y, x, x).

Theorem For the solution of initial boundary value problem (), we have the following estimate:

uyy C(C

+ (R+)) + uxx C(C

u(, x, x) = u(T, x, x), uy(, x, x) = uy(T, x, x), x

R+, x

R,

u(y, , x) = , y T, x

R.

+ (R+)) + uxx C(C

+ (R+))

M(, ) f C(C

+ (R+)),

where M(, ) is independent of f .

The proof of Theorem is based on Theorem on the structure of the fractional spaces E(A, C(R+)), Theorem on the positivity of the operator A, Theorem on coercive stability of the elliptic problem, Theorem on the structure of the fractional space E =

E(A/, E), and the following theorem on coercive stability of the nonlocal boundary value for the abstract elliptic equation.

Theorem ([, Theorem .]) Let A be a positive operator in a Banach space E and f C([, T], E ) ( < < ). Then, for the solution of nonlocal boundary value problem ()

u (y) + Au(y) = f (y), < y < T,

u() = u(T), u () = u (T)

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in a Banach space E with a positive operator A, the coercive inequality

u

C([,T],E

) + Au C([,T],E

)

M( ) f C([,T],E

)

holds, where M does not depend on and f .

5 Conclusion

In the present article, the structure of the fractional spaces E(A, C(R+)) generated by the two-dimensional elliptic dierential operator A is investigated. The positivity of this operator A in a Hlder space is established. Of course, the Banach xed point theorem and the method of the present paper enable us to establish the existence and uniqueness results which hold under some sucient conditions on the nonlinear term for the solution of the mixed problem

u(y,x,x)

y a(x, x)

u(y,x,x)

x a(x, x)

u(y,x,x)

x + u(y, x, x) = f (y, x, x, u, ux, ux, uy), < y < T, x

R+, x

R,

u(, x, x) = (x, x), u(T, x, x) = (x, x), x

R+, x

R,

u(y, , x) = , < y < T, x

R.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors read and approved the nal manuscript.

Author details

1Department of Mathematics, Fatih University, 34500 Buyukcekmece, Istanbul, Turkey. 2Department of Mathematics, ITTU, Ashgabat, Turkmenistan. 3Present address: Hacettepe Universitesi, Fen Fakultesi, Matematik Bolumu, 06800 Beytepe, Ankara, Turkey.

Acknowledgements

Some of the results of the present article were announced in the conference proceeding [11] as an extended abstract without proofs. The authors would like to thank Prof. Pavel Sobolevskii (Universidade Federal do Cear, Brasil). The second author would also like to thank The Scientic and Technological Research Council of Turkey (TUBITAK) for the nancial support.

Received: 30 October 2013 Accepted: 3 December 2013 Published: 2 January 2014

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doi:10.1186/1687-2770-2014-3Cite this article as: Ashyralyev et al.: The structure of fractional spaces generated by a two-dimensional elliptic differential operator and its applications. Boundary Value Problems 2014 2014:3.