Fractional spaces generated by the positive differential and difference operators in a Banach space

The structure of the fractional spaces

E

α,q

,(

L

q

[0, 1],

A

x

) generated by the positive differential operator

A

x

defined by the formula

A

x

u

= −

a

(

x

)

d

2

u

/

dx

2

+

δu

, with domain

D

(

A

x

) = {

u

∈

C

(2)

[0, 1] :

u

(0) =

u

(1),

u

′(0) =

u

′(1)} is investigated. It is established that for any 0 <

α

< 1/2 the norms in the spaces

E

α,q

(

L

q

[0, 1],

A

x

) and

W

q

2

α

[0, 1] are equivalent. The positivity of the differential operator

A

x

in

W

q

2α

[0, 1](0 ≤

α

< 1/2) is established. The discrete analogy of these results for the positive difference operator

A

h

x

a second order of approximation of the differential operator

A

x

, defined by the formula

$$ A_h^x u^h = \left\{ { - a\left( {x_k } \right)\frac{{u_{k + 1} - 2u_k + u_{k - 1} }} {{h^2 }} + \delta u_k } \right\}_1^{M - 1} ,u_h = \left\{ {u_k } \right\}_0^M ,Mh = 1 $$

with

u

0

=

u

M

and −

u

2

+ 4

u

1

− 3

u

0

=

u

M

−2

− 4

u

M

−1

+ 3

u

M

is established. In applications, the coercive inequalities for the solutions of the nonlocal boundary-value problem for two-dimensional elliptic equation and of the second order of accuracy difference schemes for the numerical solution of this problem are obtained.