Available at
http://pvamu.edu/aam
Appl. Appl. Math.
ISSN: 1932-9466
Applications and Applied
Mathematics:
An International Journal
(AAM)
Vol. 6, Issue 1 (June 2011) pp. 353 - 368
(Previously, Vol. 6, Issue 11, pp. 1543 – 1553)
Multidimensional Inverse Boundary Value Problem
for a System of Hyperbolic Equations
M. A. Guliev and A.M. El-Hadidi
Baku State University
Baku, Azerbaijan
mahsoup79@yahoo.com
Received: February 10, 2011; Accepted: April 1, 2011
Abstract
In the paper we investigate the solvability of the inverse multidimensional boundary value
problem for the system of hyperbolic type equations. A method is proposed to reduce the
considered problem to some non infinite system of differential equations. The proposed method
allows one to prove the existence and uniqueness theorems for the multidimensional inverse
boundary value problems in the class of the functions with bounded smoothness.
Keywords: Boundary Value Problem; Multidimensional System; Hyperbolic Equations;
existence and uniqueness theorems
MSC (2010) No.: 35A05; 31A25; 35L20
1. Introduction
The solution of various problems of physics is often necessary for restoration of the basic
characteristics of a studied phenomenon or process, if you know the mathematical formulation of
the problem. These characteristics are coefficients or unknowns in the right side of the
353
354
M.A. Guliev and A.M. El-Hadidi
differential equation, which can determine under some additional information, are called inverse
problems.
In the last twenty - thirty years, the theory of inverse problems has been enriched by considering
the problem of determining the coefficients of partial differential equations as a function of one
or several variables. The study of one-dimensional and multi-dimensional inverse problems
associated with the restoration of the coefficients of partial differential equations involved in
Lavrentev et al. (1980), Romanov (1984), Anikonov et al. (1988), Namazov (1984), Guliev
(2004), Isakov (2006) and others. The book of Lavrentev et al. (1980) is described widely recent
results on the uniqueness of integral geometry and inverse problems for differential equations in
partial derivatives. In the book of Romanov (1984), the inverse problem for second-order
equation and hyperbolic systems of first order equations, including kinematic problem of static,
dynamic, Lamb problem for the equations of the theory of elasticity and the problem of
electrodynamics.
In the paper of Anikonov et al. (1988), the example of a parabolic equation given way to explore
the problem of solvability for some nonlinear inverse problems for differential equations.
Considered here the inverse problem using the technique of Fourier transform can be reduced to
quite acceptable for the study of boundary value problem for nonlinear integro-differential
equation. In a study Bubnov (1987), author considered some inverse problems for hyperbolic
equations. The method which used based on the reduction of the inverse problem of nonlinear
infinite systems of differential equations, allows us to prove theorem existence and uniqueness
for solutions of multidimensional inverse boundary value problems in classes of functions of
finite smoothness.
In the book of Namazov (1984), author studied one-dimensional inverse problem for the
parabolic and hyperbolic type. He was assumed that the unknown coefficients of the equation
depend only on the argument t. There are theorems proved the existence and uniqueness of the
problems. In paper of Guliev (2004) using the idea of work Bubnov (1987), author investigate
the multidimensional inverse boundary value problem for integro-differential equation of
hyperbolic type in a bounded domain. In the book of Isakov (2006) using the tool of Carleman
estimates the author investigate the inverse problems for hyperbolic, parabolic and elliptic
equations and then proved the existence and uniqueness theorems for the solutions of the
considered problem.
In this paper, using the idea of work Bubnov (1987) discusses solvability of inverse boundary
value problems for hyperbolic systems of equations and then proved the existence and
uniqueness theorems for the solutions of the problem.
Note that the considered equation in this paper is called the Klein-Gordon equation. This relative
invariant quantum equation describing spineless scalar or pseudo scalar particle, which plays one
of the roles of the fundamental equation of quantum field theory. This system of equations
cannot be written as a single equation.
AAM: Intern. J., Vol. 6, Issue 1 (June 2011) [Previously, Vol. 6, Issue 11, pp. 1543– 1553]
355
2. Problem Statement and Preliminary Results
In the domain DT    0, T  consider the following problem
 2U ( x, t )
 AU ( x, t )  a ( x)U ( x, t )  b( x)V ( x, t )  f ( x, t ) ,
t 2
(1)
 2V ( x, t )
 AV ( x, t )  a1 ( x)U ( x, t )  b1 ( x)V ( x, t )  g ( x, t ) ( x, t )  DT ,
t 2
(2)
U ( x,0)   ( x),
U ( x, t )
 0,
t t 0
x  ,
(3)
V ( x,0)   ( x),
V ( x, t )
 0,
t t 0
x  ,
(4)
U ( x, t )  F ( x, t ), V ( x, t )  G ( x, t ), ( x, t )    S  0, T  ,
(5)
U ( x, T )  h( x), V ( x, T )  q( x), x   ,
(6)
U ( x, t )
 0,
t
t T
(7)
V ( x, t )
 0,
t t T
where  is bounded domain in
AU ( x, t ) 
n
 (a ( x)U
i , j 1
n
a  
i , j 1
ij i
ij
xi
Rn ,
x  ,
S    C 2,
n  3,
( x, t )) x j , aij ( x)  a ji ( x) C 4 (),
2
j
   ,   0 ,  ( x),  ( x), f ( x, t ), g ( x, t ), F ( x, t ), G ( x, t ), h( x), q( x)
are given functions, and a ( x), b( x), a1 ( x), b1 ( x), U ( x, t ), V ( x, t ) are unknown functions.
Definition:
The system U ( x, t ), V ( x, t ), a ( x), b( x), a1 ( x), b1 ( x) is called the solution of the problem (1) (7), if it satisfies to following conditions
2
1. a ( x ), b( x ), a1 ( x ), b1 ( x ) W2 () .
2. Functions U ( x, t ), and V ( x, t ) are continuous in closed domain DT together with all
356
M.A. Guliev and A.M. El-Hadidi
derivatives in the equations (1) and (2), respectively.
3. Conditions (1) - (7) are satisfied in usual sense.
Let's designate through Z l , gl ( x) eigenvalues and eigenfunctions of the problem
Ag l ( x)   Z l2 g l ( x),
g l ( x) S  0 .
(8)
Then, from the general theory of elliptic equations it follows that Z l  , l   and gl (x)

form a complete set in W21 ()  W22 ().
Condition I.
Let for
T
Z 
2
l
where 
the domain
 2n2
T
2


and for any integers
n
and
l
the following inequality be satisfied
C ( )
 1 ,
Td
= diam , Zl - eigenvalues of a problem (8) [see Bubnov (1987)].
Note. [See Bubnov (1987)]. Suppose that, for an operator Au  u xx ,   {0  x  R0 } , the
eigenfunction
g l ( x)  sin
2
g l of the problem Ag l( x )   zl2 g l ( x), g l ( x) s  0 will be the function
l
lx
, zl 
. Then, the condition I can be written in the form
R0
R0
l 2 k 2  2l 2 T 2 k 2  2l 2 T
k T
k  2l 2 T T
k C ( )






 2 
  0
2
2
2
2
2
2
R0 T
T R0 l
T R0 l R0 l
T
R0 R0 l
TR0
The last inequality is written on the basis of Liouville's theorem, provided that the
quadratic irrationality.
Lemma 1:
Let the condition I be satisfied and
 k ( x)  L2 (),  k 
k
, (k  1,2,...) .
T
Then, for k there exists a solution of the Dirichlet problem
T
is
R0
AAM: Intern. J., Vol. 6, Issue 1 (June 2011) [Previously, Vol. 6, Issue 11, pp. 1543– 1553]
357
Abk ( x)  2k bk ( x)   k ( x), bk ( x) S  0,
for which the estimation
1
 b ( x)dx    
2
k
2
1 

2
k
( x)dx
is true.
3. Reduction of the Inverse Problems to the Nonlinear Infinite System of
Differential Equations
Let us assume that operator A and functions
G ( x, t ), h( x), q( x) satisfy the following conditions
 ( x),  ( x), f ( x, t ), g ( x, t ), F ( x, t ),

1. U  W21 ()  W22 ()


 n

2
AU ( x, t ) dx  1    U x21x j ( x, t )  U 2 ( x, t ) dx,
  i , j 1


1  0;

2. f ( x, t ), g ( x, t ), Dt6 f ( x, t ), Dt6 g ( x, t )  L2 (0, T ); W22 () ,
f ( x, T ), g ( x, T ), W22 (), Dtk f ( x, t )
Dtk g ( x, t )
t 0
 Dtk g ( x, t )
t T
 0,
t 0
 Dtk f ( x, t )
t T
 0,
(k  1,3,5);


3. F ( x, t ), G ( x, t ), Dt8 F ( x, t ), Dt8G ( x, t )  L2 (0, T ); W2 2 ( S ) ,
Dtk F ( x, t )
t 0
 Dtk F ( x, t )
t T
 0, Dtk G ( x, t )
7
t 0
 Dtk G ( x, t )
t T
 0 (k  1,3,5,7);
4.  ( x),  ( x) W24 (), A( Ah( x)), A( Ag ( x))  L2 (),
x    ( x)   ( x), h( x)  q( x);
F ( x , t ) t  o   ( x ) S , G ( x , t ) t  o   ( x ) S , F ( x , t ) t T  h ( x ) S , G ( x , t ) t T  q ( x ) S ;
5. ( x)   ( x)q( x)   ( x)h( x)    0 .
The following assertion holds:
358
M.A. Guliev and A.M. El-Hadidi
Lemma 2:
Let U ( x, t ),V ( x, t ), a ( x), b( x), a1 ( x), b1 ( x) be a solution of the inverse problem (1) - (7). Then,
following relations are satisfied:
1
a ( x) 
q( x)U tt ( x, t ) t 0   ( x)U tt ( x, t ) t T   0 ( x) ,
 ( x)
1
b( x ) 
 ( x)U tt ( x, t ) t T  h( x)U tt ( x, t ) t 0   1 ( x) ,
( x)
1
a1 ( x) 
q( x)Vtt ( x, t ) t  0   ( x)Vtt ( x, t ) t T   2 ( x) ,
 ( x)
1
b1 ( x) 
 ( x)Vtt ( x, t ) t T  h( x)Vtt ( x, t ) t  0   3 ( x) ,
( x)








where
 0 ( x)   A ( x)  f ( x, o) q( x)   Ah( x)  f ( x, T )  ( x),
 1 ( x)   A ( x)  f ( x, o) h( x)   ( x) Ah( x)  f ( x, T ) ,
 2 ( x)   Aq( x)  g ( x, T )  ( x)  q ( x) A ( x)  g ( x,0)  ,
 3 ( x)  h( x) A ( x)  g ( x, o)    ( x) Aq( x)  g ( x, T )  .
The proof of a Lemma 2 follows from (1) - (7).
Lemma 3:
If the functions U k ( x), Vk ( x), k  0,1, 2, , are solutions of the following problem in the domain



U ( x) 

2
m 2
k2U k ( x)  AU k ( x)  k
 q ( x)  mU m ( x)  ( x)  (1) mU m ( x)   0 ( x) 
( x) 
m 0
m 0

~



V ( x) 
2 ~
m 2 ~
 k
  ( x)  (1) mU m ( x)  h( x)  mU m ( x)  1 ( x)  f k ( x),
( x) 
m0
m 0



U ( x) 

2 
k2Vk ( x)  AVk ( x)  k


q
(
x
)
V
(
x
)
(
x
)
(1) m m2Vm ( x)   2 ( x) 





m m
( x) 
m 0
m 0

~


V ( x) 

~
2 ~
m
 k
  ( x)  (1) mVm ( x)  h( x)  mVm ( x)   3 ( x)  g k ( x) (9)
( x) 
m 0
m 0

~
~
U k ( x)  Fk ( x), Vk ( x)  Gk ( x),
S
from the class
S
k  0,1,2,...
(10)
AAM: Intern. J., Vol. 6, Issue 1 (June 2011) [Previously, Vol. 6, Issue 11, pp. 1543– 1553]
359



 m8   AU m ( x)  dx    m12 U m ( x) dx   m6 AU m ( x)
m0
2
2
m0 

m0
2
W21 (  )





  m2 A AU m ( x)
m0



 m12  Vm ( x) dx   m8   AVm ( x)  dx   m6 AVm ( x)
m0
2
2
m 0

m0

Then, the functions

~
~
V ( x, t )  Vk ( x) cos k t ,
k o
a ( x) 



1 
~
2 ~
q
(
x
)

U
(
x
)

(
x
)
(1) k 2kU k ( x)   0 ( x),





k k
( x) 
k 0
k 0

b( x ) 



1 
~
k 2 ~

(
x
)
(
1
)

U
(
x
)
h
(
x
)
2kU k ( x)  1 ( x),






k k
( x) 
k 0
k 0

a1 ( x) 



1 
k 2~
2~
 q ( x) kVk ( x)   ( x) (1) kVk ( x)   2 ( x), and
( x) 
k 0
k 0

b1 ( x) 



1 
~
k 2~

(
x
)
(
1
)

V
(
x
)
h
(
x
)
2kVk ( x)   3 ( x)






k k
( x) 
k 0
k 0

are the solution of an inverse problem (1) - (7), where
k
k 
,
T
T
2
f k ( x)   f ( x, t ) cos  k tdt ,
T 0
T
T
2
F ( x, t ) cos  k tdt ,
T 0
g k ( x) 
2
g ( x, t ) cos k tdt ,
T 0
Gk ( x) 
2
G ( x, t ) cos k tdt , (k  0,1,2,....) .
T 0
Fk ( x) 
T
Proof:
From (9) - (11) we obtain, (1), (2), (5) and
 ,
W21 (  )
m 0
k o
L2 (  )
2

~
  2m A AVm ( x)

~
~
U ( x, t )  U k ( x) cos k t ,
2
2
L2 (  )
 , (11)
360
M.A. Guliev and A.M. El-Hadidi
~
U t ( x, t )
t 0
~
 0, U t ( x, t )
t T
~
 0, Vt ( x, t )
t 0
~
 0 Vt ( x, t )
t T
 0.
Therefore, to prove Lemma 3, it is required to establish that
~
U ( x, t )
t 0
~
  ( x ) U ( x, t )
t T
~
 h( x), V ( x, t )
t 0
~
  ( x ) V ( x, t )
t T
 q( x).
Let
~
U ( x, t )
t 0
~
 ~ ( x), V ( x, t )
t 0
 ~ ( x) .
~
Then, for the function ~ ( x)   ( x)  Z ( x), ~ ( x)   ( x)  Z ( x) from (8), (9) we have
AZ ( x) 


Z ( x)
 ( x) U tt ( x, t )  q ( x)U tt ( x, t )   0 ( x)
t T
t 0
 ( x)
~
Z ( x)
~
~

 h( x)U tt ( x, t )   ( x)U tt ( x, t )   1 ( x) ,
t 0
t T
( x)



(12)

Z ( x)
AZ ( x) 
 ( x) Vtt ( x, t )  q( x)Vtt ( x, t )   2 ( x)
t T
t 0
 ( x)
~
Z ( x)
~
~

 h( x)Vtt ( x, t )   ( x)Vtt ( x, t )   3 ( x) ,
t

t T
0
( x)

Z ( x) S  0,

~
Z ( x)  0.
(13)
S
The uniqueness of the solution of the problem (12), (13) will be established below. Then, we
prove the solvability of (9), (10).
It will be similarly shown that U ( x, t ) t T  h( x), V ( x, t ) t T  q ( x).
We investigate the solvability of (9), (10).
æ k ( x) (k  0,1, 2,....) the solutions of the following problems
Let us define by rk (x) and
A rk ( x)  0,
2
rk ( x) S  Fk ( x)
(k  0,1, 2,...),
2
rk ( x) C (  )  C1 Fk ( x) W 7 2 ( S ) ,
(14)
2
A æ k ( x)  0,
æ k ( x) S  Gk ( x)
(k  0,1, 2,...),
AAM: Intern. J., Vol. 6, Issue 1 (June 2011) [Previously, Vol. 6, Issue 11, pp. 1543– 1553]
361
2
2
æ k ( x) C (  )  C1 Gk ( x) W 7 2 ( S ) .
(15)
2
~
Let us designate U k ( x)  U k ( x)  rk ( x),
~
Vk ( x)  Vk ( x)  æ k ( x) .
Then, from (9) and (10), we have


U k ( x)  rk ( x) 
2
m 2
 U k ( x)  AU k ( x) 
q ( x)  mU m ( x)   ( x)  (1) mU m ( x)
( x)
m0
m 0

2
k

  0 ( x)  q( x)   r ( x)   ( x)  (1) m m2 rm ( x) 
m0
m 0




2
m m



Vk ( x)  æ k ( x) 
m 2
2

(
)
(

1)
(
)

(
)
(
)


(
)

(
)
(1) m m2 rm ( x)

x

U
x
h
x

U
x
x

x




1
m m
m m
( x)
m0
m0
m 0



 h( x)  2m rm ( x)  f k ( x)  2k rk ( x),
m0

k2Vk ( x)  AVk ( x) 


U k ( x)  rk ( x) 
2

q
(
x
)
V
(
x
)

(
x
)
(1) m m2Vm ( x)





m m
 ( x)
m 0
m 0




  2 ( x)  q ( x)  m2 æ m ( x)   ( x)  (1) m m2 æ m ( x) 
m0
m0





Vk ( x)  æ k ( x) 
m 2
2
m 2
 ( x)  (1) mVm ( x)  h( x)  mVm ( x)   3 ( x)   ( x)  (1) m æ m ( x)
( x)
m0
m 0
m 0



 h( x)  2mæ m ( x)  g k ( x)  2k æ k ( x), (16)
m0

U k ( x) S  0,
Vk ( x) S  0
(k  0,1,2,...) .
(17)
4. Existence and Uniqueness for the Problem (1) - (7)
Now we investigate problem (16), (17). We consider the following set of functions
B  U 0 ( x),U 1 ( x),...,U k ( x),...,, V0 ( x), V1 ( x),...,Vk ( x),..., U k ( x) S  0, Vk ( x) S  0


k  0,1,2,...,  12k U k ( x) 2L ()  U 0 ( x) 2L ( )  R ,  12k Vk ( x) 2L ()  V0 ( x) 2L ()  R ,
4
4
k 1
2
2
k 1
2
2
362
M.A. Guliev and A.M. El-Hadidi


8
k
k 1
AU k ( x)
2
L2
 AU 0 ( x)
()
2
L2 (  )
 R,


8
k
k 1
AVk ( x)
2
L2 (  )
 AV0 ( x)
2
L2 (  )

 R .

Let us show that at suitable choice R , the problem (16), (17) is solvable in B . Let
0 ( x),1 ( x),...,k ( x),..., ~0 ( x),~1 ( x),...,~k ( x),... be an arbitrary elements from B and
substitute it in a right hand side of (16). Then, we will obtain a function, which belongs to the
space L2 () . Then, from Lemma 1, we have
2
2
22  2 2   2

  2

2 2
mm ( x)   2 N 0 k ( x)   m rm ( x) 
 U ( x)dx  12   2 N0 k ( x)  
m 1

 m 1



2
k
2
2

  ( x) 
  2


2
2
2 N r ( x)   mm ( x)   2rk ( x) N 0   m2 rm ( x)   k2 ( x)  0

 m 1

 m 1

 ( x) 
2
2 2
0 k
2
2
  ( x) 
  2

  2

2 2
2 2


2
(
)
(
)
2
(
)
N
x
x
N
x
 r ( x)  0






m rm ( x) 



m m
0 k
0 k



 m 1

 m 1

 ( x) 
2
2
k
2
2
 

 

2 N æ ( x)   m2m ( x)   2 N 02 æ k2 ( x)   m2 rm ( x)   f k2 ( x)  k4 rk2 ( x) s
 m 1

 m 1

2
2
  ( x) 
  ( x)  
(18)
 ~k2 ( x) 1   æ 2k ( x) 1   dx ,
 ( x) 
  ( x)  
2
0
2
k
2
  2

  2

2 2


V
x
dx
N

x


x
N

x
k æ m ( x) 
(
)
2
(
)
(
)
2
(
)




0 k
k m



2 

1 
 m 1

 m 1

2
k
22
2
0
2
2
k
2
2
  ( x) 
 

 

2 N r ( x)   m2 m ( x)   2 N 02 rk2 ( x)   m2 æ m ( x)   k2 ( x)  2

 m 1

 m 1

 ( x) 
2
2 2
0 k
2
2
  ( x) 
  2

  2

2 2
2 2



N
x
x
N
x
2
(
)
(
)
2
(
)
 r ( x)  2






m æ m ( x) 



m m
0 k
0 k



 m 1

 m 1

 ( x) 
2
2
k
2
2
2
  ( x) 
 

 

2 N æ ( x)   m2 æ m ( x)   2 N 02 æ k2 ( x)   m2 m ( x)    k2 ( x)  3 
 m 1

 m 1

 ( x) 
2

  3 ( x) 
2
  g k2 ( x)  4k æ 2k ( x) dx, (k  0,1,2,...) ,
(19)
 æ k ( x)

 ( x) 
2
0
2
k
where
 q ( x)
N 0  max 
 ( x)
,
c()
 ( x)
( x)
,
c()
 ( x)
( x)
,
c()
h( x )
( x)

.
c() 

Now, multiplying the relation (18) by  12
k and summing over
k
from
0
to
 , we obtain
AAM: Intern. J., Vol. 6, Issue 1 (June 2011) [Previously, Vol. 6, Issue 11, pp. 1543– 1553]
363

2
   k U k ( x)dx 
12
 k 0

N 02C1C2 
2
1

44



2
   k k ( x)dx    k  Ak ( x)  dx
12
 k 0
2
8
 k 0



    k k2 ( x)dx    k Fk ( x) W 7 2     k k2 ( x)dx    k Fk ( x) W 7 2
12
 k 0

8
2
(S )
k 0

12
 k 0

12
2
(S )
k 0

   k Fk ( x) W 7 2     k rk2 ( x)dx     k  k2 ( x)dx    k  Ak ( x)  dx
8
2
(S )
k 0

 k 0

12
12
 k 0

8
 k 0

2
    k  k2 ( x)dx    k Fk ( x) W 7 2     k k2 ( x)dx    k Gk ( x) W 7 2
12
 k 0
8
2
(S )
k 0
12
 k 0
12
2
(S )
k 0
2
 22  ( x)
  12 2
    æ ( x)dx    Fk ( x) W 2   2 0
    k k ( x)dx
(S )
k 0
 k 0
 1  ( x ) C      k  0

12
k
2
k

8
k
2
7
2

 22  ( x)
  12 2

12 2

    r ( x)dx   2 1

x
dx
(
)



k æ k ( x ) dx 
  k k


 k 0
 k 0
 1  ( x ) C      k  0



16
12
22
22
 2    k f k2 ( x)dx  2    k rk2 ( x)dx ,
(20)

1
12 2
k k
1
 k 0
 k 0
where
1,
k  
k ,
k  0,
k  1, 2,.
Similarly from (19) we obtain

2
   k Vk ( x)dx 
 k 0
12

2
8
44 N 02C1C2   12 2
(
x
)
dx


 k A k ( x) dx
  k k

2

1
 k 0
  k 0






    k k2 ( x)dx   k Gk ( x) W 7 2     k  k2 ( x)dx   k Fk ( x) W 7 2
 k 0

12
8
2
(S )
k 0

 k 0

12
12
2
(S )
k 0



    k rk2 ( x)dx   k Gk ( x) W 7 2     k  k2 ( x)dx    k A k ( x) dx
 k 0

12
8
2
(S )
k 0

 k 0

12
 k 0

8
2
    k  k2 ( x)dx    k Gk ( x) W 7 2     k æ 2k ( x)dx   k Gk ( x) W 7 2
12
 k 0
8
2
(S )
k 0
12
 k 0

 22  ( x)
12
2
     k2 ( x)dx   k Gk ( x) W 7 2   2 2
(S )
k 0
 k 0
 1  ( x )

12
k
2
k 0
8
2
(S )
2
  12 2
    k k ( x)dx
C      k 0

 22  ( x)
  12 2

12


 k æ 2k ( x)dx 
    r ( x)dx   2 3

(
x
)
dx
k
 

k

 k 0
 k 0
 1  ( x ) C      k  0


12 2
k k
364
M.A. Guliev and A.M. El-Hadidi

22


2
   k g k ( x)dx 
12
2
1  k 0
22


 
2
1  k 0
16
k
æ 2k ( x)dx .
(21)
From estimations (20), (21) and (16) we obtain

2
   k U k ( x)dx 
12
 k 0

2
   k Vk ( x)dx 
12
 k 0

 11
44 N 02C1C2  R 2
44
44
 L0 R  2 L20   2 KR  2 KL0  2 L0 ,

2
1
1
1
 2
 1
(22)
 11
44 N 02 C1C 2  R 2
44
44
 L0 R  2 L20   2 KR  2 KL0  2 L0 ,

2
1
1
1
 2
 1
(23)

 R2

 L0 R  2 L20 
 2

2
2
   k AU k ( x) dx  2   k U k ( x)dx  44 N0 C1C2 
2
8
 k 0
12
 k 0
 11KR  44 KL0  44 L0 ,


(24)
 R2

 L0 R  2 L20 
 2

2
2
   k AVk ( x) dx  2   k Vk ( x)dx  44 N0 C1C2 
2
8
 k 0
12
 k 0
11KR  44 KL0  44 L0 ,
(25)
where
  ( x)
K  max  0
 ( x)
T
,
c()
1 ( x)
 ( x)
,
c()
 2 ( x)
 ( x)
,
c()


 3 ( x)
 ( x)
T

 Dt6 f ( x, t )
2

,
c() 

2
2
2
2
2
L0   F ( x, t ) W 7 2 ( S )  Dt8 F ( x, t ) 7 2 dt   G ( x, t ) W 7 2 ( S )
W
S
(
)
2
2
2
T 0
T 0
 Dt8G ( x, t )
2
7
W2 2 ( S )
 f ( x, t )
2
L2 (  )
L2 (  )
Lemma 4:
Let conditions 1-5, I and
1

2
1

22
be satisfied. Then,
35
R
R

44 N 02C1C2   L0   11KR  ,
8
2

 g ( x, t )
2
L2 (  )
 Dt6 g ( x, t )
2
L2 (  )
dt .

AAM: Intern. J., Vol. 6, Issue 1 (June 2011) [Previously, Vol. 6, Issue 11, pp. 1543– 1553]
365
88 N 02 C1C 2 L20  44 KL0  44 L0 
3
, and
11
 1



 3R
 2  1 204 N 02C1C2 
 2 L0   204 K (C1  1)  1 .

 2

 1

Then, there exists a solution of the problem (16), (17) in B and


12
k
k 0


2
U k ( x)
L2 (  )
8
k 0
AU k ( x)
2
L2 (  )
R
 ,
4


12
k
k 0

2

Vk ( x )
L2 (  )
 R, ,   AVk ( x)
k 0
8
2
L2 (  )
R
,
4
 R.
Proof:
System (16) has the form
~
EZ k  L ( Z k ),
where L is determined by the right-hand sides of the system (16), and E an elliptic operator of
the left hand sides of (16), Z k ( x)  U k ( x),Vk ( x), ( k  0,1,2,... ). But since E has a bounded
inverse, the system (16) can be written as:
Z k  L( Z k ),
(26)
where
~
L  E 1 L ,
L( Z k )  L1 U k ( x), Vk ( x) , L2 U k ( x), Vk ( x) 
 k  0,1,2,... .
Take an arbitrary vector function 0 ( x),..., k ( x),..., ~0 ( x),..., ~k ( x),...  B and substitute into
the right hand side of (16). Then, for the solutions U k ( x)  L1 (k ( x),  k ( x)),
Vk ( x)  L2 (k ( x),  k ( x)) we obtain estimations (22) - (25). Hence, under the conditions of
Lemma 4, we get that (U k ( x),...,U k ( x)...), (V0 ( x),...,Vk ( x)...)   B . This proves that, operator L
is contracting.
Let

1, k
( x), ~1, k ( x)  and
solutions will be

2, k
( x), ( 2,k ( x) 
(k  0,1, 2,)  B . Then, the corresponding
366
M.A. Guliev and A.M. El-Hadidi
U1, k ( x)  L1 (1, k ( x), ~1, k ( x)) ,
V1, k ( x)  L2 (1, k ( x), ~1, k ( x)) ,
U 2,k ( x)  L1 ( 2,k ( x), ~2,k ( x)) ,
V2,k ( x)  L2 ( 2,k ( x), ~2,k ( x)) .
Let us denote that
U k ( x)  U1, k ( x)  U 2, k ( x), Vk ( x)  V1, k ( x)  V2, k ( x),
k ( x)  1,k ( x)  2,k ( x), ~k ( x)  ~1,k ( x)  ~2,k ( x).
Let's rewrite the relation (16) for U1, k ( x), V1, k ( x), 1, k ( x), ~1, k ( x) and U 2,k ( x), V2,k ( x),
2,k ( x),  2,k ( x) . Further, subtracting the first relation to the second, under conditions of
Lemma 4, we obtain the following estimation




2
2
   k U k ( x)dx     k Vk ( x)dx     k  AU k ( x)  dx     k  AVk ( x)  dx
12
 k 0
12
 k 0
2
8
 k 0
2
8
 k 0



  12

8
12
8
2
2
      k k2 ( x)dx     k ~k2 ( x)dx     k Ak ( x) dx     k A~k ( x) dx  , (27)
 k 0
 k 0
 k 0
 k 0





where
 1

 3R


 2 L0   204 KC1  .
   2  1 204 N 02C1C2 
 2


 1

From this we obtain the Lemma 4.
Theorem:
Let conditions of Lemma 4 be satisfied. Then, there exists unique solution of problem (16), (17)
(an inverse problem (1) - (7)) for which the following estimations are true


k 0
12
k
U k ( x)
2
L2 (  )

R
,
4


k 0
12
k
Vk ( x)
2
L2 (  )

R
,
4
AAM: Intern. J., Vol. 6, Issue 1 (June 2011) [Previously, Vol. 6, Issue 11, pp. 1543– 1553]
367


8
k
k 0

AU k ( x)
L2 (  )
  k AU k ( x) W
6
2
k 0

  k AVk ( x) W
6

2
8
k 0

   k A AU k ( x)  L
1
()
2
2
k 0

2
2
L2 (  )
2
   k A AVk ( x)  L
1
()
2
k 0
 R, ,   k AVk ( x)
2
2 ()
2
k 0
2 ()
 R,
 RC , and
 RC.
Proof:
Lemma 4 implies the solvability of the problem (16), (17) in B . Thus, the first estimate of the
10
10
Theorem holds. Multiplying (16), by  k U k ( x) and  k Vk ( x), accordingly, and take the sum
over k from 0 to , then, integrating in  and using the first estimation, we obtain


10
k
U k ( x) W 1 (  )  constR,
10
k
Vk ( x) W 1 (  )  constR.
k 0


k 0
2
2
2
2
Then, from (10) and this estimation, we get


6
k
AU k ( x) W 1 (  )  constR,
6
k
AVk ( x) W 1 (  )  constR.
k 0


k 0
2
2
2
2
Similarly, we can prove the following estimations

  A AU
2
k
k 0
( x)  L
2
k
2 ()
 constR,
and

  A AV ( x)
k 0
2
k
k
2
L2 (  )
 constR.
To complete the proof of the theorem, it remains to prove that the solution of (12), (13) is equal
to zero. Indeed, under the conditions of the Theorem we have
368


M.A. Guliev and A.M. El-Hadidi
2
R
~ 2 
2
2

 
AZ ( x) dx   AZ ( x) dx  12  N 02C1C2  KC1     AZ ( x) dx   AZ ( x) dx  .
2

 



~
~
Then, AZ ( x)  0, Z ( x) S  0 and AZ ( x)  0, Z ( x) S  0, hence, Z ( x)  0, Z ( x)  0 .
4. Conclusion
In the paper we investigate the solvability for the inverse boundary value problems for the
system of hyperbolic equations. The method based on the reducing of inverse boundary value
problem to some nonlinear infinite systems of differential equations is proposed. Given method
allows one to prove existence and uniqueness theorems of multidimensional inverse boundary
value problems in the class of finite smoothness functions.
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