Vol 3, No 2 (2012)

(1)

Available online through

ISSN 2229 – 5046

A NON-LOCAL BOUNDARY VALUE PROBLEM WITH INTEGRAL CONDITIONS

FOR A FOURTH ORDER PSEUDOHYPERBOLIC EQUATION

**Azizbayov E.I.* and Y. T. Mehraliyev**

Mechanics-mathematics faculty, Baku State University, Baku, Azerbaijan

E-mail

(Received on: 26-01-12; Accepted on: 24-02-12)

________________________________________________________________________________________________

ABSTRACT

I

n the paper, the classic solution of one-dimensional boundary value problem for a pseudohyperbolic equation with non-classic boundary conditions is investigated. For that the stated problem is reduced to the not-self-adjoint boundary value problem with equivalent boundary condition. Then, using the method of separation of variables, by means of the known spectral problem the given not self-adjoint boundary value problem is reduced to an integral equation. The existence and uniqueness of the integral equation is proved by means of the contraction mappings principle and it is shown that this solution is a unique solution for a not-adjoint boundary value problem. Finally, using the equivalence, the theorem on the existence and uniqueness of a non-local boundary value problem with integral condition is proved.

2000 Mathematics Subject Classification: 35L82, 34B10, 35G30.

Keywords and Phrases: mixed problem, contracted mappings, fixed point, existence, uniqueness, classic solution,

pseudohyperbolic equation.

________________________________________________________________________________________________

1. INTRODUCTION:

Contemporary problems of natural sciences make necessary to state and investigate qualitative new problems, the striking example of which is the class of non-local problems for partial differential equations. Among non-local problems we can distinguish a class of problems with integral conditions. Such conditions appear by mathematical simulation of phenomena related to physical plasma [1], distribution of the heat [2] process of moisture transfer in capillary-simple environments [3], with the problems of demography and mathematical biology.

2. THE PROBLEM STATEMENT AND ITS REDUCTION TO THE EQUIVALENT PROBLEM:

Consider the equation [4]

) , ( ) , ( ) ( ) , ( ) , ( ) ,

(xt u xt u xt q t u xt f xt

u_tt − _ttxx − _xx = + (1)

in the domain DT ={(x,t):0≤x≤1,0≤t≤T } and state for it a problem with initial conditions

, ) ( ) 0 ,

(x x

u =ϕ ut(x,0)=ψ(x)(0≤x≤1) (2)

and non-local conditions

) , 1 ( ) , 0

( t u t

u =β (0≤t≤T), (3)

0 ) , (

1

0

=

∫

u xt dx (0≤t≤T), (4)

where β≠±1 is a given number, q(t), f(t,x),ϕ(x),ψ(x) are the given functions, u(x,t) is a sought function. Earlier, the boundary value problems with non-local integral equations were considered in the papers [1], [2] and [8]. Here, for β=0 we have an Ionkin type boundary condition [3].

________________________________________________________________________________________________

(2)

Definition: Under the classic solution of problem (1)-(4) we understand the function u(x,t) continuous in a closed domain DT together with all its derivatives contained in equation (1), and satisfying all conditions (1)-(4) in the

ordinary sense.

The following lemma is proved similarly [7].

Lemma 1: Let q(t)∈C[0,T], f(x,t)∈C(DT), ϕ(x),ψ(x)∈C[0,1], ( , ) 0

1

0

=

∫

f xt dx (0≤t≤T) and the following agreement conditions be fulfilled:

ϕ(0)−βϕ(1)=0, ( ) 0

1

0

= ϕ

∫

xdx , ϕ′(1)=ϕ′(0),

0 ) 1 ( ) 0

( −βψ =

ψ , ( ) 0

1

0

= ψ

∫

xdx , ψ′(1)=ψ′(0). (5)

Then the problem on finding the classic solution of problem (1)-(4) is equivalent to the problem on defining of the function u(x,t) from (1)-(3) and

) , 1 ( ) , 0

( t u t

ux = x (0≤t≤T). (6)

Proof: Let u(x,t) be the solution of problem (1)-(4). Integrating equation (1) with respect to x from 0 to 1, we have:

= −

− −

−

∫

( , ) ( (1, ) (0, )) ( (1, ) (0, ))

1

0 2 2

t u t u t u t u dx t x u dt

d

x x

ttx

(

)

(

,

)

(

,

)

(

0 )

1

0 1

0

T

t

dx

t

x

f

dx

t

x

u

t

q

∫

+

∫

≤

. (7)

Assuming that ( , ) 0

1

0

=

∫

f xtdx (0≤t≤T),ϕ′(1)=ϕ′(0),ψ′(1)=ψ′(0) and allowing for (2), we find: ,

0 )) , 0 ( ) , 1 ( ( )) , 0 ( ) , 1 (

(uttx t −uttx t + ux t −ux t =

, 0 ) 0 , 0 ( ) 0 , 1

( − x =

x u

u utx(1,0)−utx(0,0)=0. Hence we arrive at fulfillment of (6).

Now, assume that u(x,t) is the solution of problem (1)-(3), (6). Then allowing for (6), from (8) we find:

). 0

( 0 ) , ( ) ( ) , (

1

0 1

0 2 2

T t dx

t x u t q dx t x u dt

d ₋ ₌ _≤ _≤

∫

(8)

From (2) and ( ) 0

1

0

= ϕ

∫

x dx ,

∫

ψ =

1

0

, 0 )

(x dx it is obvious that

( ,0) ( ) 0

1

0 1

0

= ϕ =

∫

u x dx xdx , ( ,0) ( ) 0

1

0 1

0

= ψ =

∫

ut x dx x dx . (9)

Since problem (8), (9) has only a trivial solution, then ( , ) 0

1

0

=

∫

u xt dx (0≤t≤T), i.e. condition (4) is satisfied. The lemma is proved.

3. AUXILIARY FACTS:

Now, in order to investigate problem (1)-(3), (6) we cite some known facts.

Consider the following spectral problem [3] and [5]:

0 ) ( )

( +λ =

′′x X x

X (0≤x≤1), (10)

)

1 ( ) 0 ( , ) 1 ( ) 0

( X X X

(3)

Boundary value problem (10), (11) is not self-adjoint. The problem

0 ) ( )

( +λ =

′′ x Y x

Y (0≤x≤1), (12)

, ) 1 ( ) 0

( Y

Y = Y′(1)=βY′(0), (13)

will be a conjugated problem.

We denote the system of eigen and adjoint functions of problem (10), (11) in the following way [5]:

,..., sin ) ( , cos ) ( ) ( ..., , )

( 2 1 2

0 x ax b X x ax b x X x x

X = + k− = + λk k = λk (14)

where

) 1 /( ,

0 ) 1 /( ) 1 ( ,...), 2 , 1 , 0 (

2 π = = −β +β ≠ =β +β

=

λk k k a b . (15)

We choose the system of eigen and adjoint functions of the conjugated problem as follows [5]:

... , sin ) 1

( 4 ) ( , cos 4 ) ( ..., , 2 )

( 2 1 2

0 x Y x x Y x b ax x

Y = k− = λk k = − − λk . (16)

It is directly verified that the biorthogonality conditions

∫

=δ

=

1

0

) ( ) ( ) ,

(Xi Yj Xi xYj xdx ij

are fulfilled.

Here, δij is Kronecker’s symbol.

The following theorem is valid.

Theorem 1 [7]: The system of functions (14) forms a Riesz basis in the space L2(0,1) and the estimates

( )

₍₀_,₁₎

0 2 )

1 , 0

( 2

2 L

k k

L

g

R

g

x

g

r

≤

∑

≤

∞

=

, (17)

where

∫

= =

1

0

) ( ) ( )) ( ), (

(g x Y x g xY xdx

gk k k , (k=0,1,...)

1

] 1 , 0 [ 2 2

2

) ( 1 2 1 4 3 2 3 3 1

−

   

   

   



 ₊ ₊ +

    

  

+       + =

c

b ax b

b a

r ,

(

)

[ ]













₊

₋

=

1 , 0 2

1

8

C

ax

b

R

are valid for any function g(x)∈L2

( )

0,1.

Under the assumptions

[ ]

0,1, )

(x ∈C2i−1

g ( ) 2(0,1)

) 2 (

L x

g i ∈

) 1 ( )

0

( (2)

) 2

( s s

g

g =β , g(2s+1)(0)=g(2s+1)(1) ( 0, 1; 1)

______

≥ −

= i i

s

we establish the validity of the estimates:

) 1 , 0 ( ) 2 ( 2

1

2 1 2 2

2

)

(

2

2 )

(

L i

k

k i

k

g



≤

g

x











_∑

∞

= −

λ

, (18)

) 1 , 0 ( ) 1 2 ( )

2 ( 2

1

2 2 2

2

)

(

2 )

1 )(

(

2

2 )

(

L i

i

k

k i

k

g

x

b

ax

aig

x

− ∞

=

−

≤













_∑

_λ

(4)

Further, under the assumptions

], 1 , 0 [ ) (x C2i

g ∈ ( ) 2(0,1)

) 1 2 (

L x

g i+ ∈ ,

), 1 ( )

0

( (2)

) 2

( s s

g

g =β g(2s−1)(0)=g(2s−1)(1) (i≥1; s=0,i).

we prove the validity of the estimates:

) 1 , 0 ( ) 1 2 ( 2

1

2 1 2 1 2

2

) ( 2 2 ) (

L i

k

k i

k g g x

+ ∞

=

−

+ _≤

   

 

λ

∑

, (20)

) 1 , 0 ( ) 2 ( )

1 2 ( 2

1

2 2 1 2

2

)

(

)

1

2 (

)

1 )(

(

2

2 )

(

L i i

k

k i

k

g



≤

g

x

−

b

−

ax

−

a

i

+

g

x











∞ +

= +

∑

λ

. (21)

Now, denote by B2α,T [6] an aggregate of all the functions of the form

∑

∞

=

0

) ( ) ( )

, (

k

k k t X x u

t x u

Considered in DT, where each of the functions from uk(t)(k=0,1,2...) is continuous on [0,T] and

+∞

<













+













+

≡

∑

∞

= ∞

= −

2 1

1

2 ] , 0 [ 2 2

1

2 ] , 0 [ 1 2 ]

, 0 [

0

(

)

(

)

(

)

(

k

T C k k k

T C k k T

C

u

t

u

t

u

J

λ

α

λ

α ,

where α≥0. The norm in this set is defined as follows:

) ( )

, (

, 2

u J t x u

T

Bα = .

It is known that α T

B₂_, is a Banach space.

3. EXISTENCE AND UNIQUENESS OF THE SOLITION OF THE BOUNDARY VALUE PROBLEM:

Since the system (14) forms a Riesz basis in L₂(0,1)and systems (14), (16) form a system of functions biorthogonal inL2(0,1), each solution of problem (1)-(3), (6) has the form:

∑

∞

= =

0

) ( ) ( )

, (

k

k k t X x

u t x

u , (22)

where

,...) 1 , 0 ( ) ( ) , ( ) (

1

0

= =

∫

u xt Y x dx k t

uk k , (23)

Moreover, Xk(x) and Yk(x) are defined by relations (14) and (16) respectively.

Applying the method of separation of variables for determining the sought functions uk(t) (k=0,1,...), from (1), (2) we have:

) ( ) ( ) ( )

( 0 0

0 t qt u t f t

u′′ = + , (0≤t≤T) (24)

) 0 ,...; 2 , 1 ( ) ( ) ( ) ( ) ( )

( ) 1

( 2 1 2 1 2 1

2 1 2 2

T t k

t f t u t q t u t

uk k k k k

k ′′ +λ = + = ≤ ≤

λ

+ − − − − , (25)

) 0

,...; 2 , 1 ( )) ( ) ( ( 2 ) ( ) ( ) ( ) ( )

( ) 1

( 2 2 2 2 1 2 1

2 2 2

T t k

t u t u a t f t u t q t u t

u k k k k k k k k

k ′′ +λ = + − λ ′′ + = ≤ ≤

λ

+ − − , (26)

k k

u (0)=ϕ , uk′(0)=ψk (k=0,1,...), (27) where

, ) ( ) , ( ) (

1

∫

= f xtY x dx t

f ϕ =

∫

ϕ

1

) ( )

(xY xdx, ( ) ( ) ( 0,1,...)

1

= ψ

=

(5)

Solving problem (21)-(24), we have:

), 0 ( ) ; ( ) ( ) ( 0 0 0 0

0 t t t F u d t T

u t ≤ ≤ τ τ τ − + ψ + ϕ

=

∫

(28)

) 0 ,...; 2 , 1 ( ) ( sin ) ; ( ) 1 ( 1 sin 1 cos ) ( 0 1 2 2 1 2 1 2 1

2 t t t F u t d k t T

u k t k k k k k k k k

k− =ϕ − β +_β ψ − β +_β ₊_λ

∫

− τ β −τ τ = ≤ ≤ , (29)

τ

β

τ

λ

β

ψ

β

ϕ

t

F

u

t

d

t

u

_k t k k k k k k k k

k

(

;

)

sin

(

)

1 (

1 sin

1 cos

)

(

0 2 2 2 2

2

=

+

₊

∫

−

τ

β

ξ

β

ξ

λ

β

λ

ψ

β

ϕ

λ

β

λ

τ

d

t

d

t

u

F

a

t

a

k t k k k k k k k k k k k k k k k k k

)

(

sin

)

(

sin

)

;

(

)

1 (

)

1 (

2

1 cos

sin

1 sin

)

1 (

)

1 (

0 0 1 2 2 2 2 2 1 2 1 2 2 2

−













−

+

−

























−

+

−

∫ ∫

− − − , ) ( sin ) ; ( ) 1 ( 2 0 2 2

2 τ λ −τ τ

λ + β

λ

− a

∫

F u k t d

t

k k k

k ₍_k=₁_,₂_,...;₀≤_t≤_T₎₍₃₀₎

where ) ( ) ( ) ( ) ;

(tu f t q t u t

F_k = _k + _k (k=0,1,2,...).

After substitution of expressionsu0(t), u2k−1(t), u2k(t) of (28), (29), (30), respectively in (22) we have:

)

(

)

;

(

)

(

)

,

(

₀ 0 0 0

0

t

F

u

d

X

x

t

x

u

t













−

+

=

ϕ

ψ

∫

τ

∑

∞

∫

= − − −





−









−

+

1 1 2 0 1 2 2 1 2 1

2

(

;

)

sin

(

)

(

)

1 (

1 sin

1 cos

k k k t k k k k k k k

k

β

t

_β

ψ

β

t

_β

_λ

F

τ

u

β

t

τ

d

τ

X

x

ϕ

∑

∞

∫

=







−

+

1 0 2 2 2

2

(

;

)

sin

(

)

1 (

1 sin

1 cos

k k t k k k k k k k

k

β

t

_β

ψ

β

t

_β

_λ

F

τ

u

β

t

τ

d

τ

ϕ

























−

+

−

₂2 ₂ ₋₁

sin

1 sin

cos

1

₂ ₋₁

)

1 (

)

1 (

k k k k k k k k k k k

t

a

_ψ

β

ϕ

λ

β

λ

( ; )sin ( ) , ( )

) 1 ( 2 ) ( sin ) ( sin ) ; ( ) 1 ( ) 1 ( 2 2 0 2 2 2 0 0 1 2 2 2 2 2 x X d t u F a d t d t u F a k k t k k k k k t k k k k k k     τ τ − λ τ λ + β λ − τ τ − β         ξ ξ − β ξ λ + β β − λ −

∫ ∫

∫

τ − .(31)

Now, proceeding from definition of the solution of problem (1)-(3), (6) similar to [6], the following lemma is proved.

Lemma 2: If

∑

∞ = = 0 ) ( ) ( ) , ( k k k t X x u

t x

u is any solution of problem (1)-(3), (6), the functions u_k(t)(k=0,1,2...) satisfy

system (28)-(30).

Theorem 2: Let

1. q(t)∈C[0,T], β≠±1;

2. ϕ(x)∈C2[0,1], ϕ ′′′(x)∈L2(0,1), ϕ(0)=βϕ(1), ϕ′(0)=ϕ′(1), ϕ′′(0)=βϕ′′(1);

3. ( ) 2[0,1], C x ∈

(6)

Then problem (1)-(3), (6) under small values of T has a unique classic solution.

Proof: Denoting

∑

∞

= =

0

) ( ) ; (

k

k k t u X x

u P

P ,

where P0(t;u), P2k−1(t;u), P2k(t;u) equal the right hand sides of (28), (29), (30), respectively and we write equation

(31) in the form:

u

u=P . (32) We’ll study equation (32) in the space B23,T.

It is easy to see that

1 2

/

1 <βk < ,

2 1 1

0< −β2k < ,

Taking into account these relations, we have:













+

≤

∫

_[₀_, _] 0 _[₀_, _]

2

0

2 0 0

0 ] , 0 [

0

(

;

)

(

)

(

)

_C _T

(

)

_C _T

T

C

T

f

d

T

q

t

u

t

u

t

ϕ

ψ

τ

P

.

2 1

1

2 1 2 3 2

1

2 1 2 3 2

1

2 ] , 0 [ 1

2 3

)

(

2

2 )

(

2 )

)

;

(













+













≤













∑

∞

= −

∞

= −

∞

= − k

k k k

T C k

k

t

u

λ

ϕ

λ

ψ

λ

P

2

1

2 ] , 0 [ 1 2 3 ] , 0 [ 2

1

0 1

2 1

2 ( )) 2 2 () ( ( ) )

( 2

2 _

  

 

λ +

    

  

τ τ λ

+

∫∑

∑

∞

= −

∞

= − k

T C k k T C T

k

k f d Tqt u t

T ,

2 1

1

2 2 3 2

1

2 2 3 2

1

2 ] , 0 [ 2

3

)

(

20 )

(

10 )

)

;

(













+













≤













_∑

∞

= ∞

= k

k k k

T C k

k

t

u

λ

ϕ

λ

ψ

λ

P

2 1

0 1

2 2

(

)

(

20 _











+

_∫∑

∞

= T

k

f

d

T

λ

τ

2 1

1

2 ] , 0 [ 2 3 ]

, 0

[

(

)

(

20 











+

∑

∞

= k

T C k k T

C

u

t

q

T

λ

2 1

1

2 1 2 3 2

1

2 1 2 3

)

(

)

1 (

10 )

(

10 











+













+

∑

∞

= −

∞

= − k

k k k

k

a

T

aT

λ

ϕ

λ

ψ

2 1

1

2 1 2 3 2

1

2 1 2 3

)

(

)

1 (

10 )

(

10 











+













+

∑

∞

= −

∞

= − k

k k k

k

a

T

aT

λ

ϕ

λ

ψ

2 1

1

2 ] , 0 [ 1 2 3 ]

, 0 [ 2

2 1

0 1

2 1 2

)

(

)

(

10

2 )

)

(

10

2 











+













+

∑

∫∑

∞

= −

∞

= −

k

T C k k T

C T

k

k k

t

u

t

q

aT

d

f

T

aT

λ

τ

λ

.

Here, allowing for (18)-(21), we have:

3 , 2 2

2

(

)

(

,

)

(

)

(

,

)

(

)

;

(

_[₀_, _] ₍₀_,₁₎ ₍₀_,₁₎ ₍ ₎ 2 _[₀_, _]

0

T

T C T B

D L L

L T

C

a

x

aT

x

T

a

f

x

t

T

q

t

u

x

t

u

t

≤

ϕ

+

ψ

+

(7)

)

,

(

2

4 )

(

2

4 )

(

2

4 )

)

;

(

T D L x L L k T C k

k

t

u



≤

′′′

x

+

′′′

x

+

T

f

x

t











_∑

∞ = −

ψ

ϕ

λ

P

, (34)

3 , 2

)

,

(

)

(

2

_[₀_, _]

T

B T

C

u

x

t

q

T

+

) 1 , 0 ( ) 1 , 0 ( 2 1 1 2 ] , 0 [ 2 3 2 2

)

(

3 )

1 )(

(

8 )

(

3 )

1 )(

(

8 )

)

;

(

_L _L

k

T C k

k

t

u

ϕ

x

b

ax

a

ϕ

x

ψ

x

b

ax

a

ψ

x

λ



≤

′′′

−

′′

+

′′′

−

′′











_∑

∞ =

P

₍ ₎ ₍₀_,₁₎ 2 2

)

(

8 )

,

(

)

1 )(

,

(

8 T

f

_x

x

t

b

ax

af

x

t

_L _D

aT

x

_L

T

ϕ

′′′

+

−

+

₍₀_,₁₎ 2

)

(

)

1 (

8 a

+

T

ψ

′′′

x

_L

+

3 , 2 2

)

,

(

)

(

)

1 (

2

2 )

,

(

8

₍ ₎ _[₀_, _]

T

T C T B

D L

x

t

T

aT

q

t

u

x

t

f

T

aT

+

, (35)

Now, consider the operator P in the sphere

) 1 ) ( ( 3 , 2 + ≤

=K u AT

K

T B

R from

3 , 2T B , where ) 1 , 0 ( ) ( ) 1 , 0 ( ) 1 , 0

( 2 2 2

2

(

)

(

,

)

4 (

2

2 )

(

)

(

)

(

T

a

x

_L

aT

x

_L

T

a

f

x

t

_L _D

aT

x

_L

A

T

ϕ

ψ

ϕ

+

′′′

=

₍₀_,₁₎ 2

)

(

))

1 (

2

2 (

4 +

a

+

T

ψ

′′′

x

_L

+

₍ ₎ 2

)

,

(

)

2 (

4

T D L

xx

x

t

f

aT

T

+

) 1 , 0 ( 2

)

(

3 )

1 )(

(

8 ϕ

′′′

x

−

b

−

ax

−

a

ϕ

′′

x

_L

+

₍₀_,₁₎

2

)

(

3 )

1 )(

(

8 ϕ

′′′

x

−

b

−

ax

−

a

ϕ

′′

x

_L

+

) ( 2

)

,

(

)

1 )(

,

(

8

T D L

x

t

b

ax

af

x

t

f

T

−

+

(36)

It is seen from (33)-(35) that for any u,u1,u2∈KR the estimates :

3 , 2 3 , 2

)

,

(

)

(

)

(

T T B

B

A

T

B

T

u

x

t

u

≤

+

P

, (37)

3 , 2 3 , 2

)

,

(

)

(

2 1 T T B

B

T

u

x

t

u

−

P

≤

P

, (38)

where ] , 0 [

)

(

))

2

1 (

2 )

2

1 ((

)

(

T

a

T

q

t

_C _T

B

=

+

. (39)

are valid.

Then it follows from estimates (37), (38) that under sufficiently small values of T the operator P acts in the sphere

R

K

K= from 3 , 2T

B and it is contractive. Therefore, in the sphere K =K_R the operator P has a unique fixed point

{ }

u

that is a solution of equation (32).

The functionu(x,t), as an element of the spaceB23,T, is continuous and has continuous derivatives ux(x,t),uxx(x,t)

onD_T. Now, prove that utt(x,t) and uttxx(x,t) are continuous in DT. From (24)-(26) we have:

) 1 , 0 ( ] , 0 [ ] , 0 [ 0 2

)

,

(

)

(

)

,

(

)

(

L T C T

C

f

x

t

q

t

u

x

t

u

′′

≤

+

, (40)

) 1 , 0 ( ] , 0 [ ] , 0 [ 2 1 1 2 ] , 0 [ 1 2 3 2 3 , 2

)

,

(

2

6 )

,

(

)

(

1 (

3 )

)

(

L T C x B T C k T C k

k

u

t

q

t

u

x

t

f

x

t

T

+

≤













_′′

∑

∞ = −

λ

, (41)

3 , 2

)

,

(

)

(

3 (

5 )

)

(

_[₀_, _]

2 1 1 2 ] , 0 [ 2 3 T B T C k T C k

k

u

t



≤

+

q

t

u

x

t











_∑

∞

_′′

=

λ

2 1 1 2 ] , 0 [ 1 2 3 ) 1 , 0 ( ] , 0

[

2

5 (

(

)

,

(

)

,

(

2

10

2













_′′

+

∑

∞ = − k T C k k L T C

x

t

f

x

t

a

u

t

(8)

It follows from estimates (40)-(42) that utt(x,t) and uttxx(x,t) are continuous in DT.Further, it follows from (27) that

) 1 0 ( ) ( ) ( )

( ) 0 ( )

0 , (

0 0

≤ ≤ ϕ

= ϕ

=

∑

∞

= ∞

=

x x

x X x

X u x

u

k k k k

k

k ,

) 1 0 ( ) ( ) ( )

( ) 0 ( )

0 , (

0 0

≤ ≤ ψ

= ψ

= ′

=

∑

∞

= ∞

=

x x

x X x

X u x

u

k k k k

k k

t ,

since by the given theorem

+∞ < ϕ λ

∑

∞

=1 − 2 1 2 3

) (

k

k ,

∑

λ ϕ <+∞

∞

=1

2 2 3

) (

k

k ,

+∞ < ψ λ

∑

∞

=

−

1

2 1 2

3 ₎

(

k

k ,

∑

λ ψ <+∞

∞

=1

2 2

3 ₎

(

k

k ,

and the more so,

∑

ϕ <+∞ ∞

=1

k

k ,

∑

ψ <+∞ ∞

=1

k

k .

Thus, conditions (2) are fulfilled.

It is obvious that conditions (3) is fulfilled for the function

∑

∞

= ∞

=

− − ∞

=

+ +

= =

1

2 2 1

1 2 1 2 0

0 0

) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

, (

k

k k k

k

k t X x u t X x u t X x u t X x

u t x

u .

It is easy to see that

+ λ

+ ′′ λ + + ′′ = −

−

∑

∞

=1 − − −

1 2 1 2 2 1 2 2

0() [(1 ) () ()] ( )

) , ( ) , ( ) , (

k

k k

k xx

ttxx

tt xt u xt u xt u t u t u t X x

u

∑

∞

= − −

+ ′′ λ + λ

+ ′′ λ + +

1

2 1 2 1 2 2

2 2 2

) ( ))] ( ) ( ( 2 ) ( )

( ) 1 [(

k

k k

k k k

k k

k u t u t a u t u t X x (k=1,2,...;0≤t≤T). (43)

Now, if we use systems (25)-(27), equality (40) takes the form:

∑

∞

= ∞

=

−

− + =

+ =

− −

0 1

2 2

1

1 2 1 2

0( ; ) ( ; ) ( ) ( ; ) ( ) ( ; ) ( )

) , ( ) , ( ) , (

k

k k

k

k k

k

k k

xx ttxx

tt xt u xt u xt F t u F t u X x F t u X x F t u X x

u , (44)

Where the functions Xk(x)(k=0,1,2,...) are determined by relation (14), and

) ( ) ( ) ( ) ;

(tu f t q t u t

F_k = _k + _k (k=0,1,2,...).

Under the conditions of the theorem it is obvious that

+∞ <

∑

∞

=0

) ; (

k

k t u

F . (45)

Then it follows from (45) that for any fixed t∈[0,T]:

) , ( ) , ( ) ( )) , ( ( ) ( ) ; (

0

t x f t x u t q t x u F x X u t F k

k

k = ≡ +

∑

∞

=

∀x∈[0,1]. (46)

(9)

Consequently, the function u(x,t) satisfies equation (1) everywhere in DT.

So, u(x,t) is a solution of problem (1)-(3), (6), and by lemma 2 it is unique. The theorem is proved.

By means of lemma 1 we prove the following

Theorem 3: Let all the conditions of theorem 2 and agreement conditions (5) be fulfilled. Then for sufficiently small

values of T, problem (1)-(3) has a unique classic solution.

4. CONCLUSION:

The following results have been obtained:

1. The existence of the solution of a not self-adjoint boundary value problem for a fourth order pseudohyperbolic equation has been proved;

2. The uniqueness of the solution of a not self-adjoint boundary value problem for a fourth order pseudohyperbolic equation has been shown;

3. The existence of the classic solution of a non-classic boundary value problem with integral boundary for a fourth order pseudohyperbolic equation has been proved;

4. The uniqueness of the classic solution of a non-classic boundary value problem with integral boundary for a fourth order pseudohyperbolic equation has been shown.

REFERENCES:

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[2] Bouziani A. Solution forte d’un probleme mixte avec conditions non locales pour une classe d’equations hyperboliques, Bulletin de la Classe des Sciences. Academie Royale de Belgique,8(1997), 53-70.

[3] Ionkin N. I. Solutions of boundary value problem in heat conductions theory with non-local boundary conditions, Differents. Uravn., 13 (1977), 294-304.

[4] Gabov S.A., Orazov B.B. The equation ₂[ ] 0

2

= + − ∂

∂

xx xx u u

u

t and several problems associated with it, USSR Computational Mathematics and Mathematical Physics, 26 (1986), 58–64.

[5] Kasumov T.B., Mirzoyev V.S. On one generalization of Ionkin’s example, The abstracts of scientific conference devoted to 100th anniversary of the honored scientist acad. A.I.Huseynov, (2007), 87-88.

[6] Khudaverdiyev K.I., Azizbekov E.I. Investigation of classical solution of an one-dimensional not self-adjoint mixed problem for a class of semi-linear pseudopseudohyperbolic equations of fourth order, Proc. Baku State University, ph ys.-math.ser., 2 (2002), 114-121.

[7] Mehraliyev Y.T., Yusifov M.R. The solution of a boundary value problem for a second order parabolic equation with integral conditions, Proceedings of IMM NAS of Azerb., 30(2009), 91-104.

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