About Uniform Limitation of Normalized Eigen Functions of T. Regge Problem in the Case of Weight Functions, Satisfying to Lipschitz Condition

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About Uniform Limitation of Normalized Eigen Functions of T. Regge Problem in the Case of

Weight Functions, Satisfying to Lipschitz Condition

1Jwamer .K.H and ²Aigounv .G.A

1University of Sulaimani, College of Science, Department of Mathematics, Sulaimani, Iraq

2Daghestan State University, College of Mathematics, Department of Mathematical Analysis, South of Russian

Abstract

In this work, we estimate normalize eigenfunctions to the T.Regge problem whenever the weight functions satisfies Lipschitz condition.

KeyWords: Eigenfunctions, normalize, Lipschitz condition

1. INTRODUCTION

Let's consider spectral problem (q(x)∈C_{[ ]}₀_;_a ,ρ(x)∈Lip1and m≤ρ(x)≤M ⁾

(2)

) ( ) ( )

( ) ( )

(x q x y x ² x y x

y′′ + =λ ρ

− (0<x<a)

(1)

0 ) 0

( =

y , y′(a)−iλy(a)=0 (2)

1 )

( ) (

2 1

0

2  =











∫

^a^ρ ^x ^y ^x ^dx ^{, where}^λ - is spectral parameter.

(3)

The problem (1) - (2) arises in different questions of mathematical physics. T.Regge [1], who studied it (in case of ρ(x)≡1) in connection with the theory of dispersion has shown, that if q(x) in left semi-neighborhood of point a satisfies to condition q(x)~ C_µ(a−x)^µ x→a−0; µ≥0C_µ ≠0, the problem has discrete spectrum λ_n and system of eigenfunctions of problem (1) - (2) is full of century L²_{[ ]}_{0 a}_, . In work [2] is studied asymptotes of proper values and received 2 multiple decomposition in uniform converging numbers on eigenfunctions from which 2 multiple completeness of eigenfunctions of century L²_{[ ]}_{0 a}_, . In case of equation of n2 order and ρ(x)≡1 similar problem is considered in works [3] - [4]. And for ρ(x)≡/1 asymptotic of eigenvalues for more general problem is studied in works [5] - [6].

In work [7] is considered the case of weight functions close to Holder class where maximal growth rate of eigenfunctions of problem (1) - (3) is studied.

The purpose of the present work is reception of uniform estimations for normalized eigenfunctions of problem (1) - (3) in case of weight functions, satisfying to Lipschitz condition.

The following is true:

Lemma: For any ρ(x)∈Lip1^andε >⁰ there is function 2[0, ]

)

(x ∈C a

ρε

such, that ρ_ε(a)=ρ(a), ρ_ε(0)=ρ(0), x dx x dx

a a

∫

⁼

0 0

) ( )

( ρ

ρ_ε ^,

(3)

N x x

x x a

a

x ( ) ( ) ,max ( ) 2

max[0,] − ≤ [0, ] ′ ≤

∈

∈ ρ ρε ε ρε ^and

ρ_ε x ε^c

a

x ′′ ≤

∈ ( )

max[0, ] , where C is constant independent from ρ(x)^andε^.

Proof:

Let’s divide interval

[ ]

⁰^,^a ^on m equal parts ( m -arbitrary) by points0=x₀ <x₁ <...<x_m₋₁ <x_m =a, and middle of intervals [x_i₋₁,x_i] we shall designate through x′_i(such pointsm , namely x₁′,x′₂,...,x_m′ ). We shall consider broken line, that connected points(x₀,ρ(x₀)),(x₁,ρ(x₁)),...,(x_m,ρ(x_m)).

Obviously, this broken line is function graphρ₀(x), satisfying to inequalities

m a x N x

a x

) . ( ) (

max ₀

] , 0

[ − ≤

∈ ρ ρ and ρ₀′(х) ≤N there, where ρ₀′(x) exists.

Let's consider now other broken line connecting points(x₀,ρ₀(x₀)),(x₁′,ρ₀(x₁′)),(x₂′,ρ₀(x₂′))...,(x_m′ ,ρ₀(x_m′ )),(x_m,ρ₀(x_m)).

Obviously, this broken line is the schedule of function ρ₁(x), satisfying to parities ρ₁′(х) ≤N there, where ρ₁′(x) exists and

m a x N x

a x

) . ( ) (

max ₁

] , 0

[ − ≤

∈ ρ ρ ^.

On sites [x′_i,x′_i₊₁] where i=1,2,...,m−1 we shall construct curve ρ_ε(x) as

polynomials parities ₁ ₃ ⁴ ( )²

4 ) 3 8 (

) ( )

( ⁱ _i pⁱ x x_i

x p x

x

x −

− ∆

∆ − +

=ρ

ρ_ε where

2m, x a x_i − _i′=

=

∆ 2

) ( )

( ₀ ₁

0′ ′ − ′ ′+

= ⁱ ⁱ

i

x p p ρ x

. On sites [0,x′₁] and [x_m′ ,a]we shall get ρ_ε(x)=ρ₁(x) (in the same place ρ₁(x)=ρ₀(x)).

Let's put =2.[ ]+2 ε

m Na . Direct check shows, that all conditions of

lemma except for equality x dx x dx

a a

∫

⁼

0 0

) ( )

( ρ

ρ_ε are executed. In addition,

inequality ρ_ε′(x) ≤N takes place ( N and N2 in condition of lemma) now let’s

(4)

find number δ ^from ^condition ^x ^dx ^x ^dx x a

a a

∫

⁺ ⁼

0 0

) ( )

sin 1 ( )

( δ π ρ

ρ_ε ,

hence

∫

⁻

= _a

a

dx x a x

dx x x

0 0

) ( sin

) ) ( )

( (

ε ε

π ρ ρ ρ

δ . Obviously at small ε number δ is also not

enough, and function ( ) ( )(1 sin x)² x a

x ρ δ π

ρ_ε = _ε + satisfies to conditions of lemma.

Let's designate through Q_[₀_,_a_] class of continuous on

[ ]

⁰^,^a ^functions^q^(x⁾^,

satisfying to inequality _Q

a

C dx x q <

∫

¹

0

)

( , where C_Q =contand[a₀,a₁]⊆[0,a].

Let's consider countable subset

{

q_i(x) i∈N

}

≡Q[0,_a] of class Q_[₀_,_a_] satisfying to condition ^Lim_i_→_∞

∫ ∫

^{x t} ^qⁱ ^s ^dsdt ^≡ ^f^o ^x

0 0

) ( )

( , where f_o(x) function satisfying to Lipschitz condition, and convergence is uniform on

[ ]

⁰^,^a ^.

Let ρ ≠1,ρ>0,λ∈C− is complex, Im(λ)<const (that is ρ - is fixed and λ- is arbitrary of strip Im(λ)<const of complex plane).

Let's designate through y(x,λ,q) solution of Cauchy problem

. 1 ) 0 ( , 0 ) 0 (

) , 0 ( , ) ( )

( ) ( )

( ²

′ =

=

∈

= +

′′

−

y y

a x x y x

y x q x

y λ ρ

Then the following is true:

Theorem: There is constant C₀ ≡C₀(Q_[₀_,_a_]) (uniform for all classQ_[₀_,_a_]) such, that

0 2 1

0 ] 2 , 0 [

) ) , , ( (

) , ,

max ( C

dx q x y

q x y

a a

x <

∈

∫

λ ρ

λ for every value large enough by module λ.

(5)

From this theorem and previous lemma follows important consequence Consequence: Let q(x)- is continuous function, andρ(x)∈Lip1. Then solution of Cauchy problem

. 1 ) 0 ( , 0 ) 0 (

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

( ) ( )

( ²

′ =

=

≠

∈

=

′′ +

−

y y

a a x x y x

y x q x

y λ ρ ρ

Satisfies to parity < <∞

∈

∫

const

dx x y x

x y

a a x

2 1

0 ] 2 , 0 [

) ) ( ).

( (

) max (

ρ

For every value large enough λ from stripIm(λ)<const. Proof:

As solution of Cauchy problem continuously depends on weight function )

ρ(x and functional

∫

^a ^x ^y ^x ^dx

0

2 1

2 )

) , ( ) (

( ρ λ also continuously depends onρ(x).

Hence, functional

2 1

0

2 ] , 0 [

) ) , ( ).

( (

) , ( max

∫

∈ a

a x

dx x

y x

x y

λ ρ

λ

also continuously depends on weight

functionρ(x). Hence, there is number ε(R)such, that

2 1

0

2 ]

, 0 [ 2

1

0

2 ]

, 0 [

) ) , , ( ).

( (

) , , ( max 2.

1 ) ) , , ( ).

( (

) , , ( max

∫

∈

∈ ≥ _a ^x ^a

a a x

dx x

y x

x y dx

x y x

x y

ρ λ ρ

λ ρ

ρ λ

, if λ ≤R and

) ( ) ( )

(x ρ x ε R

ρ − ≤

Where R>0is arbitrary constant . Let’s take some R and by ε(R) and lemma let’s plot function ρ_ε(x) approaching ρ(x)⁽ρ⁽^x⁾−ρ⁽^x⁾ ≤ε⁽^R⁾)

) ( ) (a ρ a

ρ_ε = . Now let’s consider Cauchy problem with weight functionρ_ε(x) instead of ρ(x). In this problem 2[0, ]

)

(x ∈C a

ρε and consequently we can make

(6)

double replacement , ( ) ( ). ( ( )) )

0 (

2 y x A x x

t A

x dt

ξ η

ξ =

∫

= ^,

where ( ) ( ). ⁴( )

1 4

1

a x x

A =ρ⁻ ^ε ρ .

As a result of such replacement we shall obtain problem:

) ( ) ( )

( ) ( )).

( ) ( ) ( ( )

(ξ ³ η ξ λ²ρ η ξ

η′′ + q x A x −A′′ x A x = a

− , )

) , (

0 (

0

∫

2

∈ ^a

t A

ξ dt ,

, 0 ) 0

( =

η

) 0 ( ) 0 ( = A

η′ ^, Âs Â⁽⁰⁾=ρ⁻⁴¹⁽⁰^).ρ⁴¹⁽â⁾and

∫

â ⁼ â ⁼ â ^t ^dt

a t

a dt t

A dt

0 0

0

2 ( )

) ( 1 ))

( ) ( ( )

( ^ε

ε

ρ ρ ρ

ρ

const a a

dt t

a

=

∫

) (

0

ρ ρ

(Independent from ε), designating

) ( )]

( ) ( ) ( )[

3(x q x A x A x q_ε ξ

A − ′′ ≡ we shall obtain problem:

4 2

) 0 (

) ) (

0 ( , 0 ) 0 (

, ) , 0 ( , ) ( ) ( )

( ) ( ) (

ρ η ρ

η

ξ ξ η ρ λ ξ η ξ ξ

η _ε

a

a a

q

′ =

=

∈

=

′′ +

−

Estimated in theorem functional does not depend on value y′(0) (as all solutions of our equation, satisfying to condition y(0)=0, can be obtained from solution of problem with conditions y(0)=0,y′(0)=1, by multiplication to constant, which will be reduced in our functional) and consequently if to show, that

∫

^t^q ^d

0

) (ξ ξ

ε in regular intervals on ε and t∈[0,a] is limited for all small

>0

ε under theorem there will be constant C₀ >0such, that

(7)

, )

) ( ).

( (

)

max ( ₀

2 1

0

2

C d

a

a ≤

∫

^ρ ^η ^ξ ^ξ

ξ η

From here and from parities

∫

=

= ^x

t A x dt x

x A x y

0 2( ) )

( )), ( ( ).

( )

( η ξ ξ obviously follows, that exists C₀ >0such,

that ₀

2 1

0 ] 2 , 0 [

) ) , , ( ).

( (

) , ,

max ( C

dx x

y x

x y

a a

x <

∈

∫

ρ λ ρ

ρ λ

For every λ ≤R, and by arbitrariness R , and for all considered λ^(let’s remind, that Im(λ)<const^).

Let's estimate

∫

^t^q ^d

0

) (ξ ξ

ε . Passing to variable x in integral we shall get

′′ = +

′′ ≤

−

′′ =

−

′ =

− ′′

=

∫

−

s S

s s

t

dx x A x A x dx

A x dx q

x A x A x dx

A x q

x A x dx A x A x A x q dx x x A x A x A x q d

q

0 0

2 0

0 2

0

2 3 0

3 0

) ( ) ) (

( ) ) (

( ) (

) ) ( ( )]

( ) ( ) ( [ )

( ).

( )]

( ) ( ) ( [ )

(ξ ξ ξ

ε

].

, 0 [ ,

)]

( [ ) 0 ( ) 0 (

) ( ) ) (

( ) )] (

( [ )]

( ) ( ) [

( ) (

0

2

0 2 0

2 0

0 2

a s where dx

x A A

A

s A s A x dx

A x dx q

x A x

A x A x dx

A x q

s

S s

s S

′ ∈

′ +

′ + +

′ ≤

′ − +

∫

− −

From definition ⁴ ) (

) ) (

( x

x a A

ρρε

= follows, that

4 4

) 0 (

) ( )

0 (

) ) (

0

( ρ

ρ ρ

ρ

ε

a

A = a = ,

(8)

4 5 4 4

5 4

1

) ( 4

) ( . ) ) (

( ).

4 ( ) 1 (

x x x a

x a

x A

ε ε ε

ε ρ

ρ ρ ρ

ρ

ρ ′ =− ^′

−

′ = ⁻ ,

4

4 5

4

) 0 (

) . ( ) 0 ( 4

) 0 ( )

0 ( 4

) 0 ( . ) ) (

0

( ρ

ρ ρ

ρ

ρ _ε

ε

ε a

A a ′ = − ′

−

′ = and consequently

∫

^≤ ⁺ ^′ ⁺ ^′ ⁺ ^′

a a

t

s s a a

s s dx a

a x x d q

q

0

5 2 3

0

4 4

0 16. ( )

)]

( .[

) ( )

0 (

) . ( ) 0 ( . 4

) 0 ( )

( . 4

) ( . ) ) (

( ). (

) ) (

(

ε ε ε

ε ρ

ρ ρ ρ

ρ ρ

ρ ρ ρ

ξ ρ ξ

On lemma ρ_ε′(x) ≤2.Nand ρ(x)−ε ≤ρ_ε(x)≤ρ(x)+ε, hence

. ] ) ( [ . 4

. ) (

) 0 (

) . ( ) 0 ( ] 2

) min ( .[

2

. ) ) (

( ). (

) ) (

(

0

5 2

0 2

3

] , 0 [ 4

0 4

∫

+ −

+

− +

+

≤

∈

a a

a x t

dx x

N a

a N

x N dx a

a x x d q

q

ε ρ ρ

ρ ρ ε ρ

ρ ε ρ

ρ ρ ξ

ε ξ

As ε is not enough, consequence is proved.

Let's prove now theorem for what we shall evaluate ϕ₀′(^x,λ),ϕ(^x,λ) and )

, ( ( ) ,

( λ ϕ λ

ϕ′ x x is solution of equation (1) satisfying to entry conditions ,

1 ) , 0 ( , 0 ) , 0

( λ = ϕ′ λ =

ϕ ^andϕ0⁽^x^,λ⁾ is solution of such Cauchy problem with constant coefficientρ(x)≡ρ). As it has been established [8, p. 22]

) sin . cosh cos

. sinh ( )

,

( ₂ ₁ ₁ ₁ ₁

1 2 1

1 1

0 i x x i x x

x σ δ σ δ

δ σ

δ λ σ

ϕ − +

+

−

= − and

consequently ϕ₀′(x,λ)=(coshσ₁x.cosδ₁x−isinhσ₁x.sinδ₁x)(simple

transformations are lowered). As at greater λ parities σ₁≈ ρ.σ and δ₁≈ ρ.δ take place, then obviously ϕ′₀(x,λ) <const is regular on x∈[0,a] and λ^from considered strip. Let’s consider number

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1 0

1 0 0

0 1 0 2 0

1 1

1 0 1 0 1

0

....

) , ( ) , (

] ) ( [

...

) , ( . ] ) ( [ )

, ( ) , ( ].

) ( [ ) , ( ) , (

1

τ τ λ τ ϕ λ τ τ ϕ τ

λ τ ϕ τ

τ λ τ ϕ λ τ ϕ τ

λ ϕ λ ϕ

τ

d d q

q

x q q

d x

q q

x x

i i i

i i

i x x

o

i − −

−

− +

−

− +

=

−

∞

=

∫

∑ ∫ ∫

∫

−

Let's enter designations f_i(x)for i− member of series

∫

⁻ ⁻

=

x

d x

q q

x f

0

1 1 0 1

1( ) [ ( ) ]. ( , ) ,...)

( τ ϕ τ λ τ , As result we shall get:

∑

^∞

=

−

1

0( , ) ( )

) , (

i i x f x

x λ ϕ λ

ϕ (4)

If i≥1 , then obviously

∫

⁻ ⁻

+ =

x

i

i x q q x f d

f

0

1 1 1

0 1

1( ) [ (τ ) ]ϕ ( τ ,λ). (τ ) τ and consequently, integrating in parts, we shall obtain

{

1 0

0

1 1

0 1 1

0 0 0

1 1

0 1

] ) ( [

. )]

( ).

, ( ) ( ).

, ( [ ]

) ( [ ).

( ).

, ( )

(

1

τ

τ λ τ ϕ τ λ τ ϕ τ

λ τ

ϕ ^τ

d ds q s q

f x

ds q s q f

x x

f

x

i i

x

i i



− 

′ −

−

 −









 − −

=

∫

+

Or

∫

^′ ⁻ ⁻ ^′ ⁻ ^′ ⁻

+ =

τ τ

τ λ τ ϕ τ λ τ ϕ

0 0

1(x) [ (x , )f ( ) (x , )f ( )]. [q(s) q]dsd f

x

i i

i ,

(5)

Absolutely similarly for f₁(x)it is possible to get parity

∫

^′ ⁻ ⁻ ^′ ⁻ ^′ ⁻

= ϕ τ λ ϕ τ ϕ τ λ ϕ τ ^τ τ

0 0

1(x) [ (x , ) ( ) (x , ) ( )]. [q(s) q]dsd

f

x

(6)

(10)

Differentiating parities (5) and (6) on x we shall get parities for derivatives

, ] ) ( [ )].

( ) , (

) ( ) , ( ] [ [ ] ) ( [ ).

( ) (

0 0

0

0 0

2 1

∫

∫ ∫

′ −

−

− +

−

′ =

+

τ τ τ

τ λ τ ϕ

τ λ τ ϕ ρ λ τ

τ

dsd q q f

x

f x q

d q q x f x f

i

x x

i i

(7)

, ] ) ( [ )].

, ( ) , (

) , ( ) , ( ] [

[ ] ) ( [ ).

, ( ) (

0 0

0

0 0

2 0

1

∫

∫ ∫

′ −

−

− +

−

′ =

τ τ τ

λ τ ϕ λ τ ϕ

λ τ ϕ λ τ ϕ ρ λ τ

τ λ ϕ

dsd q q x

x q

d q q x

x f

x x

(8)

(Here, it is considered, that ϕ₀′′(^x,λ)≡(^q−λ²ρ)ϕ₀(^x,λ).)

From choice of class Q_[₀_,_a_] follows, that

∫

^τ ^q ^s −^q ^ds <^Q=^const<∞

0

] ) ( [

and consequently from (5) and (8) follows, that

] )

. [(

) ( . ,

. ) 2

( ⁰ ⁰ ₁ ⁰ ₂ ₀² ₀²

1 q C C

C Q x Q f C x C x Q

f ′ ′ ≤ + + + ′

≤ ρ λ

λ

λ ^{, where} ^C⁰

and C′₀

Such constants for which inequalities

λ λ ϕ₀( , ) C⁰

x ≤ and ϕ₀′(x,λ) ≤C₀′

are executed (as ₂

1 2 1

2 0

sin ) sinh

,

( δ σ

δ λ σ

ϕ +

≤ x+ x

x then, obviously C₀ and C′₀

exist). Using recurrent parities (5) and (7) we shall obtain, that number (4) converges and moreover

λ λ ϕ λ

ϕ( , ) 0( , ) C⁰ x

x − ≤ (evaluations f₂(x), f₃(x),..., and f₂′(x), f₃′(x),... are made consistently for i=2,3, …).

(11)

Then ϕ x λ ϕ x λ qτ q ϕ x τ λ dτ

x

) , ( ] ) ( [ ) , ( ) ,

( ₀

0

0′ + − ′ −

′ =

∫

, or, integrating in

parts, we shall get

{

^[ ⁽ ^, ⁾ ⁽ ^, ⁾ ⁽ ^, ⁾^. ⁽ ^, ^)] ^[ ⁽ ⁾ ^] ^,

] ) ( [ ).

, ( ) , ( )

, ( ) , (

0 0

0 0 0

0

τ λ

τ ϕ λ τ ϕ λ τ ϕ λ τ ϕ

λ τ ϕ λ τ ϕ λ ϕ λ ϕ

τ τ

d ds q s q x

x

ds q s q x

x x

x



− 

′′ −

′ −

−

 −





 ′ − −

′ =

′ −

∫

As ϕ₀′(0,λ)≡1 and ϕ₀′′(x−τ,λ)≡(q−λ²ρ)ϕ₀(x−τ,λ),

{

∫ ∫

∫ ∫ ∫

∫

∫ ∫







 ′ − −

−







 − − −

+

−

=

−

′ −

−

′ =

′ −

x

x x a

a

x x

d ds q s q x

q ds

q s q x

d ds q s q x

q

x ds

q s q x x

x

0 0

0

0 0 0

0 2

0 0

2

0 0

, ] ) ( [ ) , ( ) , (

] ) ( [ ).

, ( ) , ( ) (

] ) ( [ ).

, (

} ] ) ( [ )].

, ( ) , ( ) (

) , ( ) , ( [ ]

) ( [ ).

, ( ) , ( ) , (

τ λ

τ ϕ λ τ ϕ

τ λ

τ ϕ λ τ ϕ ρ λ λ

ϕ

τ λ

τ ϕ λ τ ϕ ρ λ

λ τ ϕ λ τ ϕ λ

ϕ λ ϕ λ ϕ

τ

Subtracting and adding in last integral ϕ₀′(τ,λ)to ϕ′(τ,λ) and representing integral in the form of sum of two integrals we shall obtain

∫ ∫ ∫







 − − −

− +

−

′ =

′ x − x x ^x q s qds q ^x x q s qds d

0 0 0

0 2

0( , ) ( , ). [ ( ) ] ( ) ( , ) ( , ). [ ( ) ]

) ,

( λ ϕ λ ϕ λ λ ρ ϕ τ λ ϕ τ λ τ

ϕ ^τ

τ λ

τ ϕ λ τ ϕ λ τ ϕ τ λ

τ ϕ λ τ

ϕ x ^τ q s qds d x ^τ q s qds d

x x

∫ ∫ ∫ ∫







 ′ − ′ − ′ −

 −





 ′ − ′ −

−

0 0 0

0 0

0( , )[ ( , ). [ ( ) ] ( , )[ ( , ) ( , )]. [ ( ) ]

From this equality follows, that

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∫ ∫

∫

− ′

′ −

+

′ −

′ − +

−

− +

−

′ ≤

′ −

x

x x

d ds

q s q x

d ds q s q x

d ds q s q

x q

ds q s q x

x x

0

0 0

0

0 0

0

0 0 2

0 0

) , ( ) , ( . ] ) ( [ . ) , (

] ) ( [ . ) , ( . ) , ( ]

) ( [

. ) , ( . ) , ( . ]

) ( [ . ) , ( ) , ( ) , (

τ λ τ ϕ λ τ ϕ λ

τ ϕ

τ λ

τ ϕ λ τ ϕ τ

λ τ ϕ λ τ ϕ ρ

λ λ

ϕ λ ϕ λ ϕ

τ

τ τ

And consequently, using estimations obtained before for ,

) ,

0( λ

ϕ x ϕ(x,λ) , ϕ₀′(x,λ) and considering, that

, )

( )

( ]

) ( [

0 0

∞

<

+

= +

≤

−

∫ ∫ ∫

∫

^q ^s ^q ^ds ^τ ^q ^s ^ds ^τ ^qds ^q ^τ ^q ^s ^ds ^const

τ τ we shall obtain

τ λ τ ϕ λ τ ϕ τ λ

ϕ λ

ϕ^′ ^x ⁻ ^′ ^x ^<^R⁺

∫

^x^B ^′ ⁻ ^′ ^d

0

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

) , (

, where R>0 and B(τ)>0 are limited.

Hence, by Gronwall’s lemma ∫ ≤ <∞

′ ≤

′ x − x Re const

x

d B

0

) (

0( , ) .

) , (

τ

λ τ

ϕ λ

ϕ .

As ϕ′₀(x,λ) ≤const<∞, from last inequality follows, that

∞

<

′(x,λ) ≤const

ϕ is regular on x∈[ a0, ]and λ from considered strip.

Let now ϕ(x,λ) reaches maximum ϕ_m in point x₀∈[0,a],

Maximum ϕ′(x,λ) is equal ϕ_m′ .Then function graph ϕ(x,λ) lies above triangle with top in point (x₀,ϕ_m)and lateral faces with angular coefficients

m

m ϕ

ϕ′ ,− ′ accordingly.

And consequently

x y

0 a x

0

ϕm

•

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. ] ) ( [ 3 ]

) [(

.

] ) ( ) ( . 2 [ .

] ) ( ) ( . 2 [ . )

( .

) ( )

, ( . )

, ( ) (

2 0 2

2 0 3 0 3 2 0

2

2 0 2 0 2

0

2 0 2 0 2 2

0

0 0

2 0 2

2

0 0

2

0 0

0







 + ′ − + − ′ + −

=

′ − +

′ −

−

+

′ − +

=

′ −

− +

′ − +

≥

=

≥

∫

∫ ∫

x a x x

x a a

m

dx x x x

x m

dx x x x

x m

dx x x m

dx x

m dx x

x

m m m

m m

a

x

m m

m m a

x

m m

m m m

m

a x

m m

a a

ϕ ϕ ϕ ϕ

ϕ

ϕ ϕ

ϕ ϕ ϕ

ϕ

ϕ ϕ λ

ϕ λ

ϕ ρ

From inequality we got follows, that

m m m

m a

a x

x a x x

x a a

dx m x x

x

ϕ ϕ ϕ

λ ϕ ϕ ρ

λ ϕ

− ′ +

′ − +

+ −

≤

∫

∈

] ) ( [ ) 3 .(

) . (

1 )

) , ( ) ( (

) , ( max

2 0 2

0 2 3

0 3 2 0

1

0

2 ] , 0 [

.

If to enter designations x₀ =ε.a and z

m m′ = ϕ

ϕ we shall get

4].

) 1 3 12(

) 1 2 )( 1 2 ( [

] ) 1 ( 3 [

) 1 (

] ) ( [ ) 3 .(

) (

2 2

2 2 3 3 3 3 3

2 0 2

0 2 3

0 3 0

+

− +

−

=

− + + −

+ −

=

′ =

− +

′ − +

+ −

az az

az a

z a

a a z

a a

x a x x

x a a

m m m

m

ε

ε ε ε

ε

ϕ ϕ ϕ

ϕ

Where ε∈[0,1] and z∈(0,∞). Let’s enter now designations

4 ) 1 3 12(

) 1 2 )( 1 2 ( ) ,

( z =az az− − ² + az− ² +

f ε ε and estimate from below

) , ( z

f ε . It is obvious, that

4. ] 1 4 ) 3 2 [( 3 3 1

4 ) 1 9 6 12(

) 1 2 4(

) 1 , 1 ( ) , 0 (

2

2 2 2

2

≥ +

−

=

+ +

− +

−

=

az

az z

a az

z a z

f z f

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Inside of interval 0≤ε ≤ 1 is unique critical point 2

= 1

ε , in this point we

have .

4 1 4 ) 1 3 12(

) 1 2,

(1 z = az− ² + ≥

f . Hence, for any ε∈[0,1] and z∈(0,∞) estimation

4 ) 1 , ( z ≥

f ε is fair, therefore inequality is

a a m

dx m x

x

a

a x

. 2 4 .1 .

1 )

) , ( ) ( (

) , ( max

2 1

0

2 ] , 0

[ ≤ =

∫

∈

λ ϕ ρ

λ ϕ

So theorem is proved.

Thus we have proved, that normalized eigenfunctions of problem (1) - (3) in case of weight functions satisfying to Lipschitz condition are limited in regular intervals, the obtained result is proved by statement proved in [7] at α =1^{, as}

0 )

(

lim ₀

2 1 _____

] , 0

[ = >

∞ −

→ С

x y

n n C n

a

λ α ^.

References

[1] T.Regge. Analytical properties of matrix of dispersion. Mathematics (Col.

Translations) 7 (4) (1963), 83-89.

[2] Kravitsky A.O. About decomposition in number by eigenfunctions of one non self-interfaced regional problem // Daghestan Academy of science USSR, T. 170 (6) (1966), 1255-1258.

[3] Gehtman М.М. About some analytical properties of kernel resolvent of ordinary differential operator of even order on Riemann surface, Daghestan academy of science USSR, Moscow, 201(5) (1971), 1025 - 1028.

[4] Aigunov G.А. About one boundary value problem generated by non self- adjoin differential operator of n2 order on axle, Daghestan academy of science USSR, Moscow, 213(5) (1973), 1001-1004.

[5] Aigunov G.A. Spectral problem of T.Regge type for ordinary differential operator of n2 order // Col. Functional analysis, theory of functions and their

(15)

appendices, Issue 2. P.I. Makhachkala, 1975, 21-41.

[6] Aigunov G.A., Gadzhiev T.U. Studying of asymptotic of eigenvalues of one regular boundary value problem, generated by differential equation of n2 order on interval [ a // News of high schools of North Caucasus, Rostov-on-Don, 0, ]

№5, 2008,14-19.

[7] Aigunov G.A., Jwamer K.H, Dzhalaeva G.A. Asymptotic behavior of normalized eigenfunctions of problem of T.Regge type in case of weight functions close to functions from Holder classes. Journal of mathematics collection, Makhachkala (South of Russian), 2008, 7-10.

[8] Aigunov G.A., Gadzhiev T.U. Estimation of normalized eigenfunctions of problem of T.Regge type in case of smooth coefficients. Interhighschool Col.

«FDU and their appendices », Makhachkala, 2009, 18-26.