Gen. Math. Notes, Vol. 1, No. 2, December 2010, pp. 115-129
ISSN 2219-7184; Copyright © ICSRS Publication, 2010
www.i-csrs.org
Available free online at http://www.geman.in
About Uniform Limitation of Normalized Eigen
Functions of T. Regge Problem in the Case of
Weight Functions, Satisfying to Lipschitz
Condition
1
1
Jwamer .K.H and 2Aigounv .G.A
University of Sulaimani, College of Science, Department of Mathematics,
Sulaimani, Iraq
2
Daghestan 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) ∈ Lip 1 and m ≤ ρ ( x) ≤ M )
About uniform limitation of normalized…
116
− y ′′( x) + q ( x) y ( x) = λ2 ρ ( x) y ( x)
(0 < x < a)
(1)
y ′( a) − iλy (a) = 0
y ( 0) = 0 ,
(2)
1
a
2
 ∫ ρ ( x) y ( x) 2 dx  = 1 , where λ - is spectral parameter.


0

(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 ≥ 0 C µ ≠ 0 , the problem
has discrete spectrum λn 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 2n 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) ∈ Lip 1 and ε > 0 there is function ρ ε ( x ) ∈ C 2 [ 0, a ]
such,
that
a
a
0
0
ρ ε (a ) = ρ (a ) , ρ ε (0) = ρ (0) , ∫ ρ ε ( x)dx = ∫ ρ ( x)dx ,
Jwamer .K.H and Aigounv .G.A
117
max ρ ( x) − ρ ε ( x) ≤ ε , max ρ ε′ ( x) ≤ 2 N
x∈[ 0 , a ]
x∈[ 0 , a ]
and
max ρ ε′′ ( x) ≤
x∈[ 0, a ]
c
ε
, where C is
constant independent from ρ ( x) and ε .
Proof:
[0, a ]
Let’s divide interval
on m
equal parts ( m -arbitrary) by
points 0 = x0 < x1 < ... < x m −1 < xm = a , and middle of intervals [ xi −1 , xi ] we shall
designate through xi′ (such points m , 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 graph ρ 0 ( x) , satisfying to
inequalities max ρ ( x) − ρ 0 ( x) ≤
x∈[ 0, a ]
Let's
consider
N .a
and ρ 0′ ( х) ≤ N there, where ρ 0′ ( x) exists.
m
now
other
broken
line
connecting
points ( x0 , ρ 0 ( x0 )), ( x1′ , ρ 0 ( x1′ )), ( x 2′ , ρ 0 ( x2′ ))..., ( x ′m , ρ 0 ( x ′m )), ( x m , ρ 0 ( 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 max ρ ( x) − ρ1 ( x) ≤
x∈[ 0 , a ]
N .a
.
m
On sites [ xi′ , xi′+1 ] where i = 1,2,..., m − 1 we shall construct curve ρ ε (x) as
polynomials
∆ = xi − xi′ =
ρ ε ( x) = ρ 1 ( x ) +
parities
pi
3p
( x − xi ) 4 − i ( x − xi ) 2 where
3
4∆
8∆
ρ ′ ( x ′ ) − p 0′ ( xi′+1 )
a
. On sites [0, x1′ ]
, pi = 0 i
2m
2
and [ x′m , a ] we shall
get ρ ε ( x) = ρ1 ( x) (in the same place ρ1 ( x) = ρ 0 ( x) ).
Let's put m = 2.[
Na
ε
] + 2 . Direct check shows, that all conditions of
a
lemma except for equality
∫
0
a
ρε ( x)dx = ∫ ρ ( x)dx are executed. In addition,
0
inequality ρ ε′ ( x) ≤ N takes place ( N and 2 N in condition of lemma) now let’s
About uniform limitation of normalized…
find
δ
number
118
a
condition ∫ ρε ( x) (1 + δ sin
from
0
a
hence δ =
ρ ( x) − ρ ε ( x) )dx
∫(
0
π
a
∫ sin a x
π
a
a
x)dx = ∫ ρ ( x) dx ,
0
. Obviously at small ε number δ is also not
ρ ε ( x) dx
0
enough, and function ρ ε ( x) = ρ ε ( x)(1 + δ sin
π
a
x) 2 satisfies to conditions of
lemma.
Let's designate through Q[ 0, a ] class of continuous on [0, a ] functions q(x) ,
a1
satisfying to inequality ∫ q( x)dx < C Q , where C Q = cont and [a 0 , a1 ] ⊆ [0, a] .
a0
Let's consider countable
subset
{q ( x)
i
i ∈ N } ≡ Q [ 0,a ] of class Q[ 0, a ]
x t
satisfying to condition Lim ∫ ∫ qi ( s )dsdt ≡ f o ( x) , where f o ( x) function satisfying
i →∞
0 0
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
− y ′′( x) + q ( x) y ( x) = λ2 ρ y ( x) , x ∈ (0, a )
y(0) = 0 , y ′(0) = 1.
Then the following is true:
Theorem: There is constant C 0 ≡ C 0 (Q[ 0, a ] ) (uniform for all class Q[ 0, a ] )
such, that
max
x∈[ 0, a ]
y ( x, λ , q )
a
( ∫ ρ y ( x, λ , q ) dx)
2
0
1
2
< C 0 for every value large enough by module λ .
Jwamer .K.H and Aigounv .G.A
119
From this theorem and previous lemma follows important consequence
Consequence: Let q(x) - is continuous function, and ρ ( x) ∈ Lip 1 . Then
solution of Cauchy problem
− y ′′( x) + q ( x) y ( x) = λ2 ρ y ( x) , x ∈ (0, a), ρ (a ) ≠ 1
y (0) = 0 , y ′(0) = 1.
y ( x)
Satisfies to parity max
x∈[ 0 , a ]
a
( ∫ ρ ( x). y ( x) dx)
2
1
2
< const < ∞
0
For every value large enough λ from strip Im(λ ) < const .
Proof:
As solution of Cauchy problem continuously depends on weight function
1
2
a
ρ (x) and functional ( ∫ ρ ( x) y ( x, λ ) dx) also continuously depends on ρ (x) .
2
0
max y ( x, λ )
Hence, functional
x∈[ 0, a ]
a
( ∫ ρ ( x). y ( x, λ ) dx)
2
1
2
also continuously depends on weight
0
function ρ (x) . Hence, there is number ε (R ) such, that
max y ( x, λ , ρ )
x∈[ 0 , a ]
a
2
( ∫ ρ ( x). y ( x, λ , ρ ) dx)
0
1
2
1
≥ .
2
max y ( x, λ , ρ )
x∈[ 0, a ]
a
( ∫ ρ ( x). y ( x, λ , ρ ) dx)
2
1
2
, if
λ ≤ R and
0
ρ ( x) − ρ ( x) ≤ ε ( R)
Where R > 0 is 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) )
ρ ε (a) = ρ (a) . Now let’s consider Cauchy problem with weight function ρ ε ( x)
instead of ρ (x) . In this problem ρ ε ( x) ∈ C 2 [ 0, a ] and consequently we can make
About uniform limitation of normalized…
120
x
replacement ξ = ∫
double
0
where A( x) = ρ
−
1
4
dt
, y ( x) = A( x).η (ξ ( x)) ,
A 2 (t )
1
ε
( x).ρ 4 (a) .
As a result of such replacement we shall obtain problem:
a
− η ′′(ξ ) + (q ( x) A( x) − A′′( x)). A 3 ( x)η (ξ ) = λ2 ρ (a )η (ξ ) , ξ ∈ (0, ∫
0
dt
),
A 2 (t )
η (0) = 0,
η ′(0) = A(0) ,
a
∫
0
a
dt
=
A 2 (t ) ∫0
a
=
∫
dt
ρ (a)
(
)
ρ ε (t )
As
=
1
a
ρ (a ) ∫0
A(0) = ρ
−
1
4
1
4
(0).ρ (a ) and
ρ ε (t )dt
ρ (t ) dt
0
ρ (a)
= a = const (Independent from ε ), designating
A 3 ( x)[q ( x) A( x) − A′′( x)] ≡ qε (ξ ) we shall obtain problem:
− η ′′(ξ ) + qε (ξ )η (ξ ) = λ2 ρ (a)η (ξ ) , ξ ∈ (0, a) ,
η (0) = 0 ,η ′(0) = 4
ρ (a)
ρ ( 0)
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,
t
that
∫ qε (ξ )dξ
in regular intervals on ε and t ∈ [0, a] is limited for all small
0
ε > 0 under theorem there will be constant C 0 > 0 such, that
Jwamer .K.H and Aigounv .G.A
η (ξ )
max
a
( ∫ ρ (a ).η (ξ ) dξ )
2
1
2
121
≤ C0 ,
0
From
here
and
x
y ( x) = A( x).η (ξ ( x)), ξ ( x) = ∫
0
that max
x∈[ 0 , a ]
from
parities
dt
obviously follows, that exists C 0 > 0 such,
A 2 (t )
y ( x, λ , ρ )
a
( ∫ ρ ( x). y ( x, λ , ρ ) dx)
2
< C0
1
2
0
For every λ ≤ R , and by arbitrariness R , and for all considered λ (let’s
remind, that Im(λ ) < const ).
t
Let's estimate
∫ qε (ξ )dξ . Passing to variable
x in integral we shall get
0
t
s
s
3
3
∫ qε (ξ )dξ = ∫ [q( x) A( x) − A′′( x)] A ( x).ξ ′( x)dx = ∫ [q( x) A( x) − A′′( x)] A ( x)
0
0
S
∫
0
S
∫
0
0
s
q( x)
dx − ∫ A( x) A′′( x)dx ≤
A − 2 ( x)
0
S
∫
0
q( x)
dx +
A − 2 ( x)
s
q ( x)
s
dx + [ A( x) A′( x)] 0 − ∫ [ A′( x)] 2 dx ≤
−2
A ( x)
0
s
S
s
∫ A( x) A′′( x)dx =
0
q( x)
dx + A( s ) A′( s ) +
−2
( x)
∫A
0
A(0) A′(0) + ∫ [ A′( x)]2 dx , where s ∈ [0, a].
0
From definition A( x) = 4
A(0) = 4
ρ (a)
follows, that
ρ ε ( x)
ρ (a)
ρ (a)
,
=4
ρ ε ( 0)
ρ (0)
dx
=
A 2 ( x)
About uniform limitation of normalized…
122
1
4 ρ ( a ) .ρ ′ ( x )
5
1 4
−
ε
′
,
A ( x) = − ρ (a ).ρ ε 4 ( x).ρ ε′ ( x) = −
5
4
44 ρ ε ( x)
A′(0) = −
4
t
ρ (a ) .ρ ε′ (0)
4 4 ρ 5 ε ( 0)
a
∫ qε (ξ )dξ ≤ ∫
0
0
q( x)
4
ρ (a)
=
− ρ ε′ (0) ρ (a )
and consequently
.4
4 ρ ( 0)
ρ (0)
.4 ρ ε ( x)dx +
ρ (a) .ρ ε′ ( s )
4. ρ ε ( s )
3
+
ρ ε′ (0) ρ (a ) a ρ (a) .[ ρ ε′ ( s )] 2
.
+
4.ρ (0) ρ (0) ∫0 16. ρ 5 ( s )
ε
On lemma ρ ε′ ( x) ≤ 2.N and ρ ( x) − ε ≤ ρ ε ( x) ≤ ρ ( x) + ε , hence
t
a
∫ qε (ξ )dξ ≤ ∫
0
0
q( x)
4
ρ (a)
ρ ( a ) .N
.4 ρ ( x) + ε dx +
2.[ min ρ ( x) − ε ]
3
2
+
N
ρ (a)
.
2 ρ (0) ρ (0)
x∈[ 0 , a ]
a
+∫
0
ρ (a ) .N 2
4. [ ρ ( x ) − ε ] 5
dx .
As ε is not enough, consequence is proved.
Let's prove now theorem for what we shall evaluate ϕ 0′ ( x, λ ), ϕ ( x, λ ) and
ϕ ′( x, λ ) (ϕ ( x, λ ) is solution of equation (1) satisfying to entry conditions
ϕ (0, λ ) = 0, ϕ ′(0, λ ) = 1, and ϕ 0 ( x, λ ) is solution of such Cauchy problem with
constant coefficient ρ (x) ≡ ρ ). As it has been established [8, p. 22]
ϕ 0 ( x, λ ) =
− σ 1 − iδ 1
(− sinh σ 1 x. cos δ 1 x + i cosh σ 1 x. sin δ 1 x) and
σ 12 + δ 12
consequently ϕ 0′ ( x, λ ) = (cosh σ 1 x. cos δ 1 x − i sinh σ 1 x. sin δ 1 x) (simple
transformations are lowered). As at greater λ parities σ 1≈ ρ .σ and δ 1≈ 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
Jwamer .K.H and Aigounv .G.A
123
x
∞ x
τ1
o
i =2 0
0
ϕ ( x, λ ) = ϕ 0 ( x, λ ) + ∫ [q(τ 1 ) − q ].ϕ 0 ( x − τ 1 , λ ) ϕ 0 (τ 1 , λ )dτ 1 + ∑ ∫ [q(τ 1 ) − q ].ϕ 0 ( x − τ 1 , λ ) ∫ ...
τ i −1
∫
0
[q (τ i ) − q ]ϕ 0 (τ i −1 − τ i , λ ) ϕ 0 (τ i , λ ) dτ i ....dτ 1
Let's enter designations f i ( x) for i − member of series
x
( f1 ( x) = ∫ [q (τ 1 ) − q ].ϕ 0 ( x − τ 1 , λ )dτ 1 ,...) , As result we shall get:
0
∞
ϕ ( x, λ ) − ϕ 0 ( x , λ ) = ∑ f i ( x )
i =1
(4)
If i ≥ 1 , then obviously
x
f i +1 ( x) = ∫ [q (τ 1 ) − q]ϕ 0 ( x − τ 1 , λ ) . f i (τ 1 )dτ 1 and consequently, integrating
0
in parts, we shall obtain
x
τ1
x


f i +1 ( x) = ϕ 0 ( x − τ 1 , λ ). f i (τ 1 ).∫ [q ( s ) − q ]ds  − ∫ {[ϕ 0 ( x − τ 1 , λ ). f i′(τ 1 ) − ϕ 0′ ( x − τ 1 , λ ). f i (τ 1 )].


0
0
0

x
∫ [q( s) − q]dsdτ
1
0
Or
x
τ
0
0
f i +1 ( x) = ∫ [ϕ 0′ ( x − τ , λ ) f i (τ ) − ϕ 0′ ( x − τ , λ ) f i′(τ )].∫ [q( s ) − q]dsdτ ,
(5)
Absolutely similarly for f1 ( x) it is possible to get parity
x
τ
0
0
f 1 ( x) = ∫ [ϕ 0′ ( x − τ , λ )ϕ 0 (τ ) − ϕ 0′ ( x − τ , λ )ϕ 0′ (τ )].∫ [q ( s ) − q ]dsdτ
(6)
About uniform limitation of normalized…
124
Differentiating parities (5) and (6) on x we shall get parities for
derivatives
x
x
fi′+1(x) = fi (x).∫ [q(τ ) − q]dτ +[∫ [q − λ2ρ]ϕ0 (x −τ , λ) fi (τ ) −
0
0
τ
ϕ0′ (x −τ , λ) fi′(τ )].∫ [q(τ ) − q]dsdτ ,
0
(7)
x
x
f 1′( x) = ϕ 0 ( x, λ ).∫ [q(τ ) − q ]dτ + [ ∫ [q − λ2 ρ ]ϕ 0 ( x − τ , λ )ϕ 0 (τ , λ ) −
0
0
τ
ϕ 0′ ( x − τ , λ )ϕ 0′ (τ , λ )].∫ [q (τ ) − q]dsdτ ,
0
(8)
(Here, it is considered, that ϕ 0′′ ( x, λ ) ≡ (q − λ2 ρ ) ϕ 0 ( x, λ ). )
τ
From choice of class Q[ 0, a ] follows, that
∫ [q(s) − q]ds < Q = const < ∞
0
and consequently from (5) and (8) follows, that
f1 ( x) ≤
2Q.C 0 .C 0′
λ
x , f1′( x) ≤
Q.C 0
λ
+ Q[( ρ +
q
λ
2
2
)C 02 + C 0′ ] , where C 0
and C 0′
Such constants for which inequalities ϕ 0 ( x, λ ) ≤
are executed (as ϕ 0 ( x, λ ) ≤
C0
λ
and ϕ 0′ ( x, λ ) ≤ C 0′
sinh 2 σ 1 x + sin 2 δ 1 x
then, obviously C 0 and C 0′
δ 12 + σ 12
exist). Using recurrent parities (5) and (7) we shall obtain, that number (4)
converges and moreover ϕ ( x, λ ) − ϕ 0 ( x, λ ) ≤
C0
λ
(evaluations f 2 ( x) , f 3 ( x) ,...,
and f 2′( x) , f 3′( x) ,... are made consistently for i=2,3, …).
Jwamer .K.H and Aigounv .G.A
125
x
Then ϕ ′( x, λ ) = ϕ 0′ ( x, λ ) + ∫ [q (τ ) − q]ϕ 0′ ( x − τ , λ )dτ , or, integrating in
0
parts, we shall get
x
τ


ϕ ′( x, λ ) − ϕ 0′ ( x, λ ) = ϕ 0′ ( x − τ , λ ) ϕ (τ , λ ).∫ [q ( s ) − q ]ds  −
0
0

τ

′
′
′
′
− ∫ {[ϕ 0 ( x − τ , λ ) ϕ (τ , λ ) − ϕ 0 ( x − τ , λ ) .ϕ (τ , λ )]∫ [q ( s ) − q ]ds dτ ,
0
0

x
As ϕ 0′ (0, λ ) ≡ 1 and ϕ 0′′ ( x − τ , λ ) ≡ (q − λ 2 ρ ) ϕ 0 ( x − τ , λ ) ,
x
x
0
0
ϕ ′( x, λ ) − ϕ 0′ ( x, λ ) = ϕ ( x, λ ).∫ [q ( s ) − q ]ds − ∫ {[ϕ 0′ ( x − τ , λ ) ϕ ′ (τ , λ ) −
a
(q − λ2 ρ ) ϕ 0 ( x − τ , λ ) ϕ (τ , λ )]. ∫ [q ( s ) − q]ds}dτ =
0
x
x

a

0
0

0

ϕ ( x, λ ).∫ [q( s ) − q ]ds + ∫ (λ2 ρ − q ) ϕ 0 ( x − τ , λ ) ϕ (τ , λ ). ∫ [q ( s ) − q ]ds dτ
τ


− ∫ ϕ 0′ ( x − τ , λ ) ϕ (τ , λ ) ∫ [q( s ) − q ]ds dτ ,
0
0

x
Subtracting and adding in last integral ϕ 0′ (τ , λ ) to ϕ ′ (τ , λ )
and
representing integral in the form of sum of two integrals we shall obtain
x
x

τ

0
0

0

ϕ ′( x, λ ) − ϕ 0′ ( x, λ ) = ϕ ( x, λ ).∫ [q ( s ) − q ]ds + (λ2 ρ − q ) ∫ ϕ 0 ( x − τ , λ ) ϕ (τ , λ ). − ∫ [q ( s ) − q]ds dτ
τ
x
τ




′
′
′
′
′
− ∫ ϕ0 ( x − τ , λ )[ϕ0 (τ , λ ).∫ [q(s) − q]dsdτ − ∫ ϕ0 ( x −τ , λ )[ϕ (τ , λ ) − ϕ0 (τ , λ )].∫ [q(s) − q]dsdτ
0
0
0
0


From this equality follows, that
x
About uniform limitation of normalized…
126
x
x
ϕ ′( x, λ ) − ϕ 0′ ( x, λ ) ≤ ϕ ( x, λ ) . ∫ [q ( s ) − q]ds + λ2 ρ − q .∫ ϕ 0 ( x − τ , λ ) .ϕ (τ , λ ) .
0
τ
x
0
0
0
τ
∫ [q( s) − q]ds dτ + ∫ ϕ ′ ( x − τ , λ ) .ϕ ′ (τ , λ ) . ∫ [q(s) − q]ds dτ +
0
x
τ
0
0
0
0
∫ ϕ 0′ ( x − τ , λ ) . ∫ [q(s) − q]ds .ϕ ′(τ , λ ) − ϕ 0′ (τ , λ ) dτ
And
consequently,
using
estimations
ϕ 0 ( x, λ ) , ϕ ( x, λ ) ,
ϕ 0′ ( x, λ )
τ
τ
τ
τ
0
0
0
0
obtained
and
before
considering,
for
that
∫ [q(s) − q]ds ≤ ∫ q( s)ds + ∫ qds = qτ + ∫ q( s)ds < const < ∞, we shall obtain
x
ϕ ′( x, λ ) − ϕ 0′ ( x, λ ) < R + ∫ B(τ ).ϕ ′(τ , λ ) − ϕ 0′ (τ , λ ) dτ
,
0
where R > 0 and B(τ ) > 0 are limited.
x
∫ B (τ ) dτ
Hence, by Gronwall’s lemma ϕ ′( x, λ ) − ϕ 0′ ( x, λ ) ≤ R.e 0
As
ϕ 0′ ( x, λ ) ≤ const < ∞ ,
from
last
inequality
≤ const < ∞ .
follows,
that
ϕ ′ ( x, λ ) ≤ const < ∞ is regular on x ∈ [0, a ] 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
ϕ m′ ,−ϕ m′ accordingly.
y
ϕm
0
And consequently
x0
a•
x
Jwamer .K.H and Aigounv .G.A
a
127
a
a
x0
∫ ρ ( x) ϕ ( x, λ ) dx ≥ ∫ m.ϕ ( x, λ ) dx = m.∫ ϕ ( x, λ ) dx ≥ m ∫ ϕ m + ϕ m′ ( x − x0 ) dx
2
0
2
2
0
0
2
0
x0
a
+ m. ∫ ϕ m − ϕ m′ ( x − x0 ) dx = m. ∫ [ϕ 2 m +2ϕ m .ϕ m′ ( x − x0 ) + ϕ m′ ( x − x0 ) 2 ]dx +
2
2
0
x0
a
m. ∫ [ϕ m2 − 2ϕ m .ϕ m′ ( x − x0 ) + ϕ m′ 2 ( x − x0 ) 2 ]dx
x0
3
3

ϕ m′ 2
2 (a − x0 ) + x0
2 
′
= m.ϕ a + ϕ m [
]
−
[
x
+
(
a
−
x
)
].
0
0
ϕm
3ϕ m2


2
m
From inequality we got follows, that
max ϕ ( x, λ )
x∈[ 0 , a ]
a
( ∫ ρ ( x) ϕ ( x, λ ) dx)
2
1
2
1
≤
m. a +
0
(a − x0 ) 3 + x03 ϕ m′ 2
ϕ′
.( ) − [ x02 + (a − x0 ) 2 ] m
ϕm
ϕm
3
.
If to enter designations x0 = ε .a and
a+
ϕ m′
= z we shall get
ϕm
(a − x0 ) 3 + x03 ϕ m′ 2
ϕ′
.( ) − [ x02 + (a − x0 ) 2 ] m =
3
ϕm
ϕm
a 3 (1 − ε ) 3 + ε 3 a 3 3
z − [a 2ε 2 + a 2 (1 − ε ) 2 ]z =
3
1
1
1
= a[az (az − 2)(ε − ) 2 + (az − 3) 2 + ].
2
12
4
=a+
Where ε ∈ [0,1] and z ∈ (0, ∞) . Let’s enter now designations
1
1
1
f (ε , z ) = az (az − 2)(ε − ) 2 + (az − 3) 2 + and estimate from below
2
12
4
f (ε , z ) . It is obvious, that
1 2 2
1
1
(a z − 2az ) + (a 2 z 2 − 6az + 9) +
4
12
4
1
3
3 1
= [(az − ) 2 + ] ≥ .
3
2
4
4
f (0, z ) = f (1, z ) =
About uniform limitation of normalized…
128
Inside of interval 0 ≤ ε ≤ 1 is unique critical point ε =
1
, in this point we
2
1
1
1 1
have f ( , z ) = (az − 3) 2 + ≥ . . Hence, for any ε ∈ [0,1] and z ∈ (0, ∞)
2
12
4 4
estimation f (ε , z ) ≥
1
is fair, therefore inequality is
4
max ϕ ( x, λ )
x∈[ 0 , a ]
a
( ∫ ρ ( x) ϕ ( x, λ ) dx)
2
1
2
1
≤
m . a.
0
1
4
=
2
m.a
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
_____
lim
y n ( x)
n →∞
λn
C[ 0 , a ]
1−α
2
= С0 > 0 .
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 selfadjoin differential operator of 2n 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 2n order // Col. Functional analysis, theory of functions and their
Jwamer .K.H and Aigounv .G.A
129
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 2n order
on interval [0, a ] // News of high schools of North Caucasus, Rostov-on-Don,
№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.