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Theoretical Mathematics & Applications, vol.3, no.3, 2013, 121-143 ISSN: 1792- 9687 (print), 1792-9709 (online) Scienpress Ltd, 2013
Asymptotic behaviour of
boundary condition functions
for a fourth-order boundary value problem
Emmanuel K. Essel1, Ernest Yankson2 and Samuel M. Naandam3
Abstract
In this paper, we prove that the Boundary Condition Constants for the Boundary
Value Problem
π )()()()()()()()(: 4)1(
3)2(
2)4( xxxPxxPxxPxL λφφφφφφ =+++≡
{ } 0)()(4
1
)1()1( =+≡ ∑=
−−
s
srs
srsr bnamU φφφ ( )4,1 ≤≤ sr
can be replaced by Boundary Condition Functions and that the Boundary
Condition Functions are asymptotically equivalent for large values of λ , to the
Boundary Condition Functions for the corresponding Fourier problem 𝜋, given
by
Fπ : )()()4( xx λφφ =
{ } 0)()(4
1
)1()1( =+≡ ∑=
−−
s
srs
srsr bnamU φφφ ( )4,1 ≤≤ sr .
1 Department of Mathematics and Statistics University of Cape Coast, Cape Coast Ghana. 2 Department of Mathematics and Statistics University of Cape Coast, Cape Coast Ghana. 3 Department of Mathematics and Statistics University of Cape Coast, Cape Coast Ghana. Article Info: Received: June 30, 2013. Revised: August 30, 2013. Published online : September 1, 2013
122 Asymptotic behaviour of boundary condition functions …
Mathematics Subject Classification: 35B40; 34B05
Keywords: Boundary condition functions; fourth order boundary value problem
and asymptotic behaviour
1 Introduction
Over the years quite a good number of mathematicians have studied
boundary condition functions. The use of boundary condition functions for
boundary value problems was first considered by Kodaira in [1].
In this paper Kodaira considered the replacement of boundary condition
constants of separated boundary conditions , associated with real differential
equations of arbitrary even order , by solutions of the differential equation. In [2]
E. C. Titchmarsh proved that, the self-adjoint boundary value problem:
)()()()()2( xxxqx λφφφ =+− (-∞< a ≤ x ≤ b < ); τσλ i+=∞
(1)
(1)
( ) cos ( )sin 0( )cos ( )sin 0a ab b
ϕ α ϕ α
ϕ β ϕ β
+ =
+ =
is equivalent asymptotically, for suitably large values of λ , to the corresponding
Fourier problem:
)()()2( xx λφφ =−
(1)
(1)
( ) cos ( )sin 0( )cos ( )sin 0a ab b
ϕ α ϕ α
ϕ β ϕ β
+ =
+ =
The coefficient q and the constantsα , β are real valued and [ ]baCq ,∈ . W.
N. Everitt in [3] also worked on self-adjoint boundary value problems. D. N. Offei
in [4] extended the use of boundary condition functions to non-self adjoint
boundary value problems with complex-valued coefficients and constants and
with boundary conditions separated or otherwise.
E.K. Essel, E. Yankson and S.M. Naandam 123
In [5], D. N. Offei proved that the boundary condition functions, for the
boundary value problem:
)()()()()()( 3)1(
2)3(3 xxxPxxPxiL λφφφφφ =++≡
0)()()( )1( === bba φφφ
are asymptotically equivalent, for suitably large values of λ , to the corresponding
functions, for the corresponding Fourier problem. In [9], M. B. Osei showed that
the boundary condition functions of the second order boundary value problem are
asymptotically equivalent to the boundary condition functions of the
corresponding Fourier problem of the boundary value problem.
2 Notation
In this section we give some properties of the linear differential expression L
and some notations used in subsequent sections of this paper.
1. For a suitable set of functions ),(xrφ ),41( ≤≤ r the symbol )(xΦ denotes
the 4 x 4 matrix )]([ )1( xsr
−φ )4,1( ≤≤ sr .
Thus,
)(xΦ .
)()()()()()()()()()()()(
)()()()(
)3(4
)3(3
)3(2
)3(1
)2(4
)2(3
)2(2
)2(1
)1(4
)1(3
)1(2
)1(1
4321
≡
xxxxxxxxxxxx
xxxx
φφφφφφφφφφφφφφφφ
Also )(ˆ xg represents the column vector with components
(1) ( 1)( ), ( ),..., ( ).ng x g x g x−
2. The symbol *( )xB denotes the conjugate transpose of the matrix ( )xB
whilst )(ˆ* xb denote the row vector with components (1) ( 1)( ), ( )... ( )nb x b x b x− .
3. Given the linear expression 𝐿 defined by
124 Asymptotic behaviour of boundary condition functions …
)()()()()()()()()()( 4)1(
3)2(
2)3(
1)4(
0 xxPxxPxxPxxPxxPL φφφφφφ ++++≡
)( bxa ≤≤ . The Lagrange adjoint of L is denoted by +L and defined as
ψψψψψψ )())(1()()1()()1()()1( 4)1(
3)2(
22)3(
13)4(
04 xPPPPPL +−+−+−+−≡+ .
4. (a) For suitable pairs of functions f and g
∫b
a
gLfLfg +− dx = )]([)]([ afgbfg − .
Here [fg](x) is a bilinear form in ),,,( )3()2()1( ffff and
),,,( )3()2()1( gggg given by
[ ]4 4
( 1) ( 1) *
1 1
ˆˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) j kjk
j kx B x g x f x g x xfg f x− −
= =
= =∑∑ B
where,
−−
−−−−+−−+−−−+−
=
000)(00)()(3)(0)()(2)()(2)(2)(
)()()()()()()()()()()(
)(
0
0)1(
01
0)1(
01)2(
0)1(
12
0)1(
01)2(
0)1(
12)3(
0)2(
0)2(
1)1(
2)3(
3
xPxPxPxP
xPxPxPxPxPxPxPxPxPxPxPxPxPxPxPxPxP
xB
(b) If ),(1 xP )(2 xP and )(3 xP are identically zero in some neighbourhood of
a and b and 0P is a constant independent of x then
−
−==
000000000
000
)()(
0
0
0
0
PP
PP
ba BB .
(c) The Lagrange adjoint of +L is L and for suitable pair of functions g and f
{ }∫ −=−+b
a
agfbgfdxLfggLf )}({)}({
where
{gf }(x) =∑∑= =
−−4
1
4
1
)1()1( )()()(j k
kjjk xgxfxA = )(ˆ)()(ˆ * xgxxf A .
E.K. Essel, E. Yankson and S.M. Naandam 125
The jkA are dependent on the coefficients of the differential expression +L
and ][)( jkAx =A .
5. If ( , )xϕ λ is a solution of Lϕ λϕ= and ( , )xψ λ is a solution of Lψ λψ+ =
then
{ }
{ }
2
1
2 1 1 2[ ]( ) [ ]( ) ( )
0
x
x
x x L L dx a x x b
dx
ϕψ ϕψ ψ ϕ ϕ ψ
ψλϕ ϕλψ
+− = − ≤ ≤ ≤
= −
=
∫
∫
and hence,
)]([)]([ 12 xx φψφψ = .
Thus, ))](,(),([ xxx λψλφ is independent of ],[ bax∈ .
Similarly ))}(,(),({ xxx λφλψ is independent of ],[ bax∈ . This implies that
))](,(),([ xxx λψλφ and ))}(,(),({ xxx λφλψ may be denoted by ][φψ and
}{ψφ , respectively.
6. (a) If there is a constant 𝐾 such that )()( xKxf φ≤ for 0xx ≥ we write
).(φOf =
(b) If lxxf
→)()(
φ , ∞→x where 0≠l we write φlf ~ .
.
3 Preliminaries
The boundary value problem to be considered is of the form
π : )()()()()()()()( 4)1(
3)2(
2)4( xxxPxxPxxPxL λφφφφφφ =+++≡ (1)
{ } 0)()(4
1
)1()1( =+≡ ∑=
−−
s
srs
srsr bnamU φφφ ( ),4,1 ≤≤ sr (2)
126 Asymptotic behaviour of boundary condition functions …
where the functions )(2 xP , )(3 xP , )(4 xP , the constants rsm and rsn and the
parameter λ are complex- valued. The functions )(xPr ( r = 2, 3, 4) are of the
class )4( rC − on the closed bounded interval [a, b] and )(2 xP , )(3 xP are identically
zero in a neighbourhood of both a and b .
The corresponding Fourier boundary value problem for π is given by
Fπ : )()()4( xx λφφ = (3)
{ } 0)()(4
1
)1()1( =+≡ ∑=
−−
s
srs
srsr bnamU φφφ ( )4,1 ≤≤ sr . (4)
Let )},/();,/({ λχλψ xbxa rr )41( ≤≤ r be the boundary condition
functions for π and )},/();,/({ λχλψ xbxa FrFr )41( ≤≤ r the boundary condition
functions for Fπ . Then ),/( λψ xar and ),/( λχ xbr are solutions of ψλψ =+L
such that
*)()( MBΨ aa = and *,)()( NBΧ bb =
where
−
−==
0001001001001000
)()( ba ΒB , M= ,
44434241
34333231
24232221
14131211
mmmmmmmmmmmmmmmm
and N= .
44434241
34333231
24232221
14131211
nnnnnnnnnnnnnnnn
Likewise ),/( λψ xaFr and ),/( λχ xbFr are solutions of )()()4( xx ψλψ = such
that
*)()( MBΨ aa = and .*)()( NBΧ bb =
Let )},/(),,/({ λλ xbgxaf rr )41( ≤≤ r be the boundary condition
functions for the boundary value problem
E.K. Essel, E. Yankson and S.M. Naandam 127
),()()4( xx λφφ = and )(ˆ)(ˆ 44 bIaIU φφφ += =
0000
,
where I 4 is the 4 x 4 unit matrix. Then ),/(),,/( λλ xbgxaf rr are solutions of
)()()4( xx ψλψ = such that
F(a) = B(a)I 4 = B(a) and G(b) = B(b)I 4 = B(b).
4 Proof of Theorems
We now prove five Theorems that will enable us to prove our main results in
Theorem 6.
Theorem 1
(i) ),/( λψ xaFr = ),/(4
1λxafm s
srs∑
=
.
(ii) ),/( λχ xbFr = ),/(4
1λxbgn s
srs∑
=
.
(iii) Let ( ) ( / , )s sf x f a x λ= , ( ) ( / , )s sg x f a x λ= .
Then
)(xf s = )()1( )1(1
)1( xf ss −−− 42 ≤≤ s
)(xgs = )()1( )1(1
)1( xg ss −−− .42 ≤≤ s
Proof. (i) and (ii) ),/( λxaf s and ),/( λxbgs are solutions of
)()()4( xx ψλψ = (5)
such that
F(a) = B(a)I 4 = B(a) (6)
128 Asymptotic behaviour of boundary condition functions …
i.e.,
)()()()()()()()()()()()(
)()()()(
)3(4
)3(3
)3(2
)3(1
)2(4
)2(3
)2(2
)2(1
)1(4
)1(3
)1(2
)1(1
4321
afafafafafafafafafafafaf
afafafaf
=
−
−
0001001001001000
.
Similarly G(b) = B(b)I 4 = B(b).
Let ),( λxhr = ),/(4
1λxafm s
srs∑
=
, then ),(1 λxh , ),(2 λxh , ),(3 λxh and ),(4 λxh are
solutions of (5) such that
H(a) =
)()()()()()()()()()()()(
)()()()(
)3(4
)3(3
)3(2
)3(1
)2(4
)2(3
)2(2
)2(1
)1(4
)1(3
)1(2
)1(1
4321
ahahahahahahahahahahahah
ahahahah
=
)()()()()()()()()()()()(
)()()()(
)3(4
)3(3
)3(2
)3(1
)2(4
)2(3
)2(2
)2(1
)1(4
)1(3
)1(2
)1(1
4321
afafafafafafafafafafafaf
afafafaf
.
44342414
43332313
42322212
41312111
mmmmmmmmmmmmmmmm
This implies that
H(a) = F(a)M* .
But from (6) F(a) = B(a), therefore
H(a) = B(a)M* .
Now ),/( λψ xaFr )41( ≤≤ r are solutions of the same (5) such that
*)()( Maa B=Ψ .
Hence we have ),/( λψ xaFr = ),( =λxhr ),/(4
1λxafm s
srs∑
=
).41( ≤≤ r
Similarly if
),( λxqr = ),/(4
1λxbgn s
srs∑
=
, (7)
E.K. Essel, E. Yankson and S.M. Naandam 129
then ),( λxqr )41( ≤≤ r are solutions of (5) such that Q(b) = B(b)N* and so
),/( λχ xbFr = ),( λxqr = ),/(4
1λxbgn s
srs∑
=
.
This proves (i) and (ii).
(iii) ),/( λxaf r )41( ≤≤ r are solutions of )()()4( xx ψλψ = such that
F(a) = B(a)I 4 = B(a)
)()()()()()()()()()()()(
)()()()(
)3(4
)3(3
)3(2
)3(1
)2(4
)2(3
)2(2
)2(1
)1(4
)1(3
)1(2
)1(1
4321
afafafafafafafafafafafaf
afafafaf
=
−
−
0001001001001000
(8)
The general solution of the equation )()()4( xx ψλψ = can be obtained as follows:
Let )sin(cos4 θθλρ ir +== for πθπ ≤<− , .τσρ i+=
Then,
+
+
+
=42sin
42cos4
1 kikrkπθπθρ (k = 0, 1, 2, 3).
Hence, for 𝑘 = 0 we have
+
=
4sin
4cos4
1
0θθρ ir (9)
Since the fourth roots of unity are 1, -1, i and –i we see that the four roots of the
equation λρ =4 are 0ρ ,- 0ρ , -i 0ρ and i 0ρ . Put 0ρ=P then the four roots are P,
-P , iP and –iP. The general solution of the equation
0)()( 4)4( =− xx ψρψ )( 40 λρ = (10)
is thus given by .)( )(4
)(3
)(2
)(1
axiPaxipaxpaxp eAeAeAeAx −−−−−− +++=ψ
Now ),/( λxaf r is a solution of (10) such that F(a) = B(a), and so
)(4
)(3
)(2
)(11 ),/( axiPaxipaxpaxp eAeAeAeAxaf −−−−−− +++=λ . (11)
Since from (8) 0)(1 =af ; 0)()1(1 =af ; 0)()2(
1 =af ; 1)()3(1 −=af we see that the
constants kA (k = 1, 2, 3, 4) in (11) are given by
130 Asymptotic behaviour of boundary condition functions …
04321 =+++ AAAA
04321 =−+− ipAipApApA
024
23
22
21 =−−+ pApApApA
.143
333
23
1 −=+−− AipAippApA
Solving gives
31 4
1 −−= pA ; 32 4
1 −= pA ; 333 44
1 −− −== pipi
A ; .44
1 334
−− =−= pipi
A
Substituting into (11) we find that
),/(1 λxaf = 3
41 −p )()()()( axiPaxipaxpaxp ieieee −−−−−− +−+−
),/()1(1 λxaf = 2
41 −p )()()()( axiPaxipaxpaxp eeee −−−−−− ++−−
),/()2(1 λxaf = 1
41 −p )()()()( axiPaxipaxpaxp ieieee −−−−−− −++− (12)
),/()3(1 λxaf =
41 )()()()( axiPaxipaxpaxp eeee −−−−−− −−−−
),/()4(1 λxaf = p
41 )()()()( axiPaxipaxpaxp ieieee −−−−−− +−+−
= p41 ),/(1 λxaf 4 3p
hence
),/()4(1 λxaf = p 4 ),/(1 λxaf .
Next we prove that
)(2 xf = )()1(1 xf− ; )(3 xf = )()2(
1 xf ; )(4 xf = )()3(1 xf−
Let )(1 xR = )()1(1 xf− . Then )(1 xR is a solution of (5) such that
E.K. Essel, E. Yankson and S.M. Naandam 131
)()()(
)(
)3(1
)2(1
)1(1
1
aRaRaR
aR
=
−−−−
)()()()(
)4(1
)3(1
)2(1
)1(1
afafafaf
= .
0100
(13)
We note that (13) is obtained by substituting x = a into (12). From (8)
)()()(
)(
)3(2
)2(2
)1(2
2
afafaf
af
=
0100
and so,
)()()(
)(
)3(1
)2(1
)1(1
1
aRaRaR
aR
=
−−−−
)()()(
)(
)3(1
)2(1
)1(1
1
afafaf
af
= =
0100
)()()(
)(
)3(2
)2(2
)1(2
2
afafaf
af
which implies that 1 2( ) ( / , ).R x f a x λ=
Again, )(1 xR = ),/()1(1 λxaf− implies that
),/(2 λxaf = ).,/()1(1 λxaf− (14)
Let )(2 xR = ),()2(1 xf then )(2 xR is a solution of (5) such that
)()()(
)(
)3(2
)2(2
)1(2
2
aRaRaR
aR
=
)()()()(
)5(1
)4(1
)3(1
)2(1
afafafaf
=
−
001
0
=
)()()(
)(
)3(3
)2(3
)1(3
3
afafaf
af
and so )(2 xR = ).,/(3 λxaf Thus, )(2 xR = ),/()2(1 λxaf implies that
),/(3 λxaf = ),/()2(1 λxaf . (15)
Next we let )(3 xR = ),()3(1 xf− so that )(3 xR is a solution of (5) implies that
)()()(
)(
)3(3
)2(3
)1(3
3
aRaRaR
aR
=
−−−−
)()()()(
)6(1
)5(1
)4(1
)3(1
afafafaf
=
0001
= .
)()()(
)(
)3(4
)2(4
)1(4
4
afafaf
af
Therefore )(3 xR = ).,/(4 λxaf Finally, )(3 xR = )()3(1 xf− implies that
),/(4 λxaf = ).()3(1 xf− (16)
It follows from (14), (15) and (16) that
132 Asymptotic behaviour of boundary condition functions …
)(xf s = )()1( )1(1
)1( xf ss −−− ( 42 ≤≤ s ).
Similarly )(xgs = )()1( )1(1
)1( xg ss −−− ( 42 ≤≤ s ). The proof of the theorem is
complete. □
Theorem 2 Let
),/( λxtR = ),/()( 14 λxtftP - ),/()( )1(13 λxtftP + ),/()( )2(
12 λxtftP (17)
),/( λxtS = ),/()( 14 λxtgtP - ),/()( )1(13 λxtgtP + ).,/()( )2(
12 λxtgtP
Then the boundary condition functions for the boundary value problem (1) and (2)
are given by
),/( λψ xar = ),/( λψ xaFr + ∫x
ar dttxtR )(),/( ψλ )41( ≤≤ r
),/( λχ xbr = ),/( λχ xbFr + ∫x
ar dttxtS )(),/( χλ ).41( ≤≤ r
Proof. As a function of x, ),/(1 λxtf (x, t ]),[ ba∈ is a solution of
)()4( xψ = )(xψλ such that from (8)
),/( 001 λxxf = ),/( 00)1(
1 λxxf = ),/( 00)2(
1 λxxf = 0 (18)
and
),/( 00)3(
1 λxxf = −1 where 0x ].,[ ba∈ (19)
It follows from (17), (18) and (19) that ),/( λxtR is a solution of )()4( xψ = )(xψλ
such that
0 0(1)
0 0 2 0(2)
0 0 3 0(3)
0 0 4 0
( / , ) 0
( / , ) ( )
( / , ) ( )
( / , ) ( )
R x xR x x P xR x x P xR x x P x
λ
λ
λ
λ
=
= −
= = −
(20)
If ),/( λψ xar = ),/( λψ xaFr + ∫x
ar dttxtR )(),/( ψλ ).41( ≤≤ r (21)
E.K. Essel, E. Yankson and S.M. Naandam 133
Then using the formula
( , ) ( , ) ( , )x
a
d FF x t dt x t F x xdx x
δδ
= +∫ ∫ and (20) we obtain the following equations
from (21)
(1) (1) (1)( / , ) ( ) ( / , ) ( ) ( / , ) ( )x
r Fr r ra
a x x R t x t dt R x x xψ λ ψ λ ψ λ ψ= + +∫
(2) (2) (2) (1)
(2) (2)2
( / , ) ( ) ( / , ) ( ) ( / , ) ( )
( ) ( / , ) ( ) ( ) ( )
x
r Fr r rax
Fr r ra
a x x R t x t dt R x x x
x R t x t dt P x x
ψ λ ψ λ ψ λ ψ
ψ λ ψ ψ
= + +
= + −
∫
∫
(3) (3) (3) (2) (1)2
(3) (3) (1)3 2
( / , ) ( ) ( / , ) ( ) ( / , ) ( ) [ ( ) ( )]
( ) ( / , ) ( ) ( ) ( ) [ ( ) ( )]
x
r Fr r r rax
Fr r r ra
a x x R t x t dt R x x x P x x
x R t x t dt P x x P x x
ψ λ ψ λ ψ λ ψ ψ
ψ λ ψ ψ ψ
= + + −
= + + −
∫
∫
(4) (4) (4) (3)
(1)3 2
(4) (4)4
(2)(1)3 2
( / , ) ( ) ( / , ) ( ) ( / , ) ( )
( ) ( ) [ ( ) ( )]
( ) ( / , ) ( ) ( ) ( )
[ ( ) ( )] ( ) ( ) .
x
Fr r ra
r rx
Fr r ra
r r
a x x R t x t dt R x x x
P x x P x x
x R t x t dt P x x
P x x P x x
ψ λ ψ λ ψ λ ψ
ψ ψ
ψ λ ψ ψ
ψ ψ
= + +
+ −
= + −
+ −
∫
∫
By definition, ),/( λxtR being a solution of )()4( xψ = )(xψλ implies that (4) ( / , ) ( / , )R t x R t xλ λ λ= and so
(4) (4) (4)
(2)(1)4 3 2
( ) ( ) ( / , ) ( )
( ) ( ) [ ( ) ( )] ( ) ( )
x
r Fr ra
r r r
x x R t x t dt
P x x P x x P x x
ψ ψ λ λ ψ
ψ ψ ψ
= +
− + −
∫ (22)
Substituting (21) into (22) we have (2)(4) (4) (1)
4 3 2( ) ( ) [ ( ) ( )] ( ) ( ) [ ( ) ( )] ( ) ( ) .r Fr r Fr r r rx x x x P x x P x x P x xψ ψ λ ψ ψ ψ ψ ψ = + − − + − But, )()4( xψ = )(xψλ implies that )()4( xFrψ = )(xFrψλ and hence,
134 Asymptotic behaviour of boundary condition functions …
(4) (4)
4(2)(1)
3 2
( ) ( ) [ ( ) ( )] ( ) ( )
[ ( ) ( )] ( ) ( )r Fr r Fr r
r r
x x x x P x x
P x x P x x
ψ λψ λ ψ ψ ψ
ψ ψ
= + − −
+ −
)()4( xrψ = )(xrψλ − )()(4 xxP rψ + )1(3 )]()([ xxP rψ − [ ] )2(
2 )()( xxP rψ (23)
)()4( xrψ + )()(4 xxP rψ − )1(3 )]()([ xxP rψ + [ ] )2(
2 )()( xxP rψ = )(xrψλ , hence
( )xrψ ( )41 ≤≤ r are solutions of (23) such that *)()( MBΨ aa = .
Similarly ( )xrχ ( )41 ≤≤ r are solutions of (23) such that *)()( NBΧ bb = .
Hence we conclude that )},/(),,/({ λχλψ xbxa rr are the set of boundary
conditions of π . □
Theorem 3
( ))()1()1( ),/( axssFr ePOxa −−− = σλψ as ∞→λ ( )4,1 ≤≤ sr
),/()1( λχ xbsFr
− = ( ))()1( xbs ePO −− σ as ∞→λ ( )4,1 ≤≤ sr
Proof.
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
4
1
1 2 2 3 3 4 4
(1) (2) (3)1 2 1 3 1 4 1
( / , ) ( / , )
/ , / , / , / ,
/ , / , / , / , .
Fr rs ss
ri r r r
ri r r r
a x m f a x
m f a x m f a x m f a x m f a x
m f a x m f a x m f a x m f a x
ψ λ λ
λ λ λ λ
λ λ λ λ
=
=
= + + +
= − + −
∑
For r=1 1r = and using (12) we have
{ }
{ }
{ }
{ }
3 ( ) ( ) ( ) ( )111
2 ( ) ( ) ( ) ( )12
1 ( ) ( ) ( ) ( )13
( ) ( ) ( ) ( )14
( / , )4
4
4
4
p x a p x a ip x a iP x aF
p x a p x a ip x a iP x a
p x a p x a ip x a iP x a
p x a p x a ip x a iP x a
ma x p e e ie ie
m p e e e e
m p e e ie ie
m e e e e
ψ λ − − − − − − −
− − − − − − −
− − − − − − −
− − − − − −
= − + − +
− − − + +
+ − + + −
− − − − −
Furthermore,
E.K. Essel, E. Yankson and S.M. Naandam 135
3 ( ) ( ) ( ) ( )1 1
2 ( ) ( ) ( ) ( )2
1 ( ) ( ) ( ) ( )3
( ) ( ) ( ) ( )4
( / , ) p x a p x a ip x a iP x aF
p x a p x a ip x a iP x a
p x a p x a ip x a iP x a
p x a p x a ip x a iP x a
a x K P e e ie ie
K P e e e e
K P e e ie ie
K e e e e
ψ λ − − − − − − −
− − − − − − −
− − − − − − −
− − − − − −
≤ − + − +
+ − − + +
+ − + + −
+ − − − −
{ }{ }{ }
{ }
3 ( ) ( ) ( ) ( )1
2 ( ) ( ) ( ) ( )2
1 ( ) ( ) ( ) ( )3
( ) ( ) ( ) ( )4
p x a p x a ip x a iP x a
p x a p x a ip x a iP x a
p x a p x a ip x a iP x a
p x a p x a ip x a iP x a
K P e e e e
K P e e e e
K P e e e e
K e e e e
− − − − − − −
− − − − − − −
− − − − − − −
− − − − − −
≤ + + +
+ + + +
+ + + +
+ + + +
( )
{ }
1 2 34 3 2 1
( ) ( ) ( ) ( )p x a p x a ip x a iP x a
K K P K P K P
e e e e
− − −
− − − − − −
≤ + + + ⋅
⋅ + + + (24)
Since p = τσρ i+= we see that
)( axpe − = ;)( axe −σ
)( axpe −− = ;)( axe −−σ
)( axipe − = ;)( axe −−τ
)( axipe −− = .)( axe −−τ
(25)
From (24) and (25) we find that
( ){ }1
1 2 3 ( ) ( ) ( ) ( )4 3 2 1
( / , )F
x a x a x a x a
a x
K K P K P K P e e e eσ σ τ τ
ψ λ− − − − − − − − −≤ + + + + + +
≤ ( )31
22
134
−−− +++ PKPKPKK { }( ) ( ) ( ) ( )x a x a x a x ae e e eσ σ τ τ− − − −+ + +
≤ 2 ( )31
22
134
−−− +++ PKPKPKK { }( )( ) x ax ae eτσ −− +
as ∞→λ , ∞→P and so 02 →−P ; 01 →−P ; .03 →−P
Hence, ),/(1 λψ xaF ≤ ( ))()( axax eeK −− + τσ , where 𝐾 = 2𝐾4. But from (9)
4cos41
θσ r= and 4sin41
θτ r= and so τσ > and 0≥σ . Thus,
( )( )1( / , ) x a
F a x K eσψ λ −≤ as ∞→λ and so ( )( )1( / , ) x a
F a x O eσψ λ −= as
.∞→λ
136 Asymptotic behaviour of boundary condition functions …
By similar argument
),/()1(1 λψ xaF = ( ))( axepO −σ as ∞→λ )41( ≤≤ s
and so,
),/()1(1 λψ xas
F− = ( ))()1( axs ePO −− σ as ∞→λ ).41( ≤≤ s
Also,
),/()1(2 λψ xas
F− = ( ))()1( axs ePO −− σ as ∞→λ )41( ≤≤ s
Similarly,
),/()1(1 λχ xbs
F− = ( ))()1( xbs ePO −− σ as ∞→λ )41( ≤≤ s
),/(2 λχ xbF = ( ))()1( xbs ePO −− σ as ∞→λ ).41( ≤≤ s
By combining all the result we have
),/()1( λψ xasFr
− = ( ))()1( axs ePO −− σ as ∞→λ )41( ≤≤ s (26)
),/()1( λχ xbsFr
− = ( ))()1( xbs ePO −− σ as ∞→λ ).41( ≤≤ s
□
Theorem 4
),/()1( λxtR s− = ( ))()2( txs ePO −− σ as ∞→λ )41( ≤≤ s
),/()1( λxtS s− = ( ))()2( xts ePO −− σ as ∞→λ )41( ≤≤ s and .τσ ip +=
Proof.
),/( λxtR = ),/()( 14 λxtftP − ),/()( )1(13 λxtftP + ),/()( )2(
12 λxtftP (27)
From (12)
{ }
{ }
3 ( ) ( ) ( ) ( )1
3 ( )( ) ( )( ) ( )( ) ( )( )
1( / , )414
p x a p x a ip x a iP x a
i x a i x a i i x a i i x a
f a x p e e ie ie
p e e ie ieσ τ σ τ σ τ σ τ
λ − − − − − − −
− + − − + − + − − + −
= − + − +
= − + − +
),/(1 λxtf ≤3
4
−P ( )( ) ( )( ) ( )( ) ( )( )i x a i x a i i x a i i x ae e ie ieσ τ σ τ σ τ σ τ+ − − + − + − − + −+ + +
E.K. Essel, E. Yankson and S.M. Naandam 137
If 𝑧 = 𝑥 + 𝑖𝑦 then .xz ee = Hence,
),/(1 λxtf ≤ 3
4
−P )()()()( txtxtxtx eeee −−−− +++ ττσσ
≤ 3
4
−P 2 )()( 2 txtx ee −− + τσ
≤ 3
2
−P )()( txtx ee −− + τσ as .∞→λ
3( )
1( / , )2
x tPf t x eσλ
−−≤ as .∞→λ
Similarly 2
(1) ( )1( / , )
2x tP
f t x eσλ−
−≤ as ∞→λ
and 1
(2) ( )1( / , )
2x tP
f t x eσλ−
−≤ as ∞→λ
Since )(4 tP , )(3 tP and )(2 tP are continuous in [a,b] they are bounded in [a,b] and
so
,)( 14 KtP ≤ ,)( 23 KtP ≤ .)( 32 KtP ≤ (28)
Substituting all the above into (27) we have 3 2 1
1 2 3 ( )( / , )2 2 2
x tK P K P K PR t x eσλ
− − −−
≤ − +
1 ( )( / , ) x tR t x K P eσλ − −≤ as .∞→λ (29)
or
),/( λxtR = ( ))(1 txePO −− σ as ∞→λ (30)
From (27)
),/()1( λxtR = ),/()( )1(14 λxtftP − ),/()( )2(
13 λxtftP + ),/()( )3(12 λxtftP .
138 Asymptotic behaviour of boundary condition functions …
By similar argument
),/()1( λxtR ≤ 2
21
−PK −
2
12
−PK +
2
03
−PK )( txe −σ ≤ 0−PK )( txe −σ
or
),/()1( λxtR = ( ))( txeO −σ as ∞→λ (31)
Combining (30) and (31) and generalizing we have
),/()1( λxtR s− = ( ))(2 txs ePO −− σ as ∞→λ ).41( ≤≤ s
Similarly
),/()1( λxtS s− = ( ))(2 txs ePO −− σ as ∞→λ ).41( ≤≤ s □
Theorem 5
),/( λψ xar = ( ))( axeO −σ as ∞→λ
),/( λχ xbr = ( ))( xbeO −σ as .∞→λ
Proof. Let
),/( λψ xar = ( ))(),/( axr exaF −σλ as ∞→λ (32)
then
)(trψ = ( ))()( atr etF −σ as ∞→λ (33)
and
),/( λxaFr = ),/()( λψσ xae rax−− (34)
Next we substitute ),/( λψ xar = ∫+x
arFr dttxtRxa )(),/(),/( ψλλψ into (34) to
obtain
),/( λxaFr = )( axe −−σ ∫+x
arFr dttxtRxa )(),/(),/( ψλλψ . (35)
Similarly, substituting (33) into (35) we have
E.K. Essel, E. Yankson and S.M. Naandam 139
),/( λxaFr = )( axe −−σ ∫+x
aFr xtRxa ),/(),/( λλψ )( xte −σ dttFr )( as .∞→λ
Now
),/( λxaFr ≤ .)(),/(),/( )()( dttFextRxae rxt
x
aFr
ax −−− ∫+ σσ λλψ (36)
But from (26) and (29)
),/( λψ xaFr ≤ )(1
axeK −σ ; ),/( λxtR ≤ )(12
txePK −− σ
By the mean value theorem for integrals
∫x
ar dttF )( = ))(,/( axaFr −λξ
where a <ξ < x . But (𝑥 − 𝑎) being a constant imply that
∫x
ar dttF )( = KxaFr ),/( λ
where ],,[ bax∈ K = (𝑥 − 𝑎). Substituting the above into (36) we have
),/( λxaFr ≤ ),/(131 λxaFPKK r
−+
where )(223 axKKKK −== . Therefore,
),/( λxaFr − ),/(13 λxaFPK r
− ≤ 1K .
Hence,
),/( λxaFr )1( 13
−− PK ≤ 1K
and so
),/( λxaFr ≤)1( 1
3
1−− PK
K provided that 01 1
3 >− −PK . (37)
This is true if P is large enough. Substituting (37) into (32) we have
)(4),/( ax
r eKxa −≤ σλψ as ∞→λ
where 4K = 13
1
1 −− PKK
and so
140 Asymptotic behaviour of boundary condition functions …
),/( λψ xar = )( )( axeO −σ as .∞→λ (38)
Similarly ),/( λχ xbr = )( )( xbeO −σ as .∞→λ □
Theorem 6
),/()1( λψ xasr
− ~ ),/()1( λψ xasFr
− as ∞→λ
),/()1( λχ xbsr
− ~ ),/()1( λχ xbsFr
− as ∞→λ ( ).41 ≤≤ r
Proof.
),/( λψ xar = ∫+x
arFr dttxtRxa )(),/(),/( ψλλψ (39)
),/()1( λψ xar = ∫+x
arFr dttxtRxa )(),/(),/( )1()1( ψλλψ (40)
From (30), (31) and (38)
( )( )
1 ( )
(1) ( )
( / , ) as
( / , ) as
x t
x t
R t x O P e
R t x O e
σ
σ
λ λ
λ λ
− −
−
= →∞
= →∞
(41)
)(trψ = ( ))( ateO −σ as ∞→λ
According to (35) there exist constants 1K , 2K and 3K such that
1 ( )1
(1) ( )2
( / , ) as
( / , ) as
x t
x t
R t x K P e
R t x K e
σ
σ
λ λ
λ λ
− −
−
≤ →∞
≤ →∞ (42)
)(trψ ≤ )(3
ateK −σ as ∞→λ
Substituting (42) into (39) we have
dttxtR r
x
a
)(),/( ψλ∫ ≤ dtePKK axx
a
)(131
−−
∫ σ
≤ )()(131 axePKK ax −−− σ since ∫ −=
x
a
axdt )(
= )(131 )( axeabPKK −− − σ since if bxa ≤≤ , then axab −≥−
E.K. Essel, E. Yankson and S.M. Naandam 141
= )(14
axePK −− σ as ∞→λ
∫x
ar dttxtR )(),/( ψλ = ( ))(1 axePO −− σ as .∞→λ (43)
Similarly ∫x
ar dttxtR )(),/()1( ψλ = ( ))( axeO −σ (44)
From (39) and (43)
),/( λψ xar = ( ))(1),/( axFr ePOxa −−+ σλψ as ∞→λ (45)
From (40) and (44)
),/()1( λψ xar = ( ))()1( ),/( axFr eOxa −+ σλψ as .∞→λ (46)
Combining (45) and (46) we obtain a general formula
),/()1( λψ xasr
− = ( ))()2()1( ),/( axssFr ePOxa −−− + σλψ as .∞→λ (47)
Similarly
),/()1( λχ xbsr
− = ( ))()2()1( ),/( bxssFr ePOxb −−− + σλχ as .∞→λ (48)
If follows from theorem 3 and (47) that
),/()1( λψ xasr
− ~ ),/()1( λψ xasFr
− as .∞→λ
Similarly it follows from theorem 3 and (48) that
),/()1( λχ xbsr
− ~ ),/()1( λχ xbsFr
− as ∞→λ 4,1( ≤≤ sr ).
The proof is complete. □
5 Conclusion
We have successfully proved that the boundary condition functions for the
boundary value problem in (1) and (2) are asymptotically equivalent for large
values of |𝜆|, to the boundary condition functions for the corresponding Fourier
boundary value problem for 𝜋, given by (3) and (4).
142 Asymptotic behaviour of boundary condition functions …
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