Available online throug
ISSN 2229 – 5046ON THE RIESZ BASIS AND BASIS PROPERTY OF THE EIGENFUNCTIONS
OF THE MODIFIED FRANKL PROBLEM WITH A NONLOCAL ODDNESS
CONDITION OF THE TIRED KIND
NASER ABBASI*
a,1, HAMID MOTTAGHI GOLSHAN
aa
Department of Mathematics, Lorestan University, P. O. Box 465, Khoramabad, Iran.
(Received On: 03-07-15; Revised & Accepted On: 27-07-15)
ABSTRACT
I
n the present paper, we consider a new boundaries conditions of the tired kind. we prove the basis property, completeness, and the minimality of the eigen functions with a nonlocal Oddness condition of the tired kind.Keywords and phrases: Frankl problem, Lebesgue integral, Holder inequality, Bessel equation, Sobolev space.
1. INTRODUCTION
The classical Frankl problem was considered in [3]. The problem was further developed in [2, pp.339-345], [8, pp.235-252]. The modified Frankl problem with a nonlocal boundary condition of the first kind was studied in [1, 6]. The basis property of an eigen functions of the Frankl problem with a nonlocal parity conditions in the space sobolev was studied in[7]. In the present paper, we consider a new boundaries conditions of the tired kind and prove the completeness, the basis property, and the minimality of the eigen functions in the space L2. This analysis may be of interest in itself.
2. PRELIMINARIES
Definition 2.1: In the domainD (DD1D2), we seek a solution of the modified generalizedFrankl problem
2
1 2
sgn( )
sgn(
)
0 in (
),
xx yy
u
+
y u
+
µ
x
+
y u
=
D
+∪
D
−∪
D
− (1)with the boundary conditions
(1, )
0,
0,
,
2
u
θ
=
θ
∈
π
(2)(0, )
0,
( 1, 0)
(0,1)
u
y
y
x
∂
=
∈ −
∪
∂
(3)(0, )
(0,
),
[0,1],
(0, 0)
( , 0).
ku
y
=
u
−
y y
∈
ku
+ =
u x
−
(4) whereu x y
( , )
is a regular solution in the class0 2 2
1 2 1 2
(
)
(
)
(
),
u
∈
C D
+∪
D
−∪
D
−∩
C
D
−∩
C
D
−and where
1
2
( , ) : 0
1, 0
,
2
1
( , ) :
1,
0 ,
2
1
( , ) :
1
, 0
,
2
D
r
r
D
x y
y
x
y
y
D
x y
x
y
x
x
π
θ
θ
+
−
−
=
< <
< <
−
=
− < < +
< <
=
− < < −
< <
( , 0)
( , 0),
, 0
1.
u
u
x
x
x
y
y
κ
∂
+ =
∂
− −∞ < < ∞ < <
κ
∂
∂
(5)Corresponding Author:
Naser Abbasi
*
a,1Definition 2.2: .System{ }
x
n n N∈⊂
X
is called complete inX
ifL x
[{ }
n n N∈]
=
X
.
Definition 2.3: .System{ }
x
n n N∈⊂
X
is called minimal inX
ifx
k∈
/
L x
[{ }
n n N∈],
∀ ∈
k
N
.
Remark 2.1: If the system
{ }
x
n n N∈⊂
X
minimal inL I
p( ),
then it is also minimal inL J
p( ),
forJ
⊃
I
, and if it is complete inL J
p( )
forJ
⊂
I
.
Theorem 2.5 ([5]): The eigenvalues and eigenfunctions of problem (1-5) can be written out in two series. In the first series, the eigenvalues
λ µ
=
nk2 are found from the equation4n( nk) 0,
J (6)
where
µ
nk,
n k
,
=
1, 2 ,...
, are roots of the Bessel equation (6),J
α( )
z
, is the Bessel function [4], and the eigenfunctions are given by the formula4
4 1
4 2
(
) cos(4 )
,
in
;
2
(
) cosh(4 ) ,
in
;
(
) cosh(4 ) ,
in
,
nk n nk
nk nk n nk
nk n nk
A J
r
n
D
u
kA J
n
D
kA J
R
n
D
π
µ
θ
µ ρ
ψ
µ
ϕ
+
−
−
−
=
(7)
where
x
=
r
cos ,
θ
y
=
r
sin
θ
for0
,
2 2 22
r
x
y
π
θ
≤ ≤
= +
inD
+,x
=
ρ
cosh ,
ψ
y
=
ρ
sinh ,
ψ
for,0
< <
ρ
1,
2 2 2
0,
x
y
,
ψ
ρ
−∞ < <
=
−
inD
−1, and,x
=
R
sinh ,
ϕ
y
= −
R
cosh ,
ϕ
for,0
< < +∞
ϕ
,
R
2=
y
2−
x
2inD
−2.In the second series, the eigenvalues
λ µ
=
nk2 are found from the equation
J
4(n+)(
µ
nk)
=
0.
(8) wheren k
, 1, 2 ,...
=
and the(
µ
nk)
are the roots of the Bessel equation (8).4( )
4( ) 1
4( ) 2
(
) cos 4(
)
,
in
;
2
(
)[cosh 4(
) cos 4(
)
sinh 4(
) cos 4(
)],
in
;
2
(
) cosh 4(
) [cos 4(
)
sin 4(
)
],
in
,
2
2
nk n nk
nk nk n nk
nk n nk
A J
r
n
D
u
A J
n
n
n
n
D
kA J
R
n
n
n
D
π
µ
θ
π
µ ρ
ϕ
κ
ψ
π
π
µ
ϕ
+ +
+ −
+ −
+
−
=
+
+
+
+
+
+
+
−
+
(9)
where
2
1
1
arcsin
,
0,
2
1
κ
π
κ
∆ =
∆ ∈
+
, and
1 2 2
4
0
(
)
1,
nk n nk
A
∫
J
µ
r rdr
=
1 2 2
4
0
(
)
1,
nk n nk
A
∫
J
+µ
r rdr
=
A
nk>
0
andA
nk>
0
.3. THE COMPLETENESS, THE BASIS PROPERTY, and MINIMALITY of THE EIGENFUNCTIONS Theorem 3.1: The function system
0 1
cos(4 )
, cos 4(
)
,
2
n2
nn
π
θ
n
π
θ
∞ ∞
= =
−
+
−
(10)is complete and a Riesz basis in 2
0,
2
L
π
, provided that1 1
,
4 2
−
∈
Proof: In order to prove this theorem we use the method in [1, 6] by considering convergence function
0 1
( )
cos 4
cos 4(
)
,
2
2
n n
n n
f
θ
A
n
π
θ
B
n
π
θ
∞ ∞
= =
=
−
+
+
−
∑
∑
(11)In 2
0,
2
L
π
and Riesz basis the systemsin 4(
n
)
π θ
2
+
−
for1 3
,
4 4
−
∈
.Remark 3.2: For
1
4
−
<
the system (10) is not complete but is minimal, for3
4
>
is complete but isnot minimal, andif
1
4
−
=
, is complete and minimal.Theorem 3.3: The system of eigenfunctions
4
( , )
(
) cos(4 )
,
2
nk nk n nk
u
r
θ
=
A J
µ
r
n
π
−
θ
4( )
( , )
(
)[cosh 4(
) cos 4(
)
,
2
nk nk n nk
u
r
θ
=
A J
+µ
r
n
+
ϕ
n
+
π
is complete and basis in the space 2
0,
2
L
π
, therefore0 2
( , )
nk( , )
0,
f r
u
r
rdrd
π
θ
θ
θ
=
∫
2
0
f r
( , )
u
nk( , )
r
rdrd
0,
π
θ
θ
θ
=
∫
and
0,
2
f
∈
L
π
thenf
=
0
in0,
2
π
.Proof: Using fobini theorem and Lebesgue‘s integral for any
n k
,
=
1, 2,...
we have2 0
0
f r
( , )
u
nk( , )
r
rd dr
π
θ
θ
θ
=
∫
1
2 4
0
(
rJ
n(
nkr
) ( , ) cos(4 )
0f r
n
2
d
)
dr
,
π
π
µ
θ
−
θ θ
∫
∫
Again since
0,
22
f
∈
L
π
so;1 2
2
0
| ( , ) |
0f r
d dr
.
π
θ
θ
< ∞
∫ ∫
In so much system
{
r J
4n(
µ
nkr
)}
k 1∞
= in
L
2(0,1)
is orthogonal and complete, it is enough to prove:2 2
0
( , ) cos(4 )
2
(0,1).
r
f r
n
d
L
π
π
θ
−
θ θ
∈
∫
Using the Holder inequality
2 2
2 2 2
0 0 0
1
|
( , ) co s
(4 )
|
|
( , ) |
2
2
r
f r
n
d
r
f
r
d
d
π
π
π πθ
−
θ θ
<
θ
θ
θ
∫
∫
∫
2 2
2 2
0
| ( , ) |
0| ( , ) |
,
4
r
f r
d
4
r
f r
d
π π
π
θ
θ
π
θ
θ
with the integration interval (0,1).
1 1
2 2
2 2
0
|
r
0f r
( , ) co s
(4 )
n
2
d
|
dr
| ( , ) |
4
0 0r f r
drd
.
π
π
π
πθ
−
θ θ
<
θ
θ
< ∞
∫ ∫
∫ ∫
This inequality is equivalent to
1 2
1 2
2
0
r
|
0f r
( , ) cos(4 )
n
2
d
|
dr
.
π
π
θ
θ θ
−
< ∞
∫
∫
Also system
{
r J
4n(
µ
nkr
)}
∞k=1 is orthogonal and complete in 20,
2
L
π
of relation1
2 4
0
(
r J
n(
nkr
)
r
0f r
( , ) cos(4 )
n
2
d
)
dr
0,
π
π
µ
θ
−
θ θ
=
∫
∫
imply that
2
0
( , ) cos(4 )
2
0.
r
f r
n
d
π
π
θ
−
θ θ
=
∫
According to theorem 2,we conclude that
f r
( , )
θ
=
0
inL
2(0,1).
Similarly, if we consider the above calculations forsequence
1
cos 4(
)
,
2
nn
π θ
∞
=
+
−
We have
2
0
( , ) cos 4(
)
2
0.
r
f r
n
d
π
π
θ
+
−
θ θ
=
∫
Because completeness
0
cos 4(
)
, ( , )
0
2
nn
π θ
f r
θ
∞
=
+
−
=
in2
(0,1).
L
The proof of the theorem is complete.
Theorem 3.4: The system of eigenfunctions
u
nk( , )
r
θ
andu
nk( , )
r
θ
of the problem (1)-(5) is a Riesz basis in thespace
0,
,
2
L
π
where,(
)
(
)
1 1
1 1
2 2 2 2
4 4( )
0
(
)
,
(
0)
nk n nk nk n nk
A
J
µ
r rdr
A
J
µ
r rdr
− −
+
=
∫
=
∫
.Proof: Theorem 3.3 results from Theorem 3.2 and the completeness and orthogonality of the system
{
A J
nk 4n(
µ
nkr
)}
k 1∞ =
, for
n
>
0
and{
A J
nk 4(n+)(
µ
nk) }
r
k∞=1forn
>
1
inL
2(0,1).
ACKNOWLEDGMENTS
The author is grateful to Academician E. I. Moiseev for his interest in this work.
REFERENCES
1. N.Abbasi, Basis property and completeness of the eigenfunctions of the Frankl problem. (Russian) Dokl.Akad.Nauk425 (2009), no. 3, 295-298; translation in Dokl. Math. 79 (2009), no. 2, 193-196.
2. A. V.Bitsadze, Nekotoryeklassyuravneni v chastnykhproizvodnykh. (Russian) [Some classes of partial differential equations] “Nauka”, Moscow, 1981. 448 pp.
4. H.vanHaeringen,L.P. Kok, Higher transcendental functions, Vol. II [McGraw-Hill, New York, 1953; MR 15, 419] by A.Erdlyi, W. Magnus, F. Oberhettinger and F. G. Tricomi. Math. Comp. 41 (1983), no. 164, 778. 5. E.I. Moiseev, The basis property for systems of sines and cosines. (Russian) Dokl. Akad. Nauk SSSR 275
(1984), no.4, 794–798.
6. N. Abbasi, E.I.Moiseev, Basis property of eigenfunctins of the Generalized Gasedynamic problem of Frankl with a nonlocal oddness conditions, integral transforms and special Functions.(England) 21 (2010), no. 4, 286-294.
7. E. I.Moiseev,N. Abbasi, The basis property of an eigenfunction of the Frankl problem with a nonlocal parity conditionin the space Sobolev (W1p (0,π)). Integral Transforms Spec. Funct. (England) 22 (2011), no. 6,
415–421.
8. M.M.Smirnov, Uravneniyasmeshannogotipa. (Russian) [Equations of mixed type] Izdat. “Nauka”, Moscow 1970 295pp.
9. A. Zygmund, Trigonometrical series. Dover Publications, New York, 1955.vii+329. pp.
Source of support: Nil, Conflict of interest: None Declared
