Hikmet Koyunbakan

Boundary Value Problems

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Koyunbakan Boundary Value Problems 2013, 2013:272
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RESEARCH Open Access
Inverse nodal problem for p-Laplacian
energy-dependent Sturm-Liouville equation
Hikmet Koyunbakan*
*Correspondence:
hkoyunbakan@gmail.com
Department of Mathematics, Firat
University, Elazig, 23119, Turkey
Department of Mathematics and
Statistics, Missouri University of
Science and Technology, Rolla, MO,
USA
Abstract
In this study, the inverse nodal problem is solved for p-Laplacian Schrödinger
equation with energy-dependent potential function with the Dirichlet conditions.
Asymptotic estimates of eigenvalues, nodal points and nodal lengths are given by
using Prüfer substitution. Especially, an explicit formula for a potential function is
given by using nodal lengths. Results are more general than the classical p-Laplacian
Sturm-Liouville problem. For the proofs, methods previously developed by Law et al.
and Wang et al., in 2009 and 2011, respectively, are used. In there, they solved an
inverse nodal problem for the classical p-Laplacian Sturm-Liouville equation with
eigenparameter boundary conditions.
MSC: 34A55; 34L20
Keywords: Prüfer substitution; inverse nodal problem; p-Laplacian equation
1 Introduction
Consider the following p-Laplacian eigenvalue problem for
–
[(
u′
)(p–)]′ = (p – )(λ – q(x))u(p–),  < x < , (.)
with the boundary conditions
u() = u() = , (.)
where q ∈ L(, ) is a real-valued function, p >  is a constant, u(p–) := |u|p– Sgnu and λ is
the spectral parameter []. Equation (.) is also known as a one-dimensional p-Laplacian
eigenvalue equation.Note thatwhen p = , equation (.) becomes a Sturm-Liouville equa-
tion as
–u′′ + qu = λu
and the inverse problem described in (.), (.) in the [–].
The determination of the form of a diﬀerential operator from spectral data associated
with it has enjoyed close attention from a number of authors in recent years. One such
operator is the Sturm-Liouville operator. In the typical formulation of the inverse Sturm-
Liouville problem, one seeks to recover both q and constants by giving the eigenvalueswith
another piece of spectral data. These data can take several forms, leading tomany versions
©2013 Koyunbakan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons At-
tribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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of the problem. Especially, the recent interest is a study by Hald and McLaughlin [, ]
wherein the given spectral information consists of a set of nodal points of eigenfunctions
for the Sturm-Liouville problems. These results were extended to the case of problems
with eigenparameter-dependent boundary conditions by Browne and Sleeman []. On
the other hand, Law et al. [], Law and Yang [] solved the inverse nodal problem of
determining the smoothness of the potential function q of the Sturm-Liouville problem
by using nodal data. In the past few years, the inverse nodal problem of Sturm-Liouville
problem has been investigated by several authors [, –].
When q = , consider the problem
–
(
u′(p–)
)′ = (p – )λu(p–), u() = u() = .
The eigenvalues of this problem were given as []
λn = (nπp)p, n = , , , . . . ,
where
πp = 
∫ 

dt
( – tp)/p =
π
p sin(πp )
and an associated eigenfunction is denoted by Sp. Sp and S′p are periodic functions satis-
fying the identity
[
Sp(x)
]p + [S′p(x)]p = 
for arbitrary x ∈ R. These functions are known as generalized sine and cosine functions
and for p =  become sine and cosine [].
Now, we present some further properties of Sp for deriving more detailed asymptotic
formulas. These formulas are crucial in the solution of our problem.
Lemma . []
(a) For S′p = ,
(
S′p
)′ = –
∣∣∣∣SpS′p
∣∣∣∣
p–
Sp;
(b)
(
SpS′(p–)p
)′ = ∣∣S′p∣∣p – (p – )|Sp|p =  – p|Sp|p = ( – p) + p∣∣S′p∣∣p.
According to the Sturm-Liouville theory, the zero set Xn = {x(n)j }nj= of the eigenfunction
un corresponding to λn is called the nodal set and lnj = xnj+ – xnj is deﬁned as the nodal
length of un. Using the nodal data, some uniqueness results, reconstruction and stability
of potential functions have been obtained by many authors [, , –, ].
Consider the p-Laplacian eigenvalue problem
–
[(
u′
)(p–)]′ = (p – )[λ – q(x) – λr(x)]u(p–),  < x < , (.)
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with the Dirichlet conditions
u() = u() = , (.)
or with the Neumann boundary conditions
u′() = u′() = , (.)
where q ∈ L(, ) and r ∈ W  (, ) are real-valued functions, p >  is a constant, u(p–) :=
|u|p– Sgnu and λ is the spectral parameter.
In this paper, the function r is known a priori and we try to construct the unknown
function q by the dense nodal points in the interval considered.
For p = , equation (.) becomes
–u′′ + [q + λr]u = λu. (.)
This equation is known as the diﬀusion equation or quadratic of diﬀerential pencil.
Eigenvalue equation (.) is important for both classical and quantummechanics. For ex-
ample, such problems arise in solving the Klein-Gordon equations, which describe the
motion of massless particles such as photons. Sturm-Liouville energy-dependent equa-
tions are also used for modelling vibrations of mechanical systems in viscous media (see
[]).We note that in this type of problem the spectral parameter λ is related to the energy
of the system, and this motivates the terminology ‘energy-dependent’ used for the spectral
problem of the form (.). Inverse problems of quadratic pencil have been solved by many
authors in the references [, , , –].
As in the p-Laplacian Sturm-Liouville problem, for q = r = , eigenvalues of the problem
given by (.), (.) are of the form
λn = (nπp)p
and associated eigenfunctions are denoted by Sp.
This paper is organized as follows. In Section , we give asymptotic formulas for eigen-
values, nodal points and nodal lengths. In Section , we give a reconstruction formula for
diﬀerential pencil by using nodal parameters.
2 Asymptotic estimates of nodal parameters
In this section, we study the properties of eigenvalues of p-Laplacian operator (.) with
Dirichlet conditions (.). For this, we introduce Prüfer substitution. One may easily ob-
tain similar results for Neumann problems.
We deﬁne a modiﬁed Prüfer substitution
u(x) = c(x)Sp
(
λ/pθ (x)
)
,
u′(x) = λ/pc(x)S′p
(
λ/pθ (x)
) (.)
or
u′(x)
u(x) = λ
/p S
′
p(λ/pθ (x))
Sp(λ/pθ (x))
. (.)
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Diﬀerentiating both sides of equation (.) with respect to x and applying Lemma ., one
obtains that
θ ′ =  – q
λ
Spp –

λ
rSpp . (.)
Theorem. The eigenvalues λn of the Dirichlet problem given in (.), (.) have the form
λ/pn = nπp +

p(nπp)p–
∫ 

q(t)dt + 
p(nπp)
p–
p
∫ 

r(t)dt +O
( 
n
p+
p
)
. (.)
Proof For problem (.), (.), let λ = λn, θ () =  and θ () = nπp
λ
/p
n
. Firstly, we integrate both
sides of (.) over the interval [, ]:
nπp
λ
/p
n
=  – 
λn
∫ 

q(t)Spp(t)dt –

λn
∫ 

r(t)Spp(t)dt.
Using the identity
d
dt
[
Sp
(
λ/pn θ (t)
)
S′p
(
λ/pn θ (t)
)p–] = ( – p∣∣Sp(λ/pn θ (t))∣∣p)λ/pn θ ′(t)
and Lemma .(b), we get
nπp
λ
/p
n
=  – 
λnp
∫ 

q(t)dt – 
λnp
∫ 

r(t)dt
+ 
λnp
∫ 

q(t)
λ
/p
n θ ′(t)
d
dt
[
Sp
(
λ/pn θ (t)
)
S′p
(
λ/pn θ (t)
)p–]dt
+ 
λnp
∫ 

r(t)
λ
/p
n θ ′(t)
d
dt
[
Sp
(
λ/pn θ (t)
)
S′p
(
λ/pn θ (t)
)p–]dt. (.)
Then, using integration by parts, we have
∫ 

q(t)
λ
/p
n θ ′(t)
d
dt
[
SpS′p–p
]
dt = –λ–/pn
∫ 

G
(
λ/pn θ (t)
) d
dt
( q(t)
θ ′(t)
)
dt
= O
( 
λ
/p
n
)
,
where
G
(
λ/pn θ (x)
)
= Sp
(
λ/pn θ (x)
)
S′p
(
λ/pn θ (x)
)p–
and when x = , ,
G
(
λ/pn θ (x)
)
= .
Similarly, one can show that
∫ 

r(t)
λ
/p
n θ ′(t)
d
dt
[
SpS′p–p
]
dt =O
( 
λ
/p
n
)
.
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Inserting these values in (.) and after some straightforward computations, we ob-
tain (.). 
Theorem . For problem (.), (.), the nodal points expansion satisﬁes
xnj =
j
n +
j
pnp+(πp)p
∫ xnj

q(t)dt + j
pn p–p (πp)
p–
p
∫ xnj

r(t)dt
+ 
(nπp)
p

∫ xnj

r(x)Spp dx +

(nπp)p
∫ xnj

q(x)Spp dx +O
( j
n
p+
p
)
.
Proof Let λ = λn and integrating (.) from  to xnj , we have
j · πp
λ
/p
n
= xnj –
∫ xnj

r(x)
λn
Spp dx –
∫ xnj

q(x)
λn
Spp dx.
By using the estimates of eigenvalues as

λ
/p
n
= nπp
+ p(nπp)p+
∫ 

q(t)dt + 
p(nπp) p–p
∫ 

r(t)dt +O
( 
n
p+
p
)
,
we obtain
xnj =
j
n +
j
pnp+(πp)p
∫ xnj

q(t)dt + j
pn p–p (πp)
p–
p
∫ xnj

r(t)dt
+ 
(nπp)
p

∫ xnj

r(x)Spp dx +

(nπp)p
∫ xnj

q(x)Spp dx +O
( j
n
p+
p
)
. 
Theorem . As n→ ∞,
lnj =
πp
λ
/p
n
+ pλn
∫ xnj +
xnj
r(t)dt + pλn
∫ xnj +
xnj
q(t)dt +O
( 
λ
+p
p
n
)
. (.)
Proof For large n ∈N, integrating (.) on [xnj ,xnj+] and then
πp
λ
/p
n
= lnj –

λ
∫ xnj+
xnj
r(t)Spp dt –

λ
∫ xnj+
xnj
q(t)Spp dt
or
πp
λ
/p
n
= lnj –

pλn
∫ xnj+
xnj
r(t)dt – pλn
∫ xnj+
xnj
q(t)dt
+ 
λnp
∫ xnj+
xnj

λ
/p
n θ ′(t)
d
dt
[
SpS′p–p
]
r(t)dt
+ 
λnp
∫ xnj+
xnj

λ
/p
n θ ′(t)
d
dt
[
SpS′p–p
]
q(t)dt. (.)
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By Lemma . and a similar process to that used in Theorem ., we obtain that
∫ xnj+
xnj
r(t)
λ
/p
n θ ′(t)
d
dt
[
SpS′p–p
]
dt = –
∫ (j+)πp
jπp
( q(t)
λ
/p
n θ ′(t)
)′
G(τ ) dτ
λ
/p
n θ ′(t)
= O
( 
λ
/p
n
)
,
where G(τ ) = Sp(τ )S′p(τ )(p–) and τ = λ
/p
n θ (x). Similarly, one can show that
∫ xnj+
xnj
q(t)
λ
/p
n θ ′(t)
d
dt
[
SpS′p–p
]
dt =O
( 
λ
/p
n
)
.
Inserting this value in (.), we obtain
lnj =
πp
λ
/p
n
+ pλn
∫ xnj+
xnj
r(t)dt + pλn
∫ xnj+
xnj
q(t)dt +O
( 
λ
+p
p
n
)
,
and by Theorem .,
lnj =

n +

p(nπp)p/
∫ xnj+
xnj
r(t)dt + p(nπp)p
∫ xnj+
xnj
q(t)dt +O
( 
n
+p
p
)
. 
3 Reconstruction of a potential function in the differential pencil
In this section, we give an explicit formula for a potential function. The method used in
the proof of the theorem is similar to that for classical Sturm-Liouville problems [, ].
Theorem . Let q ∈ L(, ), r ∈W  (, ) and assume r that on the interval [, ] is given
a priori. Then
q(x) = lim
n→∞pλ

n
(
λ
/p
n lnj
πp
– r(x)pλn
– 
)
for j = jn(x) =max{j : xnj ≤ x}.
Proof Applying the mean value theorem for integrals to (.), with ﬁxed n, there exists
z ∈ (xnj ,xnj+), we obtain
lnj =
πp
λ
/p
n
+ pλn
r(z)lnj +

pλn
q(z)lnj +O
( 
λ
+p
p
n
)
or
q(z) = pλn
(
πp
λ
/p
n lnj
)(
λ
/p
n lnj
πp
– r(z)pλn
λ
/p
n lnj
πp
– 
)
.
Considering (.), we can write that for n→ ∞,
λ
/p
n lnj
πp
= .
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Then
q(x) = lim
n→∞pλ

n
(
λ
/p
n lnj
πp
– rpλn
– 
)
.
This completes the proof. 
Conclusion . In Theorem ., Theorem ., Theorem . and Theorem ., taking r =
, we obtain results of the Sturm-Liouville problem given in [].
Conclusion . In Theorem ., Theorem ., Theorem . and Theorem ., taking p =
, we obtain the results of an inverse nodal problem for diﬀerential pencil [].
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author would like to thank the referees for valuable comments and suggestions on improving this paper.
Received: 15 October 2013 Accepted: 19 November 2013 Published: 12 Dec 2013
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Cite this article as: Koyunbakan: Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation.
Boundary Value Problems 2013, 2013:272


Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation

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Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation

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