R E S E A R C H
Open Access
An inverse problem related to a half-linear
eigenvalue problem
Wei-Chuan Wang
1*and Yan-Hsiou Cheng
2*Correspondence: [email protected] 1Center for General Education,
National Quemoy University, Kinmen, 892, Taiwan, ROC Full list of author information is available at the end of the article
Abstract
We study an inverse problem on the half-linear Dirichlet eigenvalue problem –(|y(x)|p–2y(x))= (p– 1)
λ
r(x)|y(x)|p–2y(x), wherep> 1 withp= 2 andris a positive function defined on [0, 1]. Using eigenvalues and nodal data (the lengths of two consecutive zeros of solutions), we reconstructr–1/p(x) and its derivatives. Our methodis based on (Law and Yang in Inverse Probl. 14:299-312, 779-780, 1998; Shen and Tsai in Inverse Probl. 11:1113-1123, 1995), and our result extends the result in (Shen and Tsai in Inverse Probl. 11:1113-1123, 1995) for the linear case to the half-linear case.
MSC: 34A55; 34B24; 47A75
1 Introduction
The subject under investigation is the half-linear eigenvalue problem consisting of
⎧ ⎨ ⎩
–(|y(x)|p–y(x))= (p– )λr(x)|y(x)|p–y(x),
y() =y() = , () wherep> withp= , andris a positive function defined on [, ]. By [–], it is well known that the problem () has countably many eigenpairs{(λn,yn(x)) :n∈N}, and the
eigenfunctionyn(x) has exactlyn– nodal points in (,), say =x(n)<x (n)
<· · ·<x (n) n–<
x(nn)= . In this paper, we intend to give the representation of the functionr(x) and its
derivatives in () by using eigenvalues and nodal points. This formation is treated as the reconstruction formula. Such a problem is called an inverse nodal problem and has at-tracted researchers’ attention. Readers can refer to [–] for the linear case (p= ), and to [, ] for the general case (p> ).
In [, ], inverse nodal problems on
–y(x)p–y(x)= (p– )λ–q(x)y(x)p–y(x) () are considered. The authors in [] studied () with Dirichlet boundary conditions
y() =y() = ,
while the authors in [] studied () with eigenparameter dependent boundary conditions
y() = , αy() +λy() = forα= .
Both of them first used the modified Prüfer substitution to derive the asymptotic expan-sion of eigenvalues and nodal points and then gave the reconstruction formula ofq(x) by using the nodal data. Note that the authors did not deal with the derivatives of q(x) in [, ]. Besides, the author in [] considered the same issue for thep-Laplacian energy-dependent Sturm-Liouville problem,
–y(x)p–y(x)= (p– )λ–q(x) – λr(x) y(x)p–y(x)
coupling with the Dirichlet boundary conditions.
In [], Shen and Tsai studied the inverse nodal problem on the string equationy+
λr(x)y= with Dirichlet boundary conditions. They employed the standard difference operatorto give a reconstruction formulas forr–/(x) and its derivatives. On the other
hand, Law and Yang [] not only studied the inverse nodal problems for the linear problem
–y+q(x)y=λy
with separated boundary conditions
⎧ ⎨ ⎩
y()cosα+y()sinα= ,
y()cosβ+y()sinβ= , ()
where ≤α,β<π, and gave a reconstruction formulas forq(x) and its derivatives, but they also mentioned that the formulas forr(x) and its derivatives in the string equation with () are still valid. They applied the difference quotient operatorδin the formulas.
The main aim and methods of this study are basically the same as the ones in [, ]. Here we employ a modified Prüfer substitution on () derived by the generalized sine function
Sp(x). The well-known properties ofSp(x) can be referred to [, , ],etc.It shall be
men-tioned thatSp is notCat odd multiples ofπp/ asp> , and notCat even multiples
ofπp/ as <p< . These lead to that the reconstruction formulas forNth derivatives,
N≥, in [, ] cannot be extended to the half-linear case (p= ) in this article.
Denote byπp the first zero ofSp(x) in the positive axis. Definef(x) =r–/p(x),jn(x)≡ max{k:x(kn)≤x}, and the nodal length(kn)≡xk(n+)–xk(n)fork= , , , . . . ,n–. The following is our first result.
Theorem Consider()and suppose r is continuous on[, ].For each x∈[, ),let j=
jn(x)for the sake of simplicity.Then the following asymptotic formula is valid:
f(x) =λ
/p n (jn)
πp
+o(). ()
Moreover,if r∈C[, ],the error term can be replaced by O( n).
Now, define the difference operatorand the difference quotient operatorδas follows:
aj≡aj+–aj, δaj≡
aj+–aj
xj+–xj
=aj
j
and δkaj≡
δk–aj+–δk–aj
j
Thisδ-operator discretizes the differential operator in a nice way. It resembles the differ-ence quotient operator in finite differdiffer-ence. For the derivatives off(x), we have the following result.
Theorem Consider()and suppose r is C on[, ].For each x∈[, ),let j=j n(x)for
the sake of simplicity.Then as n→ ∞,
f(k)(x) = λ
/p n
πp
δk(jn)+O
n
for k= , . ()
This paper is organized as follows. In Section , we give the asymptotic estimates for eigenvalues. This step makes us know such quantities well. It is necessary to specify the orders of the expansion terms in the proofs of the main results. In Section , the proofs of the main theorems are given.
2 Asymptotic estimates for eigenvalues
Before we prove the main results, we derive the eigenvalue expansion. Note that ifr(x)∈
C[, ], the last term in () can be replaced byo(n) (cf.[, p.]). In [, Theorem .], they proved the error term iso() ifr∈C[, ]. The smoothness ofr(x) increases, and the
smaller error can be derived.
Theorem Suppose that r(x)is a positive C-function defined on[, ].Then the
asymp-totic estimate yields
λ/np
r/p(t)dt=nπp+O
n
. ()
Proof Lety(x) be a solution of () and define a Prüfer-type substitution
λ/pr/p(x)y(x) =ρ(x)Sp
θ(x), y(x) =ρ(x)Spθ(x). ()
Then a direct calculation yields
θ(x) =λ/pr/p(x) + r
(x)
pr(x)Sp
θ(x)Spθ(x)p–Spθ(x), ()
ρ(x) = r
(x)ρ(x)
pr(x) Sp
θ(x)p. ()
With each eigenvalueλnof (), one can associate a Prüfer angleθn(x)≡θ(x;λn) via () if
one also specifies the initial conditionθn() = forn= , , , . . . . In particular,θn() =nπp.
Integrating both sides of () over [, ], one obtains
nπp=λ/np
r/p(t)dt+
r(t)
pr(t)Sp
θn(t)Sp
θn(t) p–
Spθn(t)
dt. ()
Letg(τ)≡Sp(τ)|Sp(τ)|p–Sp(τ). Note that ifθn(x) = is valid in some subinterval of (, ),
that the functionr(x) depends onλnin this subinterval from (). This will contradict our
original problem. Hence, the points satisfyingθn(x) = shall be isolated. Then, in ()
r(t)
pr(t)g
θn(t)
= r
(t)
pr(t)g
θn(t)
θn(t)
λ/npr/p(t) + r (t) pr(t)g(θn(t))
= r
(t)g(θ
n(t))θn(t)
pλ/npr
p+ p (t)
+ r(t)g(θn(t))
pλ/npr p+
p (t)
forθn= . Letf(t) =r–/p(t). Thenf(t) = –r(t) pr
p+ p (t)
. Dropping the function variablet, one has the following:
r prg(θn) =
–fg(θn)θn
λ/np
– f
λ/np g(θn)
= –
∞
k=
fg(θn)
λ/np k+
θn. ()
DefineG(n)(t) =θn(t)
g(τ)dτ. ThenG (n) () =G
(n)
() = sinceg(τ) is aπp-periodic
func-tion. Moreover, by integration by parts, () implies that
λ–/n pf(t)gθn(t)
θn(t)dt= –λ–/n p
f(t)G(n)(t)dt=Oλ–/n p, ()
for sufficiently largen. Substituting ()-() into (), the eigenvalue estimates () can be
derived.
3 Proofs of Theorems 1-2
Proof of Theorem Letj=jn(x). By the Sturm comparison theorem for thep-Laplacian
(cf.[, , ]etc.), one has
πp
(λnrMj)/p
≤(jn)≤ πp
(λnrmj)/p
,
whererMjandrmjare the maximal and minimal values ofron [x
(n) j ,x
(n)
j+], respectively. Then
r–/Mjp≤λ
/p n (jn)
πp ≤
rm–/jp. ()
In particular, for ≤j≤n– , we have
(jn)=O
n
. ()
Moreover, by the continuity off(x) and (), there is anξj(n)∈(x(jn),x(jn+)) such that
λ/np(jn)
πp
On the other hand, by the mean value theorem and (), one also has
f(x) –fξj(n)=f(yj)
x–ξj(n)=O
n
()
for someyjbetweenxandξj(n). Therefore, () and () complete the proof.
Before we prove Theorem , one has to derive the asymptotic behavior off(x) at the nodal points. Note that the series in () is uniformly convergent on [, ]. Set h(x) =
r/p(x) =
f(x). Integrating () over [x (n) j ,x
(n)
j+] and applying (), one has
πp=λ/np x(jn+)
x(jn)
h(t)dt–
∞
k= x(j+n)
x(jn)
λ–/n pf(t)gθn(t) k
θn(t)dt. ()
By the Taylor expansion theorem and integration by parts, we find
πp=λ/np x(j+n)
x(jn)
hx(jn)+hxj(n)t–x(jn)+h
(y(n) j )
!
t–x(jn)
dt
–λ–/n pf(t)G(n)(t)x
(n) j+
x(jn)+λ –/p n
x(jn+)
x(jn)
f(t)G(n)(t)dt
–
∞
m=
λ–nm/pf(t)mG(mn)(t)x
(n) j+
x(jn)+ ∞
m=
λ–nm/p
x(j+n)
x(jn)
f(t)m G(mn)(t)dt
for somey(jn)∈(xj(n),x(j+n)), whereG(kn)(t) =θn(t)
(g(τ))kdτ. Note that
G(n)x(jn)=G(n)x(j+n)= . Hence,
πp=λ/nph
x(jn)(jn)+ !λ
/p n h
x(jn)(jn)+ !λ
/p n h
y(jn)(jn)
+
∞
m=
λ–nm/p
x(j+n)
x(jn)
f(t)mGm(n)(t)dt–
∞
m=
λn–m/pfxj(+n) mG(mn)x(j+n)
+
∞
m=
λ–nm/pfx(jn) mG(mn)x(jn). ()
Multiplying () by f(x
(n) j )
πp , one has
fx(jn)=λ
/p n (jn)
πp
+ πp
λ/npfx(jn)hx(jn)(jn)
+ πp
λ/npfx(jn)hy(jn)(jn)
+
∞
m=
λ–nm/p
πp
fx(jn)
x(j+n)
x(jn)
–
∞
m=
λ–nm/p
πp
fx(jn)fxj(+n) mGm(n)x(jn+)
+
∞
m=
λ–nm/p
πp
fx(jn)fxj(n) mGm(n)x(jn).
For convenience, we denote
fx(jn)≡λ
/p n (jn)
πp
+Aj+Bj+Cj+Dj+Ej, ()
where
Aj=
πp
λ/npfx(jn)hx(jn)(jn),
Bj=
πp
λ/npfx(jn)hy(jn)(jn),
Cj= ∞
m=
λ–nm/p
πp
fx(jn)
x(jn+)
x(jn)
f(t)m Gm(n)(t)dt,
Dj= – ∞
m=
λ–nm/p
πp
fx(jn)fxj(+n) mG(mn)x(jn+),
Ej= ∞
m=
λ–nm/p
πp
fx(jn)fxj(n) mGm(n)x(jn).
Remark Note that the functionG(mn)is defined by the integral ofg(τ)≡Sp(τ)|Sp(τ)|p–×
Sp(τ). By the unlike property ofSp,G(mn)is not a smooth function. This is the main reason
that the result in [, Theorem .] does not hold in our case,p= .
Then the following lemmas are necessary to the proof of Theorem and the superscript will be dropped for the sake of convenience,xj=x(jn)andj=(jn).
Lemma Let f =r–/p∈C[, ].Thenδkf(x
j) =O()as k= , , andδkf(xj) =O(nk–)as
k≥.Moreover,if xjis replaced by yj∈(xj,xj+),the above result is still valid.Furthermore,
δk(
j)m=O(nm)as k= , , .
Proof The first part of the proof is followed by the mean value theorem and the asymptotic estimates for nodal length (),i.e.,
δf(xj) =
f(xj+) –f(xj)
j
=f(yj) =O() for someyj∈(xj,xj+).
It is also valid fork= , by similar arguments. On the other hand, fork= , we find
δf(xj) =
δf(xj+) –δf(xj)
j
=O()
j
and it is also valid fork> . Finally, by () and (), we findf(ξj) =f(xj) +O(n) and
δk(j)m=
πm p
λmn/p
δkf(ξj) m
=O
nm
fork= , , .
Lemma Let u∈C.Then,for k= , ,
δku(xj)(j) =O
n
.
Moreover,if v∈C,then for k= ,
δkv(xj)(j) =O
n–k
.
Proof First applying the identityδ(ajbj) =aj+δbj+bjδaj, () and Lemma , one can obtain
δu(xj)(j) = (j+)δu(xj) +u(xj)δ(j)=O
n
,
δu(xj)(j) = (j+)δ
u(xj) + δ
(j+) δ
u(xj) +u(xj)δ
(j) =O
n
.
The second part is similar to the one of the first part. So it is omitted here. The following corollary is similar to [, Lemma .]. We give the proof for the conve-nience of the readers.
Corollary Let q∈C.Define Q(xj) = xj+
xj q(t)dt.Then,for k= , ,
δkQ(xj) =O
n–k
.
Proof By the mean value theorem for integrals, for everyjthere exists someyj∈(xj,xj+)
such thatQ(xj) =q(yj)j. Then applying Lemmas -, we complete the proof.
Proof of Theorem Recall (). By Theorem and Lemma , one hasδkA
j=O(n) and
δkBj=O(n–k) fork= , . By Theorem and Corollary , one can obtainδkCj=O(n–k)
fork= , . By Theorem , Lemma and the definition ofG(mn), one hasδkDjandδkEjare
O(
n–k) fork= , . Hence, one can find
δkfx(jn)=λ
/p n
πp
δk(jn)+O
n
()
fork= , . To complete the proof, it suffices to show that
f(k)(x) =δkfx(jn)+O
n
for sufficiently largen. Obviously, () holds fork= by () and (). Whenk= , , by the Taylor theorem,
f(k–)x(j+n)=f(k–)xj(n)+f(k)xj(n)(jn)+f
(k+)(ξ(n) k+,j)
(jn)
for someξk(n+,)j∈(xj(n),x(jn+)). Thus,
f(k)x(jn)=δf(k–)x(jn)+ f
(k+)ξ(n) k+,j
(jn), ()
fork= , ;i.e., by the mean value theorem and (),
f(x) =fx(jn)+O
n
=δfx(jn)+O
n
. ()
Successively, we have
f(x) =δfx(jn)+O
n
=δ
δfx(jn)+ f
ξ(n) ,j
(jn)
+O
n
=δfx(jn)+O
n
. ()
Therefore, substituting () into ()-(), this completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The author WCW was a major contributor in writing the manuscript. The author YHC performed the literature review. Both of them performed the final editing of the manuscript. They also gave their final approval of the version to be submitted and any revised version.
Author details
1Center for General Education, National Quemoy University, Kinmen, 892, Taiwan, ROC.2Department of Mathematics and
Information Education, National Taipei University of Education, Taipei, 106, Taiwan, ROC.
Acknowledgements
The authors express their thanks to the referees for helpful comments. The first author is supported in part by the National Science Council, Taiwan under contract number NSC 102-2115-M-507-001. The author YH Cheng is supported by the National Science Council, Taiwan under contract numbers NSC 102-2115-M-152 -002.
Received: 7 December 2013 Accepted: 14 March 2014 Published:24 Mar 2014
References
1. Binding, P, Drabek, P: Sturm-Liouville theory for thep-Laplacian. Studia Sci. Math. Hung.40, 373-396 (2003) 2. Elbert, Á: A half-linear second order differential equation. In: Qualitative Theory of Differential Equations. Szeged. Coll.
Math. Soc. J. Bolyai, vol. 30, pp. 153-179 (1979)
3. Eberhart, W, Elbert, A: On the eigenvalues of a half-linear boundary value problem. Math. Nachr.213, 57-76 (2000) 4. Reichel, W, Walter, W: Sturm-Liouville type problems for thep-Laplacian under asymptotic non-resonance conditions.
J. Differ. Equ.156, 50-70 (1999)
5. McLaughlin, JR: Inverse spectral theory using nodal points as data - a uniqueness result. J. Differ. Equ.73, 354-362 (1988)
7. Law, CK, Yang, CF: Reconstructing the potential function and its derivatives using nodal data. Inverse Probl.14, 299-312 (1998); Addendum14, 779-780 (1998)
8. Law, CK, Lian, WC, Wang, WC: Inverse nodal problem and Ambarzumyan problem for thep-Laplacian. Proc. R. Soc. Edinb. A139, 1261-1273 (2009)
9. Wang, WC, Cheng, YH, Lian, WC: Inverse nodal problems for thep-Laplacian with eigenparameter dependent boundary conditions. Math. Comput. Model.54, 2718-2724 (2011)
10. Koyunbakan, H: Inverse nodal problem forp-Laplacian energy-dependent Sturm-Liouville equation. Bound. Value Probl.2013, 272 (2013)
11. Li, HJ, Yeh, CC: Sturmian comparison theorem for half-linear second-order differential equations. Proc. R. Soc. Edinb. A
125, 1193-1204 (1995)
12. Walter, W: Sturm-Liouville theory for the radialp-operator. Math. Z.227, 175-185 (1998)
10.1186/1687-2770-2014-65
