We study an inverse problem on the half-linear Dirichlet eigenvalue problem , where with and r is a positive function defined on . Using eigenvalues and nodal data (the lengths of two consecutive zeros of solutions), we reconstruct and its derivatives. Our method is 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.
The subject under investigation is the half-linear eigenvalue problem consisting of
where with , and r is a positive function defined on . By [1-4], it is well known that the problem (1) has countably many eigenpairs , and the eigenfunction has exactly nodal points in (0,1), say . In this paper, we intend to give the representation of the function and its derivatives in (1) 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 attracted researchers’ attention. Readers can refer to [5-7] for the linear case ( ), and to [8,9] for the general case ( ).
are considered. The authors in  studied (2) with Dirichlet boundary conditions
while the authors in  studied (2) with eigenparameter dependent boundary conditions
Both of them first used the modified Prüfer substitution to derive the asymptotic expansion of eigenvalues and nodal points and then gave the reconstruction formula of by using the nodal data. Note that the authors did not deal with the derivatives of in [8,9]. Besides, the author in  considered the same issue for the p-Laplacian energy-dependent Sturm-Liouville problem,
coupling with the Dirichlet boundary conditions.
In , Shen and Tsai studied the inverse nodal problem on the string equation with Dirichlet boundary conditions. They employed the standard difference operator △ to give a reconstruction formulas for and its derivatives. On the other hand, Law and Yang  not only studied the inverse nodal problems for the linear problem
with separated boundary conditions
where , and gave a reconstruction formulas for and its derivatives, but they also mentioned that the formulas for and its derivatives in the string equation with (3) are still valid. They applied the difference quotient operator δ in the formulas.
The main aim and methods of this study are basically the same as the ones in [6,7]. Here we employ a modified Prüfer substitution on (1) derived by the generalized sine function . The well-known properties of can be referred to [1,2,4], etc. It shall be mentioned that is not at odd multiples of as , and not at even multiples of as . These lead to that the reconstruction formulas for Nth derivatives, , in [6,7] cannot be extended to the half-linear case ( ) in this article.
Denote by the first zero of in the positive axis. Define , , and the nodal length for . The following is our first result.
Theorem 1Consider (1) and supposeris continuous on . For each , let for the sake of simplicity. Then the following asymptotic formula is valid:
Moreover, if , the error term can be replaced by .
Now, define the difference operator △ and the difference quotient operator δ as follows:
This δ-operator discretizes the differential operator in a nice way. It resembles the difference quotient operator in finite difference. For the derivatives of , we have the following result.
Theorem 2Consider (1) and supposeris on . For each , let for the sake of simplicity. Then as ,
This paper is organized as follows. In Section 2, 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 3, 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 if , the last term in (6) can be replaced by (cf. [, p.171]). In [, Theorem 2.5], they proved the error term is if . The smoothness of increases, and the smaller error can be derived.
Theorem 3Suppose that is a positive -function defined on . Then the asymptotic estimate yields
Proof Let be a solution of (1) and define a Prüfer-type substitution
Then a direct calculation yields
With each eigenvalue of (1), one can associate a Prüfer angle via (7) if one also specifies the initial condition for . In particular, . Integrating both sides of (8) over , one obtains
Let . Note that if is valid in some subinterval of , the term will be constant in this subinterval. This implies that the function depends on in this subinterval from (8). This will contradict our original problem. Hence, the points satisfying shall be isolated. Then, in (8)
for . Let . Then . Dropping the function variable t, one has the following:
Define . Then since is a -periodic function. Moreover, by integration by parts, (11) implies that
for sufficiently large n. Substituting (11)-(12) into (10), the eigenvalue estimates (6) can be derived. □
3 Proofs of Theorems 1-2
where and are the maximal and minimal values of r on , respectively. Then
In particular, for , we have
Moreover, by the continuity of and (13), there is an such that
On the other hand, by the mean value theorem and (14), one also has
for some between x and . Therefore, (15) and (16) complete the proof. □
Before we prove Theorem 2, one has to derive the asymptotic behavior of at the nodal points. Note that the series in (11) is uniformly convergent on . Set . Integrating (8) over and applying (11), one has
By the Taylor expansion theorem and integration by parts, we find
for some , where . Note that
Multiplying (18) by , one has
For convenience, we denote
Remark 1 Note that the function is defined by the integral of . By the unlike property of , is not a smooth function. This is the main reason that the result in [, Theorem 5.1] does not hold in our case, .
Then the following lemmas are necessary to the proof of Theorem 2 and the superscript will be dropped for the sake of convenience, and .
Lemma 1Let . Then as and as . Moreover, if is replaced by , the above result is still valid. Furthermore, as .
Proof The first part of the proof is followed by the mean value theorem and the asymptotic estimates for nodal length (14), i.e.,
It is also valid for by similar arguments. On the other hand, for , we find
and it is also valid for . Finally, by (15) and (16), we find and
Lemma 2Let . Then, for ,
Moreover, if , then for
Proof First applying the identity , (14) and Lemma 1, one can obtain
The second part is similar to the one of the first part. So it is omitted here. □
The following corollary is similar to [, Lemma 2.3]. We give the proof for the convenience of the readers.
Corollary 1Let . Define . Then, for ,
Proof By the mean value theorem for integrals, for every j there exists some such that . Then applying Lemmas 1-2, we complete the proof. □
Proof of Theorem 2 Recall (19). By Theorem 3 and Lemma 2, one has and for . By Theorem 3 and Corollary 1, one can obtain for . By Theorem 3, Lemma 1 and the definition of , one has and are for . Hence, one can find
for . To complete the proof, it suffices to show that
for sufficiently large n. Obviously, (21) holds for by (14) and (16). When , by the Taylor theorem,
for some . Thus,
for ; i.e., by the mean value theorem and (22),
Successively, we have
Therefore, substituting (20) into (23)-(24), this completes the proof. □
The authors declare that they have no competing interests.
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.
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.
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