An inverse problem related to a half-linear eigenvalue problem
© Wang and Cheng; licensee Springer. 2014
Received: 7 December 2013
Accepted: 14 March 2014
Published: 24 March 2014
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.
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 ().
coupling with the Dirichlet 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 N th 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.
Moreover, if , the error term can be replaced by .
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.
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.
for sufficiently large n. Substituting (11)-(12) into (10), the eigenvalue estimates (6) can be derived. □
3 Proofs of Theorems 1-2
for some between x and . Therefore, (15) and (16) complete the proof. □
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 1 Let . Then as and as . Moreover, if is replaced by , the above result is still valid. Furthermore, as .
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.
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. □
Therefore, substituting (20) into (23)-(24), this completes the proof. □
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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