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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 794262, 7 pages
Double Discontinuous Inverse Problems for Sturm-Liouville Operator with Parameter-Dependent Conditions
Department of Mathematics, Faculty of Arts & Science, Cumhuriyet University, 58140 Sivas, Turkey
Received 26 March 2013; Accepted 24 June 2013
Academic Editor: Dumitru Motreanu
Copyright © 2013 A. S. Ozkan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The purpose of this paper is to solve the inverse spectral problems for Sturm-Liouville operator with boundary conditions depending on spectral parameter and double discontinuities inside the interval. It is proven that the coefficients of the problem can be uniquely determined by either Weyl function or given two different spectral sequences.
Spectral problems of differential operators are studied in two main branches, namely, direct spectral problems and inverse spectral problems. Direct problems of spectral analysis consist in investigating the spectral properties of an operator. On the other hand, inverse problems aim at recovering operators from their spectral characteristics. Such problems often appear in mathematics, physics, mechanics, electronics, geophysics, and other branches of natural sciences.
First and most important results for inverse problem of a regular Sturm-Liouville operator were given by Ambartsumyan in 1929  and Borg in 1946 . Physical applications of inverse spectral problems can be found in several works (see, e.g., [3–9] and references therein).
Eigenvalue-dependent boundary conditions were studied extensively. The references [10, 11] are well-known examples for problems with boundary conditions that depend linearly on the eigenvalue parameter. In [10, 12], an operator-theoretic formulation of the problems with the spectral parameter contained in only one of the boundary conditions has been given. Inverse problems according to various spectral data for eigenparameter linearly dependent Sturm-Liouville operator were investigated in [13–17]. Boundary conditions that depend nonlinearly on the spectral parameter were also considered in [18–23].
In this study, we consider a boundary value problem generated by the Sturm-Liouville equation: subject to the boundary conditions and double discontinuity conditions where is real valued function in ; and , , are real numbers; , ,; , ; , ; and is a spectral parameter. We denote the problem (1)–(4) by , where , , and , .
It is proven that the coefficients of the problem can be uniquely determined by either Weyl function or given two different spectral sequences. The obtained results are generalizations of the similar results for the classical Sturm-Liouville operator on a finite interval.
Let the functions and be the solutions of (1) under the following initial conditions and the jump conditions (4): These solutions are the entire functions of and satisfy the relation for each eigenvalue , where .
The following asymptotics can be obtained from the integral equations given in the appendix:
The values of the parameter for which the problem has nonzero solutions are called eigenvalues, and the corresponding nontrivial solutions are called eigenfunctions.
The characteristic function and norming constants of the problem are defined as follows: It is obvious that is an entire function in and the zeros, namely, of coincide with the eigenvalues of the problem . Now, from (6) and (8), we can write
Lemma 1. See the following.(i)All eigenvalues of the problem are real and algebraically simple; that is, .(ii)Two eigenfunctions and , corresponding to different eigenvalues and , are orthogonal in the sense of
Proof. Consider a Hilbert Space , equipped with the inner product
Define an operator with the domain , and are absolutely continuous in , and such that It is easily proven, using classical methods in the similar works (see, e.g., ), that the operator is symmetric in ; the eigenvalue problem for the operator and the problem coincide. Therefore, all eigenvalues are real, and two different eigenfunctions are orthogonal.
Let us show the simplicity of the eigenvalues by writting the following equations: If these equations are multiplied by and , respectively, subtracting them side by side and finally integrating over the interval , the equality is obtained. Add and subtract in the left-hand side of the last equality, and use initial conditions (5) to get Rewrite this equality as As , is obtained by using the equality . Thus, .
3. Main Results
We consider three statements of the inverse problem for the boundary value problem ; from the Weyl function, from the spectral data , and from two spectra . For studying the inverse problem, we consider a boundary value problem , together with , of the same form but with different coefficients .
Let the function denote the solution of (1) under the initial conditions , and the jump conditions (4). It is clear that the function can be represented by Denote Then, we have The function is called Weyl function .
Theorem 2. If , then ; that is, , always everywhere in , and .
Proof. Let us define the functions and as follows:
where . If , then from (22)-(23), and are entire functions in . Denote and , where is sufficiently small number and and are square roots of the eigenvalues of the problem and , respectively. One can easily show that the asymptotics
are valid for , , and sufficiently large in . Thus, the following inequalities are obtained from (6) and (24):
According to the last inequalities and Liouville’s theorem, and . Use (23) again to take
Since and similarly , then .
On the other hand, the asymptotic expressions are valid for on the imaginary axis, where Assume that and . There are six different cases for the permutation of the numbers and . Without loss of generality, let .
From (26)-(27), we get , and , while .
Moreover, we get while . By taking limit in (29) as , we condradict . Thus, . Similarly, , and in . Hence, It can be obtained from (1), (4), and (5) that , a.e. in ; , , and ,. Consequently, .
Theorem 3. If , then .
Proof. The meromorphic function has simple poles at , and its residues at these poles are Denote , where is sufficiently small number. Consider the contour integral There exists a constant such that holds for . Use this inequality and (21) to get , for . Hence, , and so is obtained from residue theorem. Consequently, if and for all , then from (33), . Hence, Theorem 2 yields .
We consider the boundary value problem with the condition instead of (2) in . Let be the eigenvalues of the problem . It is obvious that are zeros of .
Theorem 4. If and , then .
Proof. The functions and which are entire of order can be represented by Hadamard’s factorization theorem as follows: where and are constants which depend only on and , respectively. Therefore, and , when and for all . Thus, . Moreover, since . Consequently, the equality (21) yields . Hence, the proof is completed by Theorem 2.
The solution satisfies the following integral equations.
If , if , if where .
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