Inverse spectral problem of a class of fourth-order eigenparameter-dependent boundary value problems

This paper deals with a class of inverse spectral problems of fourth-order boundary value problems with eigenparameter-dependent boundary conditions. Under the equivalent conditions of the problem and a certain type of matrix eigenvalue problem some coefficient functions are reconstructed from the given three sets of interlacing real numbers and several additional conditions. The key technique is the method of inverse matrix eigenvalue problems of a two-banded matrix.


Introduction
This paper is a generalization of [26] and [5]. In [26], a class of inverse problems of Sturm-Liouville problems with eigenparameter-dependent boundary conditions has been considered by the authors. And in [5], the authors studied the inverse fourth-order boundary value problems with finite spectrum under separated boundary conditions. Since the higher-order boundary value problems with eigenparameter-dependent boundary conditions have their own theoretical and application background, we will investigate an inverse spectral problem of the fourth-order boundary value problems with eigenparameterdependent boundary conditions. The reader may find the historical and research background on boundary value problems with eigenparameter-dependent boundary conditions in [20,26] and the references therein.
It is well known that the inverse spectral problems play an important role in many scientific fields which is motivated by recovering operators from their spectral data. Such problems have been widely studied by mathematicians, physicists, and engineers. Especially the inverse matrix eigenvalue problems and inverse Sturm-Liouville problems have been deeply studied in the last several decades [7,8,10,11,13,[22][23][24][25]. Such problems are always connected with the problems such as the vibrating systems [12], the classical moment problems [19], and quantum mechanics [18].
In recent years, boundary value problems (BVPs) of Atkinson type which have a finite spectrum have been considered by some scholars [1-4, 6, 14, 15, 17]. These problems are connected with some physical problems such as frequencies of vibrating strings and diffusion operators [21]. Besides the finite spectrum results and matrix representations of these problems, the corresponding inverse spectral problems of such a type of problems have also been investigated most recently [5,9,16,26,27]. In 2012, Kong and Zettl considered the inverse Sturm-Liouville problems with finite spectrum of Atkinson type by using the so-called matrix representations of these problems [16]. In 2018, Cai and Zheng [9] generated the results in [16] and investigated a class of inverse Sturm-Liouville problems with discontinuous boundary conditions. The inverse spectral problems of Sturm-Liouville problems with eigenparameter-dependent boundary conditions and fourth-order boundary value problems were studied by Zhang and Ao in [26,27] and [5], respectively. Since the fourth-order boundary value problems and the eigenparameter-dependent boundary value problems have their own significant background, we will discuss an inverse spectral problem on it.
Let us consider the fourth-order boundary value problem generated by the equation where the coefficients satisfy the condition r = 1/p, q, w ∈ L(I, R), here L(I, R) denotes the real valued functions which are Lebesgue integrable on I, and the eigenparameter-dependent boundary conditions (BCs) Here λ is the spectral parameter.
To present our discussions in this paper we need to introduce the following notations. Let (1.8) and assume that For a given equation (1.1) with coefficients satisfying (1.5)-(1.7), let us set Then we let σ ( α, α , β, β , γ , γ , δ, δ ) be the spectrum of the fourth-order eigenparameterdependent BVPs (1.1), (1.2). Assume k ∈ N + such that k > 2, and let M 2k be the set of 2k × 2k matrices over the reals. For any Y ∈ M 2k , we denote by σ (Y ) the set of eigenvalues of Y . Furthermore, let Y 1 be the principal submatrix obtained from Y by removing its first row and column, and Y 1 its submatrix obtained from Y by removing the last row and column. Let Y 12 be the principal submatrix obtained from Y by removing its first two rows and first two columns, and Y 12 its submatrix obtained from Y by removing the last two rows and the last two columns. For any 2 × 2 matrix F, we denote by Y 12

Equivalent matrix representations of fourth-order eigenparameter-dependent BVPs
To illustrate our main result on inverse fourth-order eigenparameter-dependent BVPs, we will show the following lemmas, which have been given in [1].
Proof The proof is similar to the proof of Lemma 2.3, hence it is omitted here.
Proof The proof is similar to the proof of Lemma 2.3, hence it is omitted here.

Related inverse matrix eigenvalue problems
We first introduce the following matrices in M 2k : where C i , i = 1, . . . , k and D i , i = 1, . . . , k -1 are 2 × 2 matrices, T denotes the transpose.
Then there exists a two-banded matrix J b ∈ M 2k such that Next, we consider the block matrices in M 2k of the form which is block symmetric except for the D 1 , D 1 , D k-1 , D k-1 entries. From Lemma 3.1, we can deduce the following theorem which includes the key technique to solve our main result. If is an 'almost' block diagonal matrix with W j = Now for each λ = λ i , i = 1, . . . , 2k, there exists a nontrivial u ∈ R 2k such that Let u = UH -1ũ and left multiplying Eq. (3.5) by L we obtain where M = LJ b UH -1 . Clearly, λ ∈ σ (M, W) and M ∈ M 2k is a block pseudo-Jacobi matrix. The same argument applies to each ξ = ξ i , i = 1, . . . , 2k -2, then we obtain the conclusion that M 12 [-E 2 (B 1 ) -1 B 2 ] = L 12 (J b ) 12 U 12 (H -1 ) 12 , W 12 = L 12 U 12 (H -1 ) 12 and ξ ∈ σ (M 12 , W 12 ). Thus, Similarly by reversing the steps above we can get Consequently, This completes the proof.

Main result
We now state our main result on the inverse spectral problem of the fourth-order eigenparameter-dependent BVPs (1.1), (1.2).