Skip to main content

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

Abstract

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 eigenparameter-dependent 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, 2225]. 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 [14, 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

$$ \bigl(py''\bigr)''+qy= \lambda wy,\quad \text{on }I=[a,b], -\infty < a< b< +\infty , $$
(1.1)

where the coefficients satisfy the condition

$$ r=1/p, q, w\in L(I,\mathbb{R}), $$

here \(L(I,\mathbb{R})\) denotes the real valued functions which are Lebesgue integrable on I, and the eigenparameter-dependent boundary conditions (BCs)

$$ A_{\lambda }Y(a)+B_{\lambda }Y(b)=0,\qquad Y=\bigl[y, y', py'', \bigl(py'' \bigr)'\bigr]^{T}, $$
(1.2)

where

Aλ=[λα1+α1λα2+α2λα3+α3λα4+α4λβ1+β1λβ2+β2λβ3+β3λβ4+β400000000],Bλ=[00000000λγ1+γ1λγ2+γ2λγ3+γ3λγ4+γ4λδ1+δ1λδ2+δ2λδ3+δ3λδ4+δ4]

with \(\alpha _{i}, \alpha _{i}', \beta _{i}, \beta _{i}', \gamma _{i}, \gamma _{i}', \delta _{i}, \delta _{i}'\in \mathbb{R}, i=1, 2, 3, 4\), and satisfying

rank(α1α2α3α4α1α2α3α4)=2,rank(β1β2β3β4β1β2β3β4)=2,rank(γ1γ2γ3γ4γ1γ2γ3γ4)=2,rank(δ1δ2δ3δ4δ1δ2δ3δ4)=2,rank(α1α2α3α4β1β2β3β4)=2,rank(α1α2α3α4β1β2β3β4)=2,rank(γ1γ2γ3γ4δ1δ2δ3δ4)=2,rank(γ1γ2γ3γ4δ1δ2δ3δ4)=2.
(1.3)

Here λ is the spectral parameter.

The fourth-order eigenparameter-dependent BVPs (1.1), (1.2) are of Atkinson type if for some positive integer \(n\geq 1\), there exists a partition of the interval \([a,b]\)

$$ a=a_{0}< b_{0}< a_{1}< b_{1}< \cdots < a_{n}< b_{n}=b, $$
(1.4)

such that

$$\begin{aligned} \begin{aligned} &r(t)=\frac{1}{p(t)}=0\quad \text{on }[a_{i},b_{i}), i=0,1,\ldots ,n,\qquad \int _{b_{i-1}}^{a_{i}}r(t)\,dt\neq 0, \\ &\int _{b_{i-1}}^{a_{i}}r(t)t\,dt\neq 0,\qquad \int _{b_{i-1}}^{a_{i}}r(t)t^{2}\,dt\neq 0,\quad i=1,2, \ldots ,n; \end{aligned} \end{aligned}$$
(1.5)
$$\begin{aligned} &q(t)=0\quad \text{on }[b_{i-1},a_{i}), i=1,2,\ldots ,n, \end{aligned}$$
(1.6)

and

$$ \begin{aligned}& w(t)=0 \quad\text{on }[b_{i-1},a_{i}), i=1,2, \ldots ,n,\qquad \int _{a_{i}}^{b_{i}}w(t)\,dt>0, \\ &\int _{a_{i}}^{b_{i}}w(t)t\,dt>0,\qquad \int _{a_{i}}^{b_{i}}w(t)t^{2}\,dt>0,\quad i=0,1,\ldots ,n, \end{aligned} $$
(1.7)

and BC (1.2) satisfies the conditions (1.3).

To present our discussions in this paper we need to introduce the following notations. Let

$$ \begin{aligned} &r_{i}= \int ^{a_{i}}_{b_{i-1}}r(t)\,dt,\qquad \hat{r}_{i}= \int ^{a_{i}}_{b_{i-1}}r(t)t\,dt,\qquad \check{r}_{i}= \int ^{a_{i}}_{b_{i-1}}r(t)t^{2}\,dt,\quad i=1,2,\ldots ,n, \\ &q_{i}= \int _{a_{i}}^{b_{i}}q(t)\,dt,\qquad \hat{q}_{i}= \int _{a_{i}}^{b_{i}}q(t)t\,dt,\qquad \check{q}_{i}= \int _{a_{i}}^{b_{i}}q(t)t^{2}\,dt,\quad i=0,1,\ldots ,n, \\ &w_{i}= \int _{a_{i}}^{b_{i}}w(t)\,dt, \qquad \hat{w}_{i}= \int _{a_{i}}^{b_{i}}w(t)t\,dt,\qquad \check{w}_{i}= \int _{a_{i}}^{b_{i}}w(t)t^{2}\,dt,\quad i=0,1,\ldots ,n, \end{aligned} $$
(1.8)

and assume that

$$ \theta _{i}=\hat{r}_{i}^{2}-r_{i} \check{r}_{i}\neq 0,\quad i=1,2,\ldots ,n. $$

For a given equation (1.1) with coefficients satisfying (1.5)–(1.7), let us set

$$ \begin{aligned}& \vec{\alpha }=(\alpha _{1},\alpha _{2},\alpha _{3},\alpha _{4}),\qquad \vec{\alpha '}=\bigl(\alpha _{1}',\alpha _{2}',\alpha _{3}',\alpha _{4}'\bigr), \qquad \vec{\beta }=(\beta _{1},\beta _{2},\beta _{3},\beta _{4}), \\ & \vec{\beta '}=\bigl(\beta _{1}',\beta _{2}',\beta _{3}',\beta _{4}'\bigr),\qquad \vec{\gamma }=(\gamma _{1},\gamma _{2},\gamma _{3},\gamma _{4}),\qquad \vec{\gamma '}=\bigl(\gamma _{1}',\gamma _{2}',\gamma _{3}',\gamma _{4}'\bigr),\\ & \vec{ \delta }=(\delta _{1},\delta _{2},\delta _{3}, \delta _{4}),\qquad \vec{\delta '}=\bigl(\delta _{1}',\delta _{2}',\delta _{3}',\delta _{4}'\bigr), \qquad\vec{0}=(0,0,0,0). \end{aligned} $$

Then we let \(\sigma (\vec{\alpha }, \vec{\alpha '}, \vec{\beta }, \vec{\beta '}, \vec{\gamma }, \vec{\gamma '}, \vec{\delta }, \vec{\delta '})\) be the spectrum of the fourth-order eigenparameter-dependent BVPs (1.1), (1.2).

Assume \(k\in \mathbb{N}^{+}\) such that \(k>2\), and let \(\mathbb{M}_{2k}\) be the set of \(2k\times 2k\) matrices over the reals. For any \(Y\in \mathbb{M}_{2k}\), we denote by \(\sigma (Y)\) 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 \(2\times 2\) matrix F, we denote by \(Y_{12}[F]\) the matrix of \(Y_{12}\) plus F in the \((1, 1), (1, 2), (2, 1), (2, 2)\) entries. We denote by \(Y^{12}[F]\) the matrix of \(Y^{12}\) plus F in the \((2k-3, 2k-3), (2k-3, 2k-2), (2k-2, 2k-3), (2k-2, 2k-2)\) entries.

For any \(Y, Z\in \mathbb{M}_{2k}\), we say that \(\lambda ^{*}\) is an eigenvalue of the matrix-pair \((Y,Z)\) if there exists a nontrivial vector \(u\in \mathbb{R}^{2k}\) such that \((Y-\lambda ^{*}Z)u=0\). We denote by \(\sigma (Y,Z)\) the set of eigenvalues of \((Y,Z)\). Clearly, \(\lambda ^{*}\in \sigma (Y)\) if and only if \(\lambda ^{*}\in \sigma (Y,I _{2k})\), where \(I_{2k}\) is the identity matrix in \(\mathbb{M}_{2k}\).

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].

Lemma 2.1

([1])

Let \(x_{1}, x_{2}\in \mathbb{R}\)and \(x_{1}\neq x_{2}\). Then, for any \(\eta _{0}, \eta _{1}, \eta _{2}\in \mathbb{R}\), there exists a unique polynomial function \(P(t)=\tau _{2}t^{2}+\tau _{1}t+\tau _{0}\), such that

$$ \int ^{x_{2}}_{x_{1}}P(t)\,dt=\eta _{0},\qquad \int ^{x_{2}}_{x_{1}}P(t)t\,dt=\eta _{1},\qquad \int ^{x_{2}}_{x_{1}}P(t)t^{2}\,dt=\eta _{2}. $$
(2.1)

Denote the polynomial constructed in Lemma 2.1 by \(\chi (x_{1},x _{2},\eta _{0},\eta _{1},\eta _{2})\) and define the piecewise polynomial functions \(\bar{r}(t)=\frac{1}{\bar{p}(t)}, \bar{q}(t)\) and \(\bar{w}(t)\) on \([a,b]\) by

$$\begin{aligned} &\bar{r}(t)=\frac{1}{\bar{p}(t)}= \textstyle\begin{cases} \chi (b_{i-1},a_{i},r_{i},\hat{r}_{i},\check{r}_{i}), & t\in [b_{i-1},a_{i}), i=1,2,\ldots ,n; \\ 0,& t\in [a_{i},b_{i}), i=0,1,\ldots ,n; \end{cases}\displaystyle \\ &\bar{q}(t)=\textstyle\begin{cases} \chi (a_{i},b_{i},q_{i},\hat{q}_{i},\check{q}_{i}), &t\in [a_{i},b_{i}), i=0,1,\ldots ,n; \\ 0, & t\in [b_{i-1},a_{i}), i=1,2,\ldots ,n; \end{cases}\displaystyle \\ &\bar{w}(t)=\textstyle\begin{cases} \chi (a_{i},b_{i},w_{i},\hat{w}_{i},\check{w}_{i}), & t\in [a_{i},b_{i}), i=0,1,\ldots ,n; \\ 0,& t\in [b_{i-1},a_{i}), i=1,2,\ldots ,n. \end{cases}\displaystyle \end{aligned}$$

Then we can state the following lemma.

Lemma 2.2

([1])

Assume \(r(t), q(t), w(t) \in L(I, \mathbb{R})\)satisfy (1.5)(1.7). Then the fourth-order eigenparameter-dependent BVPs (1.1), (1.2) have exactly the same eigenvalues as the fourth-order eigenparameter-dependent BVPs consisting of the equation with piecewise polynomial functions

$$ \bigl(\bar{p}y''\bigr)''+ \bar{q}y=\lambda \bar{w}y ,\quad \textit{on }I=[a,b], $$
(2.2)

and the same BC (1.2).

It can be inferred from Lemma 2.2, for a fixed BC (1.2) and a given partition of the interval \([a,b]\), there is a family of fourth-order eigenparameter-dependent BVPs of Atkinson type which has exactly the same eigenvalues as fourth-order eigenparameter-dependent BVP (2.2), (1.2). Such a family is called the equivalent family of the fourth-order eigenparameter-dependent BVPs (2.2), (1.2). It is clear that, for an equivalent family of the fourth-order eigenparameter-dependent BVPs (2.2), (1.2), the parameters \(r_{i}, \hat{r}_{i}, \check{r}_{i}\) and \(q_{i}, \hat{q} _{i}, \check{q}_{i}, w_{i}, \hat{w}_{i}, \check{w}_{i}\) defined by (1.8) are all the same.

Lemma 2.3

([1])

Consider the BC (1.2) with \(\alpha _{i}, \alpha _{i}', \beta _{i}, \beta _{i}', \gamma _{i}, \gamma _{i}', \delta _{i}, \delta _{i}'\in \mathbb{R}, i=1, 2, 3, 4\), satisfying the conditions (1.3). Define an \((n+3)\times (n+3)\)block tridiagonal matrix

$$ \mathbf{P}= \begin{bmatrix} B_{1} & B_{2} & & & & & \\ E_{2} & -P_{1} & P_{1} & & & & \\ & P_{1} & -P_{1}-P_{2} & P_{2} & & & \\ & & \cdots & \cdots & \cdots & & \\ & & & P_{n-1} & -P_{n-1}-P_{n} & P_{n} & \\ & & & & P_{n} & -P_{n} & -E_{2} \\ & & & & & B_{3} & B_{4} \end{bmatrix}, $$
(2.3)

a block diagonal matrix

$$ \mathbf{Q}=\operatorname{diag}(O,Q_{0},Q_{1},\ldots ,Q_{n},O), $$
(2.4)

and an ‘almost’ block diagonal matrix

$$ \mathbf{W}= \begin{bmatrix} -B_{1}' & -B_{2}' & & & & & \\ & W_{0} & & & & & \\ & & W_{1} & & & & \\ & & & \cdots & & & \\ & & & & W_{n-1} & & \\ & & & & & W_{n} & \\ & & & & & -B_{3}' & -B_{4}' \end{bmatrix}, $$
(2.5)

where \(E_{2}, O\), \(P_{j}, j=1,\ldots ,n\)and \(Q_{j}\), \(W_{j}, j=0,1, \ldots ,n\), are \(2\times 2\)matrices with the rule

E2=[0110],O=[0000],Pj=[rjθjrˆjθjrˆjθjrˇjθj],j=1,2,3,,n,Qj=[qjqˆjqˆjqˇj],Wj=[wjwˆjwˆjwˇj],j=0,1,,n,

and \(B_{i}, B_{i}', i= 1, 2, 3, 4\), are \(2\times 2\)matrices with the rule

$$\begin{aligned} &B_{1}= \begin{bmatrix} \alpha _{3} & a\alpha _{3}+\alpha _{4} \\ \beta _{3} & a\beta _{3}+\beta _{4} \end{bmatrix},\qquad B_{2}= \begin{bmatrix} \alpha _{1} & a\alpha _{1}+\alpha _{2} \\ \beta _{1} & a\beta _{1}+\beta _{2} \end{bmatrix}, \qquad B_{3}= \begin{bmatrix} \gamma _{3} & b\gamma _{3}+\gamma _{4} \\ \delta _{3} & b\delta _{3}+\delta _{4} \end{bmatrix},\\ & B_{4}= \begin{bmatrix} \gamma _{1} & b\gamma _{1}+\gamma _{2} \\ \delta _{1} & b\delta _{1}+\delta _{2} \end{bmatrix}, \qquad B_{1}'= \begin{bmatrix} \alpha _{3}' & a\alpha _{3}'+\alpha _{4}' \\ \beta _{3}' & a\beta _{3}'+\beta _{4}' \end{bmatrix},\qquad B_{2}'= \begin{bmatrix} \alpha _{1}' & a\alpha _{1}'+\alpha _{2}' \\ \beta _{1}' & a\beta _{1}'+\beta _{2}' \end{bmatrix}, \\ &B_{3}'= \begin{bmatrix} \gamma _{3}' & b\gamma _{3}'+\gamma _{4}' \\ \delta _{3}' & b\delta _{3}'+\delta _{4}' \end{bmatrix},\qquad B_{4}'= \begin{bmatrix} \gamma _{1}' & b\gamma _{1}'+\gamma _{2}' \\ \delta _{1}' & b\delta _{1}'+\delta _{2}' \end{bmatrix}. \end{aligned}$$

Then we have the same eigenvalues of the fourth-order eigenparameter-dependent BVPs (1.1), (1.2) and the matrix-pair \((\mathbf{P}+\mathbf{Q},\mathbf{W})\), i.e.

$$ \sigma \bigl(\vec{\alpha }, \vec{\alpha '}, \vec{\beta }, \vec{ \beta '}, \vec{\gamma }, \vec{\gamma '}, \vec{\delta }, \vec{\delta '}\bigr) =\sigma (\mathbf{P}+\mathbf{Q},\mathbf{W}). $$

Lemma 2.4

Consider the BC (1.2) with \(\alpha _{i}', \beta _{i}',\gamma _{i}, \gamma _{i}', \delta _{i}, \delta _{i}'\in \mathbb{R}, i=1, 2, 3, 4\), satisfying the conditions (1.3). Define an \((n+2)\times (n+2)\)block tridiagonal matrix

P12[E2(B1)1B2]=[P1E2(B1)1B2P1P1P1P2P2Pn1Pn1PnPnPnPnE2B3B4],
(2.6)

a block diagonal matrix

$$ \mathbf{Q}_{12}=\operatorname{diag}(Q_{0},Q_{1}, \ldots ,Q_{n},O), $$
(2.7)

and an ‘almost’ block diagonal matrix

$$ \mathbf{W}_{12}= \begin{bmatrix} W_{0} & & & & & \\ & W_{1} & & & & \\ & & \cdots & & & \\ & & & W_{n-1} & & \\ & & & & W_{n} & \\ & & & & -B_{3}' & -B_{4}' \end{bmatrix}, $$
(2.8)

where \(E_{2}, O, P_{j}, j=1,\ldots ,n\), \(Q_{j}\), \(W_{j}, j=0,1, \ldots ,n\), and \(B_{i}, i= 3, 4, B_{i}', i= 1, 2, 3, 4\)are \(2\times 2\)matrices as given in Lemma 2.3.

Then \(\sigma (\vec{0}, \vec{\alpha '}, \vec{0}, \vec{\beta '}, \vec{\gamma }, \vec{\gamma '}, \vec{\delta }, \vec{\delta '}) =\sigma (\mathbf{P}_{12}+\mathbf{Q}_{12},\mathbf{W}_{12})\).

Proof

The proof is similar to the proof of Lemma 2.3, hence it is omitted here. □

Lemma 2.5

Consider the BC (1.2) with \(\alpha _{i}, \alpha _{i}', \beta _{i}, \beta _{i}', \gamma _{i}', \delta _{i}'\in \mathbb{R}, i=1, 2, 3, 4\), satisfying the conditions (1.3). Define an \((n+2)\times (n+2)\)block tridiagonal matrix

P12[E2(B4)1B3]=[B1B2E2P1P1P1P1P2P2Pn1Pn1PnPnPnPn+E2(B4)1B3],
(2.9)

a block diagonal matrix

$$ \mathbf{Q}^{12}=\operatorname{diag}(O,Q_{0},Q_{1}, \ldots ,Q_{n}), $$
(2.10)

and an ‘almost’ block diagonal matrix

$$ \mathbf{W}^{12}= \begin{bmatrix} -B_{1}' & -B_{2}' & & & & \\ & W_{0} & & & & \\ & & W_{1} & & & \\ & & & \cdots & & \\ & & & & W_{n-1} & \\ & & & & & W_{n} \end{bmatrix}, $$
(2.11)

where \(E_{2}, O, P_{j}, j=1,\ldots ,n\), \(Q_{j}\), \(W_{j}, j=0,1, \ldots ,n\), and \(B_{i}, i= 1, 2; B_{i}', i= 1, 2, 3, 4\)are \(2\times 2\)matrices as given in Lemma 2.3.

Then \(\sigma (\vec{\alpha }, \vec{\alpha '}, \vec{\beta }, \vec{\beta '}, \vec{0}, \vec{\gamma '}, \vec{0}, \vec{\delta '}) = \sigma (\mathbf{P}^{12}+\mathbf{Q}^{12},\mathbf{W}^{12})\).

Proof

The proof is similar to the proof of Lemma 2.3, hence it is omitted here. □

Remark 2.1

In Lemmas 2.3, 2.4, 2.5, it is easy to conclude that

$$ (\mathbf{P}+\mathbf{Q})_{12}=\mathbf{P}_{12}+ \mathbf{Q}_{12}, \qquad( \mathbf{P}+\mathbf{Q})^{12}= \mathbf{P}^{12}+\mathbf{Q}^{12}. $$

Related inverse matrix eigenvalue problems

We first introduce the following matrices in \(\mathbb{M}_{2k}\):

$$ \begin{bmatrix} \mathcal{C}_{1} & \mathcal{D}_{1} & & & & \\ \mathcal{D}_{1}^{T} & \mathcal{C}_{2} & \mathcal{D}_{2} & & & \\ & \cdots & \cdots & \cdots & & \\ & & & \mathcal{D}_{k-2}^{T} & \mathcal{C}_{k-1} & \mathcal{D}_{k-1} \\ & & & & \mathcal{D}_{k-1}^{T} & \mathcal{C}_{k} \end{bmatrix}, $$
(3.1)

where \(\mathcal{C}_{i}, i=1,\ldots ,k\) and \(\mathcal{D}_{i}, i=1, \ldots ,k-1\) are \(2\times 2\) matrices, T denotes the transpose.

Definition 3.1

([22])

A matrix \(\mathbf{J}_{b}\in \mathbb{M}_{2k}\) in the form of (3.1) is called a two-banded matrix if Ci=[c`ic´ic´ic¯i], \(\grave{c}_{i}, \acute{c}_{i}, \bar{c}_{i}\neq 0\), \(i=1,\ldots ,k\); Di=[d`i0d´id¯i], \(\grave{d}_{i}, \acute{d}_{i}, \bar{d}_{i}\neq 0\), \(i=1,\ldots ,k-1\).

Lemma 3.1

([22])

Let \(\{\lambda _{i}:i=1,\ldots ,2k\}\), \(\{\mu _{i}:i=1, \ldots ,2k-1\}\)and \(\{\xi _{i}:i=1,\ldots ,2k-2\}\)be three sets of real numbers satisfying the interlacing property

$$ \begin{aligned}& \lambda _{1}\leq \mu _{1}\leq \lambda _{2}\leq \mu _{2}\leq \cdots \leq \lambda _{2k-1}\leq \mu _{2k-1}\leq \lambda _{2k}, \\ &\mu _{1}\leq \xi _{1}\leq \mu _{2}\leq \xi _{2}\leq \cdots \leq \mu _{2k-2} \leq \xi _{2k-2}\leq \mu _{2k-1}. \end{aligned} $$
(3.2)

Then there exists a two-banded matrix \(\mathbf{J}_{b}\in \mathbb{M} _{2k}\)such that

$$ \begin{aligned}& \sigma (\mathbf{J}_{b})=\{\lambda _{i}:i=1,\ldots ,2k\}, \qquad \sigma \bigl(( \mathbf{J}_{b})_{1} \bigr)=\{\mu _{i}:i=1,\ldots ,2k-1\}, \\ &\sigma \bigl((\mathbf{J}_{b})_{12}\bigr)=\{\xi _{i}:i=1,\ldots ,2k-2\}. \end{aligned} $$

Next, we consider the block matrices in \(\mathbb{M}_{2k}\) of the form

$$ \begin{bmatrix} C_{1} & D_{1} & & & & \\ D_{1}' & C_{2} & D_{2} & & & \\ & \cdots & \cdots & \cdots & & \\ & & & D_{k-2}^{T} & C_{k-1} & D_{k-1}' \\ & & & & D_{k-1} & C_{k} \end{bmatrix}, $$
(3.3)

which is block symmetric except for the \(D_{1}, D_{1}', D_{k-1}, D _{k-1}'\) entries.

Definition 3.2

A matrix \(\mathbf{M}\in \mathbb{M}_{2k}\) in the form of (3.3) is called a \(2\times 2\) block pseudo-Jacobi matrix if Ci=[cicˆicˆicˇi], \(c_{i}, \hat{c}_{i}, \check{c}_{i}\neq 0\), \(i=2,\ldots ,k-1\); Di=[didˆidˆidˇi], \(d_{i}, \hat{d}_{i}, \check{d}_{i}\neq 0\), \(i=2,\ldots ,k-2\), and C1=[c11(1)c12(1)c21(1)c22(1)], D1=[d11(1)d12(1)d21(1)d22(1)], D1=[d11(1)d12(1)d21(1)d22(1)], Dk1=[d11(k1)d12(k1)d21(k1)d22(k1)], Dk1=[d11(k1)d12(k1)d21(k1)d22(k1)], Ck=[c11(k)c12(k)c21(k)c22(k)] with each \(D_{i}, i=1,\ldots ,k-1\), and \(C_{1}, C_{k}, D _{1}', D_{k-1}'\) are \(2\times 2\) nonsingular matrices.

From Lemma 3.1, we can deduce the following theorem which includes the key technique to solve our main result.

Theorem 3.1

Let \(\{\lambda _{i}:i=1,\ldots ,2k\}\), \(\{\mu _{i}:i=1,\ldots ,2k-1\}\)and \(\{\xi _{i}:i=1,\ldots ,2k-2\}\)be three sets of real numbers satisfying the interlacing property (3.2).

If

W=[B1B2W0W1Wk4Wk3B3B4]

is an ‘almost’ block diagonal matrix with Wj=[wjwˆjwˆjwˇj],j=0,1,,k3, Bi=[b11(i)b12(i)b21(i)b22(i)],i=1,2,3,4satisfying \(w_{i}, \hat{w}_{i}, \check{w}_{i}>0, \det (W_{i})>0\), \(i=0,1,\ldots ,k-3\), \(\frac{w_{i}\hat{w}_{i+1}- \hat{w}_{i}w_{i+1}}{w_{i+1}\mathrm{det}(W_{i})}=\frac{\acute{d}_{i+2}}{ \grave{d}_{i+2}}\), \(i=0,1,\ldots ,k-4\), where \(\acute{d}_{i+2}, \grave{d}_{i+2}, i=0,1,\ldots ,k-4\)are entries of \(\mathbf{J}_{b}\)as mentioned in Lemma 3.1, and \(\mathrm{det}(B_{i}')\neq 0, i=1,2,3,4\). Then there exists a block pseudo-Jacobi matrix \(\mathbf{M}\in \mathbb{M}_{2k}\)such that

$$ \begin{aligned} &\sigma (\mathbf{M},\mathbf{W})=\{\lambda _{i}:i=1,\ldots ,2k\}, \\ &\sigma \bigl(\mathbf{M}_{12}\bigl[-E_{2}\bigl(B_{1}' \bigr)^{-1}B_{2}'\bigr],\mathbf{W}_{12} \bigr)= \{\xi _{i}:i=1,\ldots ,2k-2\}. \end{aligned} $$
(3.4)

Proof

Since \(\{\lambda _{i}:i=1,\ldots ,2k\}\), \(\{\mu _{i}:i=1,\ldots ,2k-1\}\) and \(\{\xi _{i}:i=1,\ldots ,2k-2\}\) are three sets of real numbers satisfying the interlacing property (3.2), hence by Lemma 3.1, there exists a two-banded matrix \(\mathbf{J}_{b}\in \mathbb{M}_{2k}\) such that

$$ \begin{aligned}& \sigma (\mathbf{J}_{b})=\{\lambda _{i}:i=1,\ldots ,k\},\qquad \sigma \bigl(( \mathbf{J}_{b})_{1} \bigr)=\{\mu _{i}:i=1,\ldots ,k-1\}, \\ &\sigma \bigl((\mathbf{J}_{b})_{12}\bigr)=\{\xi _{i}:i=1,\ldots ,k-2\}. \end{aligned} $$

Now for each \(\lambda =\lambda _{i}, i=1,\ldots ,2k\), there exists a nontrivial \(u\in \mathbb{R}^{2k}\) such that

$$ (\mathbf{J}_{b}-\lambda I_{2k})u=0. $$
(3.5)

Set

$$\mathbf{H}= \begin{bmatrix} I_{2} & -(B_{1}')^{-1}B_{2}' & & & \\ & I_{2} & & & \\ & & \cdots & & \\ & & & I_{2} & \\ & & & -(B_{4}')^{-1}B_{3}' & I_{2} \end{bmatrix}, $$

then one has \(\mathbf{W}\mathbf{H}=\mathbf{L}\mathbf{U}\), with \(\mathbf{L}=\operatorname{diag}(L_{1}, R^{T}_{0}, R^{T}_{1}, \ldots , R^{T}_{k-3}, L_{2})\), \(\mathbf{U}=\operatorname{diag}(U_{1}, R _{0}, R_{1},\ldots , R_{k-3}, U_{2})\), where \(U_{1}=(R^{T}_{0})^{-1}(\mathcal{D}^{T}_{1})^{-1}E_{2}, U_{2}=(R ^{T}_{k-3})^{-1}(\mathcal{D}_{k-1})^{-1}(-E_{2})\), L1=B1(U1)1,L2=B4(U2)1,Ri=[wiwˆiwi0wˇiwiwˆi2wi],i=0,1,,k3, \(\mathcal{D}^{T}_{1}, \mathcal{D}_{k-1} \in \mathbf{J}_{b}\).

Let \(u=\mathbf{U}\mathbf{H}^{-1}\tilde{u}\) and left multiplying Eq. (3.5) by L we obtain

$$ \bigl(\mathbf{L}\mathbf{J}_{b}\mathbf{U}\mathbf{H}^{-1}- \lambda \mathbf{L} \mathbf{U}\mathbf{H}^{-1}\bigr)\tilde{u}=0,\quad \text{i.e.}, \quad ( \mathbf{M}-\lambda \mathbf{W})\tilde{u}=0, $$

where \(\mathbf{M}=\mathbf{L}\mathbf{J}_{b}\mathbf{U}\mathbf{H}^{-1}\). Clearly, \(\lambda \in \sigma (\mathbf{M},\mathbf{W})\) and \(\mathbf{M} \in \mathbb{M}_{2k}\) is a block pseudo-Jacobi matrix. The same argument applies to each \(\xi =\xi _{i}, i=1,\ldots ,2k-2\), then we obtain the conclusion that \(\mathbf{M}_{12}[-E_{2}(B_{1}')^{-1}B_{2}']= \mathbf{L}_{12}(\mathbf{J}_{b})_{12}\mathbf{U}_{12}(\mathbf{H}^{-1})_{12}\), \(\mathbf{W}_{12}=\mathbf{L}_{12}\mathbf{U}_{12}(\mathbf{H}^{-1})_{12}\) and \(\xi \in \sigma (\mathbf{M}_{12},\mathbf{W}_{12})\). Thus,

$$ \sigma (\mathbf{J}_{b})\subset \sigma (\mathbf{M},\mathbf{W})\quad\text{and}\quad \sigma \bigl((\mathbf{J}_{b})_{12}\bigr)\subset \sigma \bigl( \mathbf{M}_{12}\bigl[-E_{2}\bigl(B _{1}' \bigr)^{-1}B_{2}'\bigr],\mathbf{W}_{12} \bigr). $$

Similarly by reversing the steps above we can get

$$ \sigma (\mathbf{J}_{b})\supset \sigma (\mathbf{M},\mathbf{W}) \quad\text{and}\quad\sigma \bigl((\mathbf{J}_{b})_{12}\bigr)\supset \sigma \bigl( \mathbf{M}_{12}\bigl[-E_{2}\bigl(B _{1}' \bigr)^{-1}B_{2}'\bigr],\mathbf{W}_{12} \bigr). $$

Consequently,

$$ \sigma (\mathbf{J}_{b})=\sigma (\mathbf{M},\mathbf{W})\quad \text{and}\quad \sigma \bigl(( \mathbf{J}_{b})_{12}\bigr)=\sigma \bigl( \mathbf{M}_{12}\bigl[-E_{2}\bigl(B_{1}' \bigr)^{-1}B _{2}'\bigr],\mathbf{W}_{12} \bigr). $$

This completes the proof. □

Corollary 3.1

When \(\mathbf{M}_{12}[-E_{2}(B_{1}')^{-1}B_{2}']\)and \(\mathbf{W}_{12}\)are replaced by \(\mathbf{M}^{12}[E_{2}(B_{4}')^{-1}B_{3}']\)and \(\mathbf{W}^{12}\), respectively, then the similar conclusion of Theorem 3.1still holds.

Corollary 3.2

When \(w_{i}, \hat{w}_{i}, \check{w}_{i}>0\)are replaced by \(w_{i}, \hat{w}_{i}, \check{w}_{i}<0\)for \(i=1,\ldots ,k\), then the similar conclusion of Theorem 3.1still holds.

Main result

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

Theorem 4.1

Let \(\{\lambda _{i}:i=1,\ldots ,2k\}\), \(\{\mu _{i}:i=1,\ldots ,2k-1\}\)and \(\{\xi _{i}:i=1,\ldots ,2k-2\}\)be three sets of real numbers satisfying the interlacing property (3.2). Let \(n=k-3\). Then, for any \(-\infty < a< b<+\infty \), any partition (1.4), any \(w\in L(I, \mathbb{R})\)satisfying (1.7), any real numbers \(\alpha _{i}', \beta _{i}', \gamma _{i}', \delta _{i}', i=1, 2, 3, 4\)satisfying \(\alpha _{1}'\beta _{2}'-\alpha _{2}'\beta _{1}'\neq 0\), \(\alpha _{3}'\beta _{4}'-\alpha _{4}'\beta _{3}'\neq 0\), \(\gamma _{1}'\delta _{2}'-\gamma _{2}'\delta _{1}'\neq 0\), \(\gamma _{3}'\delta _{4}'-\gamma _{4}'\delta _{3}'\neq 0\), we have the following:

(a) There exist \(r,q\in L(I,\mathbb{R})\)satisfying (1.5) and (1.6), the values of eigenparameter-dependent BCs \(\alpha _{i}, \beta _{i}, \gamma _{i}, \delta _{i}, i=1, 2, 3, 4\), such that the associated equivalent family of the fourth-order eigenparameter-dependent BVPs (1.1), (1.2) has the spectrum

$$ \begin{aligned}& \sigma \bigl(\vec{\alpha }, \vec{\alpha '}, \vec{\beta }, \vec{\beta '}, \vec{\gamma }, \vec{ \gamma '}, \vec{\delta }, \vec{\delta '}\bigr)=\{ \lambda _{i}:i=1,\ldots ,2k\}, \\ &\sigma \bigl(\vec{0}, \vec{\alpha '}, \vec{0}, \vec{\beta '}, \vec{\gamma }, \vec{\gamma '}, \vec{\delta }, \tilde{\delta '}\bigr)=\{\xi _{i}:i=1,\ldots ,2k-2\}. \end{aligned} $$

(b) There exist \(r,q\in L(I,\mathbb{R})\)satisfying (1.5) and (1.6), the values of eigenparameter-dependent BCs \(\alpha _{i}, \beta _{i}, \gamma _{i}, \delta _{i}, i=1, 2, 3, 4\), such that the associated equivalent family of the fourth-order eigenparameter-dependent BVPs (1.1), (1.2) has the spectrum

$$ \begin{aligned}& \sigma \bigl(\vec{\alpha }, \vec{\alpha '}, \vec{\beta }, \vec{\beta '}, \vec{\gamma }, \vec{ \gamma '}, \vec{\delta }, \vec{\delta '}\bigr)=\{ \lambda _{i}:i=1,\ldots ,2k\}, \\ &\sigma \bigl(\vec{\alpha }, \vec{\alpha '}, \vec{\beta }, \vec{ \beta '}, \vec{0}, \vec{\gamma '}, \vec{0}, \vec{\delta '}\bigr)=\{\xi _{i}:i=1, \ldots ,2k-2\}. \end{aligned} $$

Proof

(a) For any \(-\infty < a< b<+\infty \) and a given partition (1.4) of \([a,b]\), define \(w_{i}=\int _{a_{i}}^{b_{i}}w(t)\,dt, \hat{w}_{i}=\int _{a_{i}} ^{b_{i}}w(t)t\,dt, \check{w}_{i}=\int _{a_{i}}^{b_{i}}w(t)t^{2}\,dt, i=0,1, \ldots ,n\),Wi=[wiwˆiwˆiwˇi],i=0,1,,n, satisfying the conditions \(\frac{w_{i} \hat{w}_{i+1}-\hat{w}_{i}w_{i+1}}{w_{i+1}\mathrm{det}(W_{i})}=\frac{ \acute{d}_{i+2}}{\grave{d}_{i+2}}\), \(i=0,1,\ldots ,n-1\). For any real numbers \(\alpha _{i}', \beta _{i}', \gamma _{i}', \delta _{i}', i=1, 2, 3, 4\), satisfying \(\alpha _{1}'\beta _{2}'-\alpha _{2}'\beta _{1}'\neq 0\), \(\alpha _{3}'\beta _{4}'-\alpha _{4}'\beta _{3}' \neq 0\), \(\gamma _{1}'\delta _{2}'-\gamma _{2}'\delta _{1}'\neq 0\), \(\gamma _{3}'\delta _{4}'-\gamma _{4}'\delta _{3}'\neq 0\), define B1=[α3aα3+α4β3aβ3+β4], B2=[α1aα1+α2β1aβ1+β2],B3=[γ3bγ3+γ4δ3bδ3+δ4],B4=[γ1bγ1+γ2δ1bδ1+δ2]. Then define

W=[B1B2W0W1Wn1WnB3B4].

By (1.7), \(w_{i}>0, \hat{w}_{i}>0, \check{w}_{i}>0, i=0,1, \ldots ,n\), and \(\mathrm{det}(B_{i}')\neq 0, i=1,2,3,4\). Since \(k=n+3\), by Theorem 3.1, there exists a block pseudo-Jacobi matrix \(\mathbf{M}\in \mathbb{M}_{2n+6}\) in the form of (3.3) satisfying \(D_{1}'=-D_{k-1}'=E_{2}\) such that

$$ \begin{aligned}& \sigma (\mathbf{M},\mathbf{W})=\{\lambda _{i}:i=1,\ldots ,2n+6\}, \\ &\sigma \bigl(\mathbf{M}_{12}\bigl[-E_{2}\bigl(B_{1}' \bigr)^{-1}B_{2}'\bigr],\mathbf{W}_{12} \bigr)= \{\xi _{i}:i=1,\ldots ,2n+4\}. \end{aligned} $$

Let

$$ \begin{aligned} &\theta _{i}=\frac{1}{\hat{d}_{i}^{2}-\check{d}_{i}\,d_{i}}, \quad i=1,\ldots ,n; \\ &r_{i}=\theta _{i}d_{i}, \qquad \hat{r}_{i}= \theta _{i}\hat{d}_{i},\qquad \check{r}_{i}=\theta _{i}\check{d}_{i},\quad i=1,\ldots ,n; \\ &q_{i}=c_{i+2}+\frac{r_{i}}{\theta _{i}}+\frac{r_{i+1}}{\theta _{i+1}},\qquad \hat{q}_{i}=\hat{c}_{i+2}+\frac{\hat{r}_{i}}{\theta _{i}}+ \frac{ \hat{r}_{i+1}}{\theta _{i+1}},\\ & \check{q}_{i}=\check{c}_{i+2}+ \frac{ \check{r}_{i}}{\theta _{i}}+\frac{\check{r}_{i+1}}{\theta _{i+1}},\quad i=1, \ldots ,n-1, \\ &q_{0}=c_{2}+\frac{r_{1}}{\theta _{1}},\qquad \hat{q}_{0}= \hat{c}_{2}+\frac{\hat{r}_{1}}{\theta _{1}},\qquad \check{q}_{0}= \check{c}_{2}+\frac{\check{r}_{1}}{\theta _{1}}, \\ &q_{n}=c_{n+2}+\frac{r_{n}}{\theta _{n}},\qquad \hat{q}_{n}= \hat{c}_{n+2}+\frac{\hat{r}_{n}}{\theta _{n}},\qquad \check{q}_{n}= \check{c}_{n+2}+\frac{\check{r}_{n}}{\theta _{n}}; \\ &\alpha _{1}=d^{(1)}_{11}, \qquad \alpha _{2}=d^{(1)}_{12}-a*d^{(1)}_{11},\qquad \beta _{1}=d^{(1)}_{21},\qquad \beta _{2}=d^{(1)}_{22}-a*d^{(1)}_{21}, \\ &\alpha _{3}=c^{(1)}_{11},\qquad \alpha _{4}=c^{(1)}_{12}-a*c^{(1)}_{11},\qquad \beta _{3}=c^{(1)}_{21},\qquad \beta _{4}=c^{(1)}_{22}-a*c^{(1)}_{21}, \\ &\gamma _{1}=d^{(k-1)}_{11},\qquad \gamma _{2}=d^{(k-1)}_{12}-b*d^{(k-1)}_{11},\qquad \delta _{1}=d^{(k-1)}_{21},\qquad \delta _{2}=d^{(k-1)}_{22}-b*d^{(k-1)}_{21}, \\ &\gamma _{3}=c^{(k)}_{11}, \qquad \gamma _{4}=c^{(k)}_{12}-b*c^{(k)}_{11},\qquad \delta _{3}=c^{(k)}_{21},\qquad \delta _{4}=c^{(k)}_{22}-b*c^{(k)}_{21}, \end{aligned} $$

and define \(\mathbf{P}, \mathbf{Q}, \mathbf{P}_{12}[-E_{2}(B_{1}')^{-1}B _{2}']\) and \(\mathbf{Q}_{12}\) by (2.3), (2.4), (2.6) and (2.7), respectively. Clearly, \(r_{i}, \hat{r}_{i}, \check{r}_{i}\neq 0, i=1,\ldots ,n\). It is easy to see that \(\mathbf{M}=\mathbf{P}+\mathbf{Q}\) and \(\mathbf{M}_{12}[-E_{2}(B_{1}')^{-1}B _{2}']=\mathbf{P}_{12}[-E_{2}(B_{1}')^{-1}B_{2}']+\mathbf{Q}_{12}\). With the notation in (2.5) and (2.8), we also have \(( \mathbf{W})_{12}=\mathbf{W}_{12}\). Therefore,

$$\begin{aligned} &\sigma (\mathbf{P}+\mathbf{Q},\mathbf{W})=\{\lambda _{i}:i=1,\ldots ,2n+6 \}, \\ &\sigma \bigl(\mathbf{P}_{12}\bigl[-E_{2}\bigl(B_{1}' \bigr)^{-1}B_{2}'\bigr]+\mathbf{Q}_{12}, \mathbf{W}_{12}\bigr)=\{\xi _{i}:i=1,\ldots ,2n+4\}. \end{aligned}$$

By Lemmas 2.3 and 2.4 we see that for the fourth-order eigenparameter-dependent BVPs (1.1), (1.2)

$$ \begin{aligned} &\sigma \bigl(\vec{\alpha }, \vec{\alpha '}, \vec{\beta }, \vec{\beta '}, \vec{\gamma }, \vec{ \gamma '}, \vec{\delta }, \vec{\delta '}\bigr)=\{ \lambda _{i}:i=1,\ldots ,2n+6\}, \\ &\sigma \bigl(\vec{0}, \vec{\alpha '}, \vec{0}, \vec{\beta '}, \vec{\gamma }, \vec{\gamma '}, \vec{\delta }, \tilde{\delta '}\bigr)=\{\xi _{i}:i=1,\ldots ,2n+4\}. \end{aligned} $$

We observe that the choice of \(r_{i}, \hat{r}_{i}, \check{r}_{i}, i=1, \ldots ,n\) and \(q_{i}, \hat{q}_{i}, \check{q}_{i}, i=0,\ldots ,n\) and all \(r, q\in L(I,\mathbb{R})\) by this choice form an equivalent family of the fourth-order eigenparameter-dependent BVPs. This completes the proof.

(b) The proof is similar using Corollary 3.1, and Lemmas 2.3, 2.5, hence the details are omitted here. □

To illustrate our main result we show the following algorithm and example.

Algorithm 1

Step 1.:

Input three sets of real numbers \(\lambda _{1}, \lambda _{2}, \ldots , \lambda _{2k}\), \(\mu _{1}, \mu _{2}, \ldots , \mu _{2k-1}\)and \(\xi _{1}, \xi _{2}, \ldots , \xi _{2k-2}\).

Step 2.:

Construct the matrix \(\mathbf{J}_{b}\)by the proof of Problem 2.6 in [22].

Step 3.:

Set \(a,b\)and the partition of \([a,b]\). Input \(\bar{w}(t)\)and \(\alpha _{i}', \beta _{i}', \gamma _{i}', \delta _{i}' \in \mathbb{R}, i=1,2,3,4\). SetWandHto obtainLandU. Compute \(\mathbf{M}=\mathbf{L}\mathbf{J}_{b}\mathbf{U}\mathbf{H}^{-1}\).

Step 4.:

Obtain \(\alpha _{i}, \beta _{i}, \gamma _{i}, \delta _{i} \in \mathbb{R}, i=1,2,3,4\), and \(r_{i}, \hat{r}_{i}, \check{r}_{i}, i=1, \ldots ,n\), \(q_{i}, \hat{q}_{i}, \check{q}_{i}, i=0,\ldots ,n\)by Theorem 4.1.

Example 1

Given three sets of real numbers satisfying the interlacing property (3.2): \(\lambda _{1}=-3.8397, \lambda _{2}=-3.0377, \lambda _{3}=-0.0540, \lambda _{4}=2.5830, \lambda _{5}=5.5791, \lambda _{6}=7.0055, \lambda _{7}=11.3818, \lambda _{8}=15.3820; \mu _{1}=-3.0842, \mu _{2}=-1.9058, \mu _{3}=2.3327, \mu _{4}=3.2462, \mu _{5}=6.2319, \mu _{6}=9.4212, \mu _{7}=13.7579; \xi _{1}=-2.7411, \xi _{2}=-1.3502, \xi _{3}=3.1316, \xi _{4}=5.7604, \xi _{5}=8.0532, \xi _{6}=13.1461\). Let \(\alpha _{1}'=-10, \alpha _{2}'=-12, \alpha _{3}'=-5, \alpha _{4}'=-14, \beta _{1}'=-4, \beta _{2}'=-2, \beta _{3}'=-2, \beta _{4}'=-7, \gamma _{1}'=9, \gamma _{2}'=-33, \gamma _{3}'=-3, \gamma _{4}'=7, \delta _{1}'=3, \delta _{2}'=-15, \delta _{3}'=-1, \delta _{4}'=1, n=1\). Given an interval \(I=[-2,3]\) and a partition of it: \(-2<0<1<3\). Define the piecewise polynomial function on I by

$$ \bar{w}(t)= \textstyle\begin{cases} 0.7624t^{2}-0.1236t-0.1401, & t\in [-2,0), \\ 0, & t\in [0,1), \\ -3.2811t^{2}+12.3742t-9.0306, & t\in [1,3]. \end{cases} $$

Then we can obtain the fourth-order eigenparameter-dependent boundary value problem:

$$ \textstyle\begin{cases} (\bar{p}y'')''+\bar{q}y=\lambda \bar{w}y, \quad \text{on }I=[-2,3], \\ (-10\lambda +483.2857)y(-2)+(-12\lambda +1587.3571)y'(-2) \\ \quad{}+(-5\lambda +36.1429)(py')(-2)+(-14\lambda +104.1429)(py'')'(-2)=0, \\ (-4\lambda +29.7143)y(-2)+(-2\lambda +81.1429)y'(-2) \\ \quad{}+(-2\lambda +6.8571)(py')(-2)+(-7\lambda +22.8571)(py'')'(-2)=0, \\ (9\lambda -522.0000)y(3)+(-33\lambda +1380.9000)y'(3) \\ \quad{}+(-3\lambda +76.5000)(py'')(3)+(7\lambda -96.4500)(py'')'(3)=0, \\ (3\lambda -14.0000)y(3)+(-15\lambda +74.9667)y'(3) \\ \quad{}+(-\lambda +5.5000)(py'')(3)+(\lambda -6.1500)(py'')'(3)=0, \end{cases} $$

where the piecewise polynomial functions \(\bar{r}=\frac{1}{\bar{p}}\) and on \(I=[-2,3]\) are as follows:

$$\begin{aligned} &\bar{r}(t)= \textstyle\begin{cases} 0, & t\in [-2,0), \\ 211.2683t^{2}-201.4703t+30.9248, & t\in [0,1), \\ 0, & t\in [1,3]. \end{cases}\displaystyle \\ & \bar{q}(t)= \textstyle\begin{cases} 8.5514t^{2}-9.2826t-12.0102,& t\in [-2,0), \\ 0, & t\in [0,1), \\ -80.7883t^{2}+325.0663t-291.8755,& t\in [1,3]. \end{cases}\displaystyle \end{aligned}$$

Hence the spectrum of the reconstructed fourth-order eigenparameter-dependent BVPs (1.1), (1.2) is

$$ \textstyle\begin{cases} \sigma (\vec{\alpha }, \vec{\alpha '}, \vec{\beta }, \vec{\beta '}, \vec{\gamma }, \vec{\gamma '}, \vec{\delta }, \vec{\delta '}) \\ \quad=\{-3.8397, -3.0377, -0.0540, 2.5830, 5.5791, 7.0055, 11.3818, 15.3820 \}, \\ \sigma (0, \vec{\alpha '}, 0, \vec{\beta '}, \vec{\gamma }, \vec{\gamma '}, \vec{\delta }, \vec{\delta '}) \\ \quad=\{-2.7411, -1.3502, 3.1316, 5.7604, 8.0532, 13.1461\}. \end{cases} $$

References

  1. 1.

    Ao, J.J., Bo, F.Z.: Matrix representations of fourth order boundary value problems with eigenparameter-dependent boundary conditions. Acta Math. Sinica (Chin. Ser.) 60, 427–438 (2017) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Ao, J.J., Sun, J., Zettl, A.: Finite spectrum of 2nth order boundary value problems. Appl. Math. Lett. 42, 1–8 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ao, J.J., Sun, J., Zhang, M.Z.: The finite spectrum of Sturm–Liouville problems with transmission conditions. Appl. Math. Comput. 218, 1166–1173 (2011)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Ao, J.J., Sun, J., Zhang, M.Z.: Equivalence of fourth order boundary value problems and matrix eigenvalue problems. Results Math. 63, 581–595 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Ao, J.J., Zhang, L.: Inverse spectral problem of fourth-order boundary value problems with finite spectrum. Math. Methods Appl. Sci. 42, 4472–4479 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Atkinson, F.V.: Discrete and Continuous Boundary Problems. Academic Press, New York (1964)

    Google Scholar 

  7. 7.

    Bebinao, N., Providência, J.D.: Inverse problems for pseudo-Jacobi matrices: existence and uniqueness results. Inverse Probl. 27, 025005 (2011)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Borg, G.: Eine Umkehrung der Sturm–Liouvilleschen Eigenwertaufgabe. Acta Math. 78, 1–96 (1946)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cai, J.M., Zheng, Z.W.: Inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type. Appl. Math. Comput. 327, 22–34 (2018)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Freiling, G., Yurko, V.A.: Inverse Sturm–Liouville Problems and Their Applications. NOVA Science Publishers, New York (2001)

    Google Scholar 

  11. 11.

    Fu, S.Z., Xu, Z.B., Wei, G.S.: The interlacing of spectra between continuous and discontinuous Sturm–Liouville problems and its application to inverse problems. Taiwan. J. Math. 16, 651–663 (2012)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Gladwell, M.: Inverse Problems in Vibrations. Kluwer Academic, Dordrecht (2004)

    Google Scholar 

  13. 13.

    Hochstadt, H.: On some inverse problems in matrix theory. Arch. Math. 18, 201–207 (1967)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kong, Q., Volkmer, H., Zettl, A.: Matrix representations of Sturm–Liouville problems with finite spectrum. Results Math. 54, 103–116 (2009)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Kong, Q., Wu, H., Zettl, A.: Sturm–Liouville problems with finite spectrum. J. Math. Anal. Appl. 263, 748–762 (2001)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Kong, Q., Zettl, A.: Inverse Sturm–Liouville problems with finite spectrum. J. Math. Anal. Appl. 386, 1–9 (2012)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Mirzaei, H., Ghanbari, K.: Matrix representation of a sixth order Sturm–Liouville problem and related inverse problem with finite spectrum. Bull. Iran. Math. Soc. 41, 1031–1043 (2015)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Shieh, C.T.: Some inverse problems on Jacobi matrices. Inverse Probl. 20, 589–600 (2004)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Simon, B.: The classical moment problem as a self-adjoint finite difference operator. Adv. Math. 137, 82–203 (1998)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Tretter, C.: Boundary eigenvalue problems with differential equations \(N\eta =\lambda P\eta \) with λ-polynomial boundary conditions. J. Differ. Equ. 170, 408–471 (2001)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Volkmer, H.: Eigenvalue problems of Atkinson, Feller and Krein and their mutual relationship. Electron. J. Differ. Equ. 2005, 48 (2005)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Xu, S.F.: An Introduction to Inverse Algebraic Eigenvalue Problems. Peking University Press, Beijing (1998)

    Google Scholar 

  23. 23.

    Yang, C.F., Yang, X.P.: An interior inverse problem for the Sturm–Liouville operator with discontinuous conditions. Appl. Math. Lett. 22, 1315–1319 (2009)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Yurko, V.A.: An inverse problem for higher order differential operators on star-type graphs. Inverse Probl. 23, 893–903 (2007)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Yurko, V.A.: Inverse problems for second order integro-differential operators. Appl. Math. Lett. 74, 1–6 (2017)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Zhang, L., Ao, J.J.: On a class of inverse Sturm–Liouville problems with eigenparameter dependent boundary conditions. Appl. Math. Comput. 2019, 362 (2019). https://doi.org/10.1016/j.amc.2019.06.067

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Zhang, L., Ao, J.J.: Inverse spectral problem for Sturm–Liouville operator with coupled eigenparameter dependent boundary conditions of Atkinson type. Inverse Probl. Sci. Eng. 27, 1689–1702 (2019)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the referees for their helpful comments and detailed suggestions. These have significantly improved the presentation of this paper.

Availability of data and materials

No data were used to support this study.

Funding

This work was supported by National Natural Science Foundation of China (Grant Nos. 11661059, 11301259), Natural Science Foundation of Inner Mongolia (Grant No. 2017JQ07).

Author information

Affiliations

Authors

Contributions

The authors equally contributed this work. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Ji-jun Ao.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflicts of interest.

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ao, J., Zhang, L. Inverse spectral problem of a class of fourth-order eigenparameter-dependent boundary value problems. Adv Differ Equ 2020, 44 (2020). https://doi.org/10.1186/s13662-020-2519-0

Download citation

MSC

  • 34A55
  • 34B09
  • 15A29

Keywords

  • Inverse spectral problems
  • Inverse matrix eigenvalue problems
  • Fourth-order problems
  • Eigenparameter-dependent boundary conditions
  • Block pseudo-Jacobi matrix
  • Two-banded matrix