Eigenvalues of fourth-order differential operators with eigenparameter dependent boundary conditions

1.   Introduction

Sturm-Liouville problem originates from various fields such as physics, engineering, finance and medicine, and it has been widely researched ^[1,2]. Nowadays, differential equations with boundary conditions depending on eigenparameter are also widely used in acoustic scattering, quantum mechanics theory and so on. Particularly, more and more researchers have paid close attention to Sturm-Liouville problems with boundary conditions depending on eigenparameter, the distribution of eigenvalues, asymptotic of eigenvalues and eigenfunctions, oscillation theory and inverse spectral theory of such problrm are deeply researched, and many results are obtained. Up to now, it has become an important research topic and has made great progress ^[3,4,5,6,7]. In recent years, the fourth-order differential operators with eigenparameter dependent boundary conditions appear in elastic beam models, the heat conduct problem and so on are also gained great progress. For more details, we refer the readers to ^[8,9,10,11].

In the last two decades, the dependence of eigenvalues on coefficients and parameters of differential operator has attracted lots of the attention by many researchers. In ^[12], Kong and Zettl obtained that the eigenvalues of regular Sturm-Liouville problems are differentiable functions with respect to all the data and they gave expressions for their derivatives. Later, this problem was extended to Sturm-Liouville operators with discontiuity, third-order and fourth-order differential operators etc. ^{[13,14,15,16,17,18]}. Recently, Zhang and Li in ^[19] showed that the eigenvalues of Sturm-Liouville problems with eigenparameter dependent boundary conditions are differentiable functions of all the data. In ^[20], Zinsou considered the dependence of eigenvalues of a general fourth-order differential equation with transmission conditions and obtained similar results. These results provide a theoretical support for the numerical calculation of eigenvalues and eigenfunctions ^[21,22].

Inspired by the above mentioned results, a natural question is that whether similar results still true for fourth-order boundary value problems when eigenparameter appear in the the boundary conditions? In this paper, we give a confirm answer. As we know, the problems with spectral parameter arise from several physical or other applied problems, for instance, the free bending vibrations of rod ^[23,24]. Therefore, in this paper, we try to discuss the dependence of eigenvalues of fourth-order differential equations with eigenparameter dependent boundary conditions. It is worth mentioning that we consider such a problem with both endpoints depending on the spectral parameter $\mu$. Compared with the problem with spectral parameter at one end, the inner product and space constructed are different, and it is more troublesome in the process of deriving the differential expression of the eigenvalues with respect to the coefficient matrix of the boundary conditions with spectral parameter. The main result is that each of the eigenvalues of the fourth-order boundary value problem can be embedded in a continuous eigenvalue branch. Furthermore, we obtain the differential expression of the eigenvalues with respect to all data in the sense of ordinary or Fréchet derivatives.

The rest of this paper is organized as follows. In Section 2, we introduce a fourth-order boundary value problems and define a new self-adjoint operator $\mathcal{F}$ such that the eigenvalues of such a problem coincide with those of $\mathcal{F}$. In Section 3, we discuss the continuity of the eigenvalues and eigenfunctions. In Section 4, we give the differential expressions of the eigenvalues with respect to each of parameters.

2.   Fourth-order boundary value problem

We consider the fourth-order differential equation

on $[a, b]$, with eigenparameter dependent boundary conditions at endpoints

where $-\infty < a < b < +\infty$, $\mu \in \mathbb{C}$ is the spectral parameter,

Note that the quasi-derivatives associated to (2.1) are

Let the weighted Hilbert space be defined as

with inner product $\langle f, g\rangle_{1} = \int_{a}^{b}f(x)\bar{g}(x)w(x)\mathrm{d}x$ for any $f, g\in H_{1}$. We define a new Hilbert space

with the inner product

for $F = (f, f_{1}, f_{2}, f_{3}, f_{4})^{T}, G = (g, g_{1}, g_{2}, g_{3}, g_{4})^{T}\in H$. Define an operator $\mathcal{F}$ as

with the domain

Lemma 2.1. The operator $\mathcal{F}$ is a self-adjoint operator in H.

Proof. The proof is similar to that of ^[25], the equation we considered is more complicated and the derivatives in boundary conditions are quasi-derivatives, here we omit the details.

Lemma 2.2. ^[25]The spectrum of $\mathcal{F}$ consists of isolated eigenvalues, which coincide with those of the fourth-order boundary value problems (2.1)–(2.5). Furthermore, all the eigenvalues are real-valued.

3.   Continuity of eigenvalues and eigenfunctions

Let $\chi_{1}(x, \mu), \chi_{2}(x, \mu), \chi_{3}(x, \mu), \chi_{4}(x, \mu)$ be the linearly independent solutions of Eq (2.1) satisfying the initial conditions

We define their Wronskian as

Lemma 3.1. The number $\mu$ is an eigenvalue of operator $\mathcal{F}$ if and only if

where

Proof. By [26,Theorem 1.8], we see that boundary value problems (2.1)–(2.5) is well-posed. Let $\mu$ be an eigenvalue of (2.1)–(2.5), then there exists a non-trivial solution

of (2.1), where $c_{1}, c_{2}, c_{3}, c_{4}$ are not all zero. Since $f(x, \mu)$ satisfies the boundary conditions (2.2)–(2.5), we have

By the initial condition (3.1), we have

Since not all $c_{1}, c_{2}, c_{3}, c_{4}$ are zero, we get that $\det\left(M+N\Phi(b, \mu)\right) = 0$.

On the other hand, if $\Delta(\mu) = 0$, then Eq (3.2) has non-zero solution $c_{1}, c_{2}, c_{3}, c_{4}$. Let

then $f(x, \mu)$ satisfies (2.1)–(2.5) and thus $\mu$ is an eigenvalue. This completes the proof.

Now, we consider the Banach space

with norm

for any $\xi = (\frac{1}{p}, q, q_{0}, w, \gamma_{1}, \gamma_{2}, \tau_{1}, \tau_{2}, \alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2})\in B$. Let

Theorem 3.1. Let $\tilde{\xi} = (\frac{1}{\tilde{p}}, \tilde{q}, \tilde{q}_{0}, \tilde{w}, \tilde{\gamma_{1}}, \tilde{\gamma_{2}}, \tilde{\tau_{1}}, \tilde{\tau_{2}}, \tilde{\alpha_{1}}, \tilde{\alpha_{2}}, \tilde{\beta_{1}}, \tilde{\beta_{2}})\in\Omega$ and $\mu(\tilde{\xi})$ be an isolated eigenvalue of (2.1)–(2.5) with $\tilde{\xi}$. Then $\mu$ is continuous on $\tilde{\xi}$. That is, given any $\varepsilon > 0$, there exists a $\delta > 0$ such that the problems (2.1)–(2.5) has exactly an isolated eigenvalue $\mu(\xi)$ satisfying

if $\xi = (\frac{1}{p}, q, q_{0}, w, \gamma_{1}, \gamma_{2}, \tau_{1}, \tau_{2}, \alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2})$ satisfies

Proof. By Lemma 3.1, $\mu(\tilde{\xi})$ is an eigenvalue of $(2.1)$–$(2.5)$ if and only if $\Delta(\tilde{\xi}, \mu(\tilde{\xi})) = 0$. For any $\xi\in\Omega$, $\Delta(\xi, \mu)$ is an entire function of $\mu$ and is continuous on $\xi$ (see [27,Theorems 2.7 and 2.8]). It is easy seen that $\Delta(\tilde{\xi}, \mu)$ is not a constant in $\mu$ because $\mu(\tilde{\xi})$ is an isolated eigenvalue. Therefore, there exists $\rho_{0} > 0$ such that $\Delta(\tilde{\xi}, \mu)\neq0$ for $\mu\in S_{\rho_{0}} : = \{\mu\in\mathbb{C} :|\mu-\mu(\tilde{\xi})| = \rho_{0}\}$. By the continuity of the roots of an equation as a function of parameters (see [28,(9.17.4)]), the statement follows.

By a normalized eigenvector $(m, m_{1}, m_{2}, m_{3}, m_{4})^{T}\in H$, we mean $m$ satisfies the problems $(2.1)$–$(2.5)$, $m_{1} = m(a), m_{2} = m^{[1]}(a), m_{3} = \tau_{1}m(b)-\gamma_{1}m^{[3]}(b), \; m_{4} = \beta_{1}m^{[1]}(b)-\alpha_{1}m^{[2]}(b)$, and

Now we give a result for normalized eigenfunctions.

Theorem 3.2. Assume that $\mu(\xi)$ is an eigenvalue of (2.1)–(2.5) with $\xi\in\Omega$ and $(m, m_{1}, m_{2}, m_{3}, m_{4})^{T}\in H$ is the corresponding normalized eigenvector for $\mu(\xi)$. Then there exists a normalized eigenvector $(n, n_{1}, n_{2}, n_{3}, n_{4})^{T}\in H$ for $\mu(\tilde{\xi})$ with $\tilde{\xi}\in\Omega$, which is specified in Theorem 3.1, such that

as $\tilde{\xi}\rightarrow \xi$ both uniformly on $[a, b]$.

Proof. (i) We know that $\mu(\xi)$ is an isolated eigenvalue of multiplicity $j (j = 1, 2, 3, 4)$ for all $\xi$ in some neighborhood $\mathcal{N}$ of $\tilde{\xi}$ in $\Omega$. Suppose $\mu(\xi)$ is simple. Let $(f(x, \xi), f_{1}(\xi), f_{2}(\xi), f_{3}(\xi), f_{4}(\xi))^{T}$ be an eigenvector for $\mu(\xi)$ with

By Theorem 3.1, there exists $\mu(\tilde{\xi})$ such that

Define the boundary condition matrix as

then

By Theorem 3.2 of ^[12], we can obtain an eigenfunction $f(x, \tilde{\xi})$ for $\mu(\tilde{\xi})$ such that $\|f(x, \tilde{\xi})\| = 1$ and

as $\tilde{\xi}\rightarrow \xi$ both uniformly on $[a, b]$. Then we obtain

Let

Then (3.3) holds by (3.4) and (3.5).

(ii) Assume that $\mu(\xi)$ is an eigenvalue of multiplicity $j(j = 2, 3, 4)$. Then we can choose eigenfunctions of $\mu(\xi)$ such that all of them satisfy the same initial conditions at $c_{0}$ for some $c_{0}\in[a, b]$ since a linear combination of $j$ linearly independent eigenfunctions can be chosen to satisfy arbitrary initial conditions.

Similarly, we obtain (3.3) as (i), This completes the proof.

4.   Differential expression of eigenvalues

In this section, we focus on giving the derivative formulas of the eigenvalues for all the parameters. First we give the definition of the Fréchet derivative.

Definition 4.1. A map $\mathcal{F}$ from a Banach space X into Banach space Y is differentiable at a point $x\in X$, if there exists a bounded linear operator $\mathrm{d}\mathcal{F}_{x}: X\rightarrow Y$ such that for $k\in X$

Theorem 4.1. Assume that $\mu(\xi)$ is an eigenvalue of (2.1)–(2.5) with $\xi\in\Omega$ and $(m, m_{1}, m_{2}, m_{3}, m_{4})^{T}\in H$ is the corresponding normalized eigenvector for $\mu(\xi)$. Suppose $\mu(\xi)$ is a simple eigenvalue or $\mu(\sigma)$ is an eigenvalue of multiplicity $j\; (j = 2, 3, 4)$ for each $\sigma$ in some neighborhood $\mathcal{N}\subset\Omega$ of $\xi$. Then $\mu$ is differentiable with respect to all the data in $\xi$.

(1) Let all the data of $\xi$ be fixed except the boundary condition parameter matrix

and $\mu(K_{1}): = \mu(\xi)$. Then

for all L satisfying $\det(K_{1}+L) = \det K_{1} = \rho_{1}.$

(2) Let all the data of $\xi$ be fixed except the boundary condition parameter matrix

and $\mu(K_{2}): = \mu(\xi)$. Then

for all L satisfying $\det(K_{2}+L) = \det K_{2} = \rho_{2}.$

(3) Let all the data of $\xi$ be fixed except $p$ and $\mu(\frac{1}{p}): = \mu(\xi)$. Then

(4) Let all the data of $\xi$ be fixed except $q$ and $\mu(q): = \mu(\xi)$. Then

(5) Let all the data of $\xi$ be fixed except $q_{0}$ and $\mu(q_{0}): = \mu(\xi)$. Then

(6) Let all the data of $\xi$ be fixed except $w$ and $\mu(w): = \mu(\xi)$. Then

Proof. Let all the data of $\xi$ be fixed except one and $\mu(\tilde{\xi})$ be the eigenvalue satisfying Theorem 3.1 when $\|\tilde{\xi}-\xi\| < \varepsilon$ for sufficiently small $\varepsilon > 0$. For the above six cases, we replace $\mu(\tilde{\xi})$ by $\mu(K_{1}+L), \; \mu(K_{2}+L), \; \mu(\frac{1}{p}+k), \; \mu(q+k), \; \mu(q_{0}+k), \; \mu(w+k)$, respectively. Let $(n, n_{1}, n_{2}, n_{3}, n_{4})^{T}$ be the corresponding normalized eigenvector.

(1) By (2.1) we have

It follows from (4.7) and (4.8) that

Integrating from $a$ to $b$ implies that

According to the boundary condition (2.2), we have

Thus

Analogously, the boundary condition (2.3) implies that

Let $K_{1}+L = \left(\begin{array}{cc} \tilde{\tau_{1}} & \tilde{\tau_{2}}\\ \tilde{\gamma_{1}} & \tilde{\gamma_{2}} \end{array}\right).$ Then according to the boundary condition (2.4), we have

Thus

Analogously, the boundary condition (2.5) implies that

From (4.9)–(4.13), we get

Dividing both sides of (4.14) by $L$ and taking the limit as $L\rightarrow0$, by Theorem 3.2, we get

Then (4.1) follows. In a similar discussion, we can obtain (4.2).

(2) For $k\in L^{1}[a, b]$, let $\frac{1}{p}+k = \frac{1}{\tilde{p}}$. Using (2.1) and integration by parts, we have

where $\bar{n}^{[2]} = \tilde{p}\bar{n}''$, $\bar{n}^{[3]} = (\tilde{p}\bar{n}'')'-q\bar{n}'$. Then by (2.2)–(2.5), we obtain

Dividing both sides of (4.15) by $k$ and taking the limit as $k\rightarrow0$, by Theorem 3.2 we get

Then (4.3) follows. Similar to the proof of (4.3), we can obtain (4.4).

(3) For $k\in L^{1}[a, b]$. By (2.1), we have

Using the boundary conditions (2.2)–(2.5), we have

Then (4.5) follows. The proof of (4.6) is similar as that of (4.5), hence we omit the details.

Theorem 4.2. Let $\mu(\xi)$ be an eigenvalue of (2.1)–(2.5) with $\xi\in\Omega$ and $(m, m_{1}, m_{2}, m_{3}, m_{4})^{T}\in H$ be a normalized eigenvector for $\mu(\xi)$. Assume that $\mu(\xi)$ is a simple eigenvalue or $\mu(\sigma)$ is an eigenvalue of multiplicity $j\; (j = 2, 3, 4)$ for each $\sigma$ in some neighborhood $\mathcal{N}\subset\Omega$ of $\xi$. Then $\mu$ is differentiable with respect to the data in $\xi$.

(1) Let all the data of $\xi$ be fixed except $\tau_{1}$ and $\mu(\tau_{1}): = \mu(\xi)$. Then

where $\mu\gamma_{1}-\gamma_{2}\neq0$.

(2) Let all the data of $\xi$ be fixed except $\tau_{2}$ and $\mu(\tau_{2}): = \mu(\xi)$. Then

where $\mu\gamma_{1}-\gamma_{2}\neq0$.

(3) Let all the data of $\xi$ be fixed except $\gamma_{1}$ and $\mu(\gamma_{1}): = \mu(\xi)$. Then

where $\mu\tau_{1}-\tau_{2}\neq0$.

(4) Let all the data of $\xi$ be fixed except $\gamma_{2}$ and $\mu(\gamma_{2}): = \mu(\xi)$. Then

where $\mu\tau_{1}-\tau_{2}\neq0$.

(5) Let all the data of $\xi$ be fixed except $\alpha_{1}$ and $\mu(\alpha_{1}): = \mu(\xi)$. Then

where $\mu\beta_{1}+\beta_{2}\neq0$.

(6) Let all the data of $\xi$ be fixed except $\alpha_{2}$ and $\mu(\alpha_{2}): = \mu(\xi)$. Then

where $\mu\beta_{1}+\beta_{2}\neq0$.

(7) Let all the data of $\xi$ be fixed except $\beta_{1}$ and $\mu(\beta_{1}): = \mu(\xi)$. Then

where $\mu\alpha_{1}+\alpha_{2}\neq0$.

(8) Let all the data of $\xi$ be fixed except $\beta_{2}$ and $\mu(\beta_{2}): = \mu(\xi)$. Then

where $\mu\alpha_{1}+\alpha_{2}\neq0$.

Proof. (1) For $k\in L^{1}[a, b]$. Using (2.1) and integration by parts, we have

It follows from (2.2)–(2.5) that

and

Combining (4.25)–(4.29), we obtain

Dividing both sides of (4.30) by $k$ and taking the limit as $k\rightarrow0$, by Theorem 3.2, we get

where $\mu\gamma_{1}-\gamma_{2}\neq0$. Then (4.17) follows. The proofs of (4.18)–(4.24) are similar as that of (4.17), hence we omit the details.

5.   Conclusions

This paper gives the dependence of eigenvalues of a fourth-order differential operator with eigenparameter dependent boundary conditions. The novelty lies in the fact that the fourth-order differential operator we considered has eigenparameter dependent boundary conditions at two endpoints. By a newly defined operator $\mathcal{F}$ such that the eigenvalues of the fourth-order boundary problem being consistent with those of $\mathcal{F}$, we give the differential expressions of the eigenvalues with respect to all data.

Acknowledgments

This research is supported by NSF of Shandong Province (Nos. ZR2019MA034, ZR2020QA009, ZR2020QA010) and PSF of China (No. 2020M682139). The authors are grateful to the referees for his/her careful reading and very helpful suggestions which improved and strengthened the presentation of this manuscript.

Conflict of interest

All authors declare no conflicts of interest in this paper.