The Uniform Lipschitz Continuity of Eigenvalues of Sturm–Liouville Problems with Respect to the Weighted Function

Abstract

The present paper is concerned with the uniform boundedness of the normalized eigenfunctions of Sturm–Liouville problems and shows that the sequence of eigenvalues is uniformly local Lipschitz continuous with respect to the weighted functions.

1. Introduction

Consider the regular Sturm–Liouville problem with the self-adjoint separated boundary conditions

$$-y^{″}\left(x\right)+q\left(x\right)y\left(x\right)=\lambda w\left(x\right)y\left(x\right),x\in [0,1],$$

(1)

$${\mathbf{B}}_{\mathbf{1}}y:=cos{\theta }_{1}y\left(0\right)-sin{\theta }_1{y}^{{}^{\prime }}\left(0\right)=0,{\mathbf{B}}_{\mathbf{2}}y:=cos{\theta }_{2}y\left(1\right)-sin{\theta }_2{y}^{{}^{\prime }}\left(1\right)=0,$$

(2)

where ${\theta }_1\in [0,\pi ),{\theta }_2\in (0,\pi ]$, $\lambda$ is the spectral parameter, $q\in L^1([0,1],\mathbb{R})$,

$$w\left(x\right)\ge 0,{\int }_0^{1}w\left(x\right)\mathrm{d}x>0,w\in L^1[0,1].$$

(3)

From the spectral theory of the Sturm–Liouville problems, it is known that the Sturm–Liouville operator is symmetric and all of the eigenvalues of the (1) and (2) are real, isolated with no finite accumulation point, bounded below, and can be ordered to satisfy

$$-\infty <{\lambda }_1<{\lambda }_2<{\lambda }_3<\dots {\lambda }_n<\dots ,$$

(4)

and ${\lambda }_n\to +\infty$ as $n\to +\infty$ (see [1,2]).

When it comes to eigenvalue problems, we typically focus on their properties and the behavior of their eigenfunctions. The continuity and differentiability of eigenvalues are important properties in classical spectral theory. The continuity of eigenvalues can tell us how to find continuous eigenvalues in the parameter space, helping us to understand their properties. Meanwhile, the differentiability of eigenvalue problems can allow us to gain deeper insights into how eigenvalues change. Beyond mathematics, the continuity and differentiability of eigenvalues are also widely applied in other fields. For instance, in quantum mechanics [3,4,5,6,7], it can help us better understand the behavior of physical systems, such as the behavior of electrons in a lattice. In engineering [8,9,10,11,12], it can also help us better design and optimize complex systems.

As one of the important properties of eigenvalues in classical spectral theory, the continuity and differentiability of eigenvalues for the Sturm–Liouville problems, with respect to the parameters in the equation (the potentials and the weights), or in the boundary conditions, have been widely studied by many authors. Zettl et al. proved the continuity of the eigenvalues with respect to the coefficients in the equation and the boundary conditions in the usual sense. The continuous eigenvalue branch was constructed, and the differential formula for the continuous eigenvalue branch is provided (see [13,14,15]). Meirong Zhang et al. proved the strong continuity of the eigenvalues and the corresponding eigenfunctions on the weak topology space of the coefficient functions (see [16,17,18,19]). Such strong continuity has been applied efficiently to solve the extremal problems and the optimal recovery problems in spectral theory [20,21,22]. Recently, Jiangang Qi and Xiao Chen discussed a new kind of continuity of eigenvalues, which is the uniform local Lipschitz continuity of the eigenvalue sequence ${\left\{{\lambda }_n\left(q\right)\right\}}_{n\ge 1}$ with respect to $q\left(x\right)$ (see [23]) under the restrictions that $w\left(x\right)$ is monotone and has a positive lower bound. This kind of continuity of eigenvalues indicates that the eigenvalues possess good properties, not only from a single point of view, but also from the whole point of view.

In this paper, we continue to study the uniform local Lipschitz continuity of the eigenvalue sequence with respect to the weighted functions. To this end, we first prove the uniform boundedness of normalized eigenfunctions of the Sturm–Liouville problems (1) and (2), see Theorem 3 below. In our results, the monotonicity restriction on the weight function, which is used in the results of [23], is removed. Furthermore, in order to ensure the uniform boundedness of normalized eigenfunctions, we show that the restriction “positive lower bound” on the weight is not necessary, see Theorem 4 below. Meanwhile, a counterexample is constructed to indicate that the sequence of normalized eigenfunctions is not uniformly bounded if $w\left(x\right)$ has no positive lower bound. With the aid of Theorems 3 and 4, we prove the main result of this paper, Theorem 5.

The present paper demonstrates that, under some appropriate restrictions, the sequence of eigenvalues ${\left\{{\lambda }_n\left(w\right)\right\}}_{n>1}$ possesses the desired continuity mentioned earlier, and these restrictions can be easily satisfied. This finding enriches the classical spectral theory and offers a fresh tool to facilitate further investigation into the Sturm–Liouville eigenvalue problem, such as the asymptotic distribution formula for eigenvalues with non-constant weights.

The arrangement of the present paper is as follows. In Section 2, we introduce the Pr$\ddot{\mathrm{u}}$fer transformation and some preliminary knowledge of eigenvalues and eigenfunctions. The proofs of Theorems 3 and 4 are given in Section 3. By using several auxiliary lemmas, the main result of Theorem 5 is proved in Section 4.

2. Preliminary Knowledge

This section introduces the Pr$\ddot{\mathrm{u}}$fer transformation and some preliminary knowledge of eigenvalues and eigenfunctions of Sturm–Liouville problems.

2.1. Pr$\ddot{u}$fer Transformation

The Pr$\ddot{u}$fer transformation is widely used in studying the distribution of eigenvalues and the oscillation theory of Sturm–Liouville problems; for more details, the reader may refer to [24,25,26].

Consider (1) and (2), let

$$y(x;\lambda )=\rho (x;\lambda )sin\theta (x;\lambda ),y^{{}^{\prime }}(x;\lambda )=\sqrt{\lambda }\rho (x;\lambda )cos\theta (x;\lambda ),$$

(5)

where $\lambda \ge 0,\rho >0,\theta \in \mathbb{R}$. Then $\rho ,\theta$ satisfies the following differential equations

$${\theta }^{\prime }(x;\lambda )=\sqrt{\lambda }({cos}^2\theta (x;\lambda )+w\left(x\right){sin}^2\theta (x;\lambda ))-\frac{1}{\sqrt{\lambda }}q\left(x\right){sin}^2\theta (x;\lambda ),$$

(6)

$${\rho }^{\prime }(x,\lambda )=\frac{\sqrt{\lambda }}{2}(1-w\left(x\right)+\frac{q\left(x\right)}{\lambda })\rho (x;\lambda ){sin}^2\theta (x;\lambda ).$$

(7)

Equations (6) and (7) are usually called the Pr$\ddot{u}$fer equations. The above transformation (5) transforms Equation (1) into the one-order differential system of $\rho ,\theta$.

2.2. Continuity and Differentiability of Eigenvalues and Eigenfunctions

Let $\varphi (x;{\lambda }_n)$ be the eigenfunction of (1) and (2) corresponding to the nth eigenvalue ${\lambda }_n$, such that

$${\int }_0^{1}w\left(x\right){|\varphi (x;{\lambda }_n)|}^2\mathrm{d}x=1.$$

(8)

Note that, after fixing the sign of the initial value, such eigenfunction $\varphi (x;{\lambda }_n)$ is unique. Throughout this paper, an eigenfunction $\varphi (x;{\lambda }_n)$ is normalized if $\varphi (x;{\lambda }_n)$ satisfies (8) and $\varphi (0;{\lambda }_n)>0$, or ${\varphi }^{\prime }(0;{\lambda }_n)>0$ if $\varphi (0;{\lambda }_n)=0$.

The following theorem gives the differentiability of eigenvalues with respect to the weighted function

Theorem 1. 

Consider the eigenvalue problems (1) and (2), for any integer $n\ge 1$ and $w\in L^1([0,1],\mathbb{R})$, there exists a neighborhood $U\left(w\right)$ with respect to w, such that the map

$${\lambda }_n:U\to \mathbb{R},w\in U\mapsto {\lambda }_n\left(q\right)\in R$$

is differentiable at w. When q is fixed, the $Fr\acute{e}chet$ derivative of ${\lambda }_n$ with respect to w satisfies

$$\frac{\partial {\lambda }_n\left(w\right)}{\partial w}\left(h\right)=-{\lambda }_n\left(w\right){\int }_0^1{\varphi }_n^2(x,w)h\left(x\right)\mathrm{d}x,h\in L^1([0,1],\mathbb{R}),$$

(9)

where ${\varphi }_n$ is the normalized eigenfunction of problems (1) and (2).

The proof can be found in [27]. Theorem 1 can be viewed as a special case of a well-known theorem [1] (Theorem 4.2); for more eigenvalues of differentiability, the reader may refer to [27]. The following theorem shows the continuity of eigenvalues, eigenfunctions, and the Pr$\ddot{\mathrm{u}}$fer argument $\theta$ with respect to $w\left(x\right)$.

Theorem 2. 

Assume that $\lambda \left(w_0\right)$ is an isolated eigenvalue of problems (1) and (2). Then, given any $ϵ>0$, there exists a $\delta >0$, such that if w satisfies

$${\int }_0^1|w-w_0|<\delta$$

(10)

then there is an eigenvalue $\lambda \left(w\right)$, eigenfunction $\varphi (x,w)$, and Pr$\ddot{\mathrm{u}}$fer argument θ satisfying

$$|\lambda \left(w\right)-\lambda \left(w_0\right)|<ϵ,|\varphi (x,w)-\varphi (x,w_0)|<ϵ,|\theta (x,w)-\theta (x,w_0)|<ϵ$$

(11)

for all $x\in [0,1]$, where θ is defined as in (5).

The proof can be found in [1], Theorem 3.1.

2.3. Definition of Uniformly Local Lipschitz Continuity

Definition 1. 

Let Ω be a subset of $L^1[0,1]$. The eigenvalue sequence $\{{\lambda }_n\left(w\right):n\ge 1\}$ of problems (1) and (2) is said to be uniformly locally Lipschitz continuous with respect to weight functions in $L^1[0,1]$, in the following sense:

for every $w_1,w_2\in \Omega$, there exists $M=M(w_1,w_2)>0$, such that

$$|ln|{\lambda }_n\left(w_2\right)|-ln|{\lambda }_n\left(w_1\right)||\le M(w_1,w_2){\parallel w_2-w_1\parallel }_1,n\ge 1$$

(12)

whenever ${\lambda }_n\left(w_1\right){\lambda }_n\left(w_2\right)\ne 0$.

Remark 1. 

For fixed q, the zero eigenvalue of problems (1) and (2) does not depend on the weight function, precisely, for some $n\ge 1,$ if ${\lambda }_n\left(w_1\right)=0$, then ${\lambda }_n\left(w_2\right)=0$.

Remark 2. 

Let N be the integer, such that ${\lambda }_n>0$ if $n>N$, we remark that N is independent of w. It can be proved by the min–max principle (Rayleigh–Rize principle); the reader can refer to [28] (Proposition 2.6).

3. The Uniform Boundedness of Normalized Eigenfunctions

In this section, we study the uniform boundedness of normalized eigenfunctions with respect to the weight w. Throughout this section, ${\lambda }_n$ and ${\varphi }_n$ are always the n-th eigenvalue and normalized eigenfunction of (1) and (2), respectively.

In order to simplify our proof process in this section, we introduce the following lemma.

Lemma 1. 

Consider the eigenvalue problems (1) and (2). If the normalized eigenfunctions are bounded uniformly when $q\left(x\right)=0$, then for each $q\left(x\right)\in L^1[0,1]$, the corresponding normalized eigenfunctions of problems (1) and (2) are bounded uniformly, too.

Proof. 

Since the proof of Lemma 1 is the same as the proof in [23] (Proposition 3.5), we omit it here. □

We take $q\left(x\right)\equiv 0$ in the following discussion. The next lemma gives a dominant of ${\int }_0^1{{\varphi }_n^{{}^{\prime }}(x;{\lambda }_n)}^2\mathrm{d}x$ by the boundary conditions.

Lemma 2. 

Assume that condition (3) holds and w has a positive lower bound $w_0>0$. Then for $n\ge max\{2,N\}$

$${\int }_0^1{{\varphi }_n^{{}^{\prime }}(x;{\lambda }_n)}^2\mathrm{d}x\le 2{\lambda }_n+\frac{8}{w_0}{\left(|{cot}^{*}{\theta }_1|+|{cot}^{*}{\theta }_2|\right)}^2.$$

(13)

where ${cot}^{*}\theta =0$ for $\theta =0$; ${cot}^{*}\theta =cot\theta$ for $\theta \ne 0$.

Proof. 

The normalized eigenfunction ${\varphi }_n$ of problems (1) and (2) satisfies

$$-{\varphi }_n^{″}={\lambda }_{n}w{\varphi }_n,x\in [0,1]{\mathbf{B}}_{\mathbf{1}}{\varphi }_n=0,{\mathbf{B}}_{\mathbf{2}}{\varphi }_n=0.$$

(14)

Multiplying both sides of Equation (14) by ${\varphi }_n$ and integrating by parts on the interval $[0,1]$ gives

$${\varphi }_n^{\prime }\left(0\right){\varphi }_n\left(0\right)-{\varphi }_n^{\prime }\left(1\right){\varphi }_n\left(1\right)+{\int }_0^1{{\varphi }_n^{{}^{\prime }}}^2\mathrm{d}x={\lambda }_n{\int }_0^{1}w{\varphi }_n^2\mathrm{d}x.$$

(15)

From the boundary condition ${\mathbf{B}}_{\mathbf{1}}{\varphi }_n={\mathbf{B}}_{\mathbf{2}}{\varphi }_n=0$ and ${\int }_0^{1}w\left(x\right){\varphi }_n^2\left(x\right)\mathrm{d}x=1$, we have

$${\varphi }_n^2\left(0\right){cot}^{*}{\theta }_1-{\varphi }_n^2\left(1\right){cot}^{*}{\theta }_2+{\int }_0^1{{\varphi }_n^{{}^{\prime }}}^2\mathrm{d}x={\lambda }_n.$$

(16)

Since $n\ge 2$, ${\varphi }_n\left(x\right)$ has zero in $(0,1)$, say ${\varphi }_n\left(x_0\right)=0$. Then for $x\in [0,1]$,

$${\varphi }_n^2\left(x\right)={\varphi }_n^2\left(x\right)-{\varphi }_n^2\left(x_0\right)=2{\int }_{x_0}^x{\varphi }_n\left(\xi \right){\varphi }_n^{\prime }\left(\xi \right)d\xi ,$$

and, hence, we arrive at

$$|{\varphi }_n{\left(x\right)|}^2=2\left|{\int }_{x_0}^x{\varphi }_n\left(\xi \right){\varphi }_n^{\prime }\left(\xi \right)d\xi \right|\le 2{\int }_0^1|{\varphi }_n\left(\xi \right){\varphi }_n^{\prime }\left(\xi \right)|d\xi \le 2\epsilon \parallel {\varphi }_n^{\prime }{\parallel }_2^2+\frac{2}{\epsilon }{\parallel {\varphi }_n\parallel }_2^2$$

for arbitrary $\epsilon >0$. Since $w\left(x\right)\ge w_0>0$ and ${\int }_0^{1}w{\varphi }_n^2=1$, we have

$$\parallel {\varphi }_n{\parallel }_2^2={\int }_0^1{\varphi }_n^2\le \frac{1}{w_0}{\int }_0^{1}w{\varphi }_n^2=\frac{1}{w_0}.$$

The above inequalities give that

$$|{\varphi }_n{\left(x\right)|}^2\le 2\epsilon {\int }_0^1{|{\varphi }_n^{{}^{\prime }}|}^2+\frac{2}{\epsilon w_0}.$$

(17)

Putting (17) into (16), we obtain

$$\parallel {\varphi }_n^{{}^{\prime }}{\parallel }_2^2\le {\lambda }_n+(|{cot}^{*}{\theta }_1|+|{cot}^{*}{\theta }_2|)(2\epsilon {\int }_0^1|{\varphi }_n^{{}^{\prime }}{|}^2+\frac{2}{\epsilon w_0}).$$

(18)

Thus, $\epsilon$ satisfies $(|{cot}^{*}{\theta }_1|+|{cot}^{*}{\theta }_2|)2\epsilon =1/2$. One sees that (13) holds. Then the proof is complete. □

The following is the key lemma for the proof of Theorem 3 in this section.

Lemma 3. 

Let θ be defined as (5) and assume $w\left(x\right)$ is a step function with a positive lower bound $w_0>0$. Then, for $x\in [0,1]$ and $\lambda >0$,

$$\sqrt{\lambda }\left|{\int }_0^x(1-w\left(t\right))sin2\theta (t;\lambda )\mathrm{d}t\right|\le \frac{\underset{0}{\overset{1}{V}}\left(w\right)}{w_0}+{2\parallel w\parallel }_{\infty }+2|ln{w}_0|.$$

(19)

where ${\parallel w\parallel }_{\infty }=max\{w\left(x\right):x\in [0,1]\}$.

Proof. 

Set $w\left(x\right)={\alpha }_j>0$ for $x\in \Delta j=(x_{j-1},x_j]$, $j=1,2\dots m$, where, $0=x_0<x_1<x_2<\dots <x_m=1$. Then by the assumption,

$$min\{{\alpha }_j:1\le j\le m\}=w_0>0.$$

(20)

For $\lambda >0$, define

$$G(x;\lambda )=\sqrt{\lambda }{\int }_0^x(1-w\left(t\right))sin2\theta (t;\lambda )\mathrm{d}t.$$

(21)

It follows from (6) that (the case for $q\left(x\right)\equiv 0$)

$${\theta }^{{}^{\prime }}=\sqrt{\lambda }({cos}^2\theta +w{sin}^2\theta ).$$

Note that $w\left(x\right)={\alpha }_j$ for $x\in (x_{j-1},x_j]$. Set ${\beta }_j=1-{\alpha }_j$. Then,

$$\begin{gathered} \sqrt{\lambda }{\int }_{x_{j-1}}^x(1-w)sin2\theta \mathrm{d}t={\int }_{x_{j-1}}^x\frac{{\beta }_j\mathrm{d}{sin}^2\theta }{{cos}^2\theta +{\alpha }_j{sin}^2\theta }={\int }_{x_{j-1}}^x\frac{{\beta }_j\mathrm{d}{sin}^2\theta }{1+({\alpha }_j-1){sin}^2\theta } \\ =-{\int }_{x_{j-1}}^x\frac{\mathrm{d}({\alpha }_j-1){sin}^2\theta +1}{({\alpha }_j-1){sin}^2\theta +1}=ln\frac{{cos}^2\theta \left(x_{j-1}\right)+{\alpha }_j{sin}^2\theta \left(x_{j-1}\right)}{{cos}^2\theta \left(x\right)+{\alpha }_j{sin}^2\theta \left(x\right)}, \end{gathered}$$

(22)

and hence,

$$\begin{array}{cc}\hfill & G(x;\lambda )=\sqrt{\lambda }\left[\sum _{j=1}^{k-1}{\int }_{x_{j-1}}^{x_j}(1-w)sin2\theta \mathrm{d}t+{\int }_{x_{k-1}}^x(1-w)sin2\theta \mathrm{d}t\right]\hfill \\ \hfill & =ln({cos}^2\theta \left(0\right)+{\alpha }_1{sin}^2\theta \left(0\right))+\sum _{j=2}^{k}ln\frac{{cos}^2\theta \left(x_{j-1}\right)+{\alpha }_j{sin}^2\theta \left(x_{j-1}\right)}{{cos}^2\theta \left(x_{j-1}\right)+{\alpha }_{j-1}{sin}^2\theta \left(x_{j-1}\right)}\hfill \\ \hfill & -ln({cos}^2\theta \left(x\right)+{\alpha }_k{sin}^2\theta \left(x\right)).\hfill \end{array}$$

(23)

Since for $\alpha >0$, it holds that

$$min\{1,\alpha \}\le {cos}^2\theta \left(x\right)+\alpha {sin}^2\theta \left(x\right)\le 1+\alpha ,$$

and, hence,

$$\left|ln\frac{{cos}^2\theta \left(0\right)+{\alpha }_1{sin}^2\theta \left(0\right)}{{cos}^2\theta \left(x\right)+{\alpha }_k{sin}^2\theta \left(x\right)}\right|\le {2ln[1+\parallel w\parallel }_{\infty }]+2|ln{w}_0{|\le 2\parallel w\parallel }_{\infty }+2|ln{w}_0|.$$

(24)

In addition, one sees that if $\frac{{\alpha }_j}{{\alpha }_{j-1}}\ge 1$, then

$$1\le \frac{{cos}^2\theta \left(x_{j-1}\right)+{\alpha }_j{sin}^2\theta \left(x_{j-1}\right)}{{cos}^2\theta \left(x_{j-1}\right)+{\alpha }_{j-1}{sin}^2\theta \left(x_{j-1}\right)}\le \frac{{\alpha }_j}{{\alpha }_{j-1}}.$$

(25)

If $\frac{{\alpha }_j}{{\alpha }_{j-1}}<1$, then

$$1>\frac{{cos}^2\theta \left(x_{j-1}\right)+{\alpha }_j{sin}^2\theta \left(x_{j-1}\right)}{{cos}^2\theta \left(x_{j-1}\right)+{\alpha }_{j-1}{sin}^2\theta \left(x_{j-1}\right)}>\frac{{\alpha }_j}{{\alpha }_{j-1}}.$$

(26)

As a result, inequalities (25) and (26) yield that

$$\left|\sum _{j=2}^{k}ln\frac{{cos}^2\theta \left(x_{j-1}\right)+{\alpha }_j{sin}^2\theta \left(x_{j-1}\right)}{{cos}^2\theta \left(x_{j-1}\right)+{\alpha }_{j-1}{sin}^2\theta \left(x_{j-1}\right)}\right|\le \sum _{j=2}^k\left|ln\frac{{\alpha }_j}{{\alpha }_{j-1}}\right|.$$

(27)

Since $w\left(x\right)$ is a step function, we have ${\sum }_{j=2}^m\left|{\alpha }_j-{\alpha }_{j-1}\right|\le \underset{0}{\overset{1}{V}}\left(w\right)$. Set

$$V_j=min\{{\alpha }_j,{\alpha }_{j-1}\},{\mathsf{\Lambda }}_j=max\{{\alpha }_j,{\alpha }_{j-1}\}.\frac{{\mathsf{\Lambda }}_j}{V_j}=1+{\epsilon }_j,2\le j\le m.$$

Recall that $w\left(x\right)\ge w_0,x\in [0,1]$. Hence,

$$\begin{gathered} \underset{0}{\overset{1}{V}}\left(w\right)\ge \sum _{j=2}^n\left|{\alpha }_j-{\alpha }_{j-1}\right|=\sum _{j=2}^n({\mathsf{\Lambda }}_j-V_j) \\ =\sum _{j=2}^n((1+{\epsilon }_j)V_j-V_j)=\sum _{j=2}^n{\epsilon }_j{V}_j\ge w_0\sum _{j=2}^n{\epsilon }_j, \end{gathered}$$

and, hence,

$$\sum _{j=2}^k\left|ln\frac{{\alpha }_j}{{\alpha }_{j-1}}\right|=\sum _{j=2}^{k}ln\frac{{\mathsf{\Lambda }}_j}{V_j}=\sum _{j=2}^{k}ln(1+{\epsilon }_j)\le \sum _{j=2}^n{\epsilon }_j\le \frac{\underset{0}{\overset{1}{V}}\left(w\right)}{w_0}.$$

(28)

Therefore, from (24), (27), and (28), we conclude that

$$|G(x;\lambda )|\le \frac{\underset{0}{\overset{1}{V}}\left(w\right)}{w_0}+{2\parallel w\parallel }_{\infty }++2|ln{w}_0|$$

(29)

for $x\in [0,1]$ and $\lambda >0$. This completes the proof of Lemma 3. □

The following result is a consequence of Lemma 3.

Lemma 4. 

Let θ be defined as in (5). Assume that $q\left(x\right)\equiv 0$ and $w\left(x\right)$ is a bounded variation function with a positive lower bound $w_0>0$. If ${\lambda }_n>0$, then for $x\in [0,1]$,

$$|\sqrt{{\lambda }_n}{\int }_0^x(1-w\left(t\right))sin2\theta (t;{\lambda }_n)\mathrm{d}t|\le \frac{\underset{0}{\overset{1}{V}}\left(w\right)}{w_0}+{2\parallel w\parallel }_{\infty }+2|ln{w}_0|.$$

(30)

Proof. 

Since $w\left(x\right)$ is a bounded variation function, we know that $w\left(x\right)$ is bounded and the number of discontinuous points is, at most, countable; hence, $w\left(x\right)$ is Riemann integrable. Thus, there exists a sequence of step function $\{w_m\left(x\right):m\ge 1\}$, such that $w_m\left(x\right)\to w\left(x\right)in{L}^1[0,1]$ as $m\to \infty$ and

$$w_m\left(x\right)\ge inf\{w\left(x\right):x\in [0,1]\},\parallel w_m{\parallel }_{\infty }\le {\parallel w\parallel }_{\infty },\underset{0}{\overset{1}{V}}\left(w_m\right)\le \underset{0}{\overset{1}{V}}\left(w\right).$$

(31)

Let ${\lambda }_{nm}$ be the nth eigenvalue of problems (1) and (2) with w replaced by $w_m\left(x\right)$. By Lemma 3 and (31), for ${\lambda }_{nm}>0$,

$$|\sqrt{{\lambda }_{nm}}{\int }_0^x(1-w_m\left(t\right))sin2\theta (t,{\lambda }_{nm})\mathrm{d}t|\le \frac{\underset{0}{\overset{1}{V}}\left(w\right)}{w_0}+{2\parallel w\parallel }_{\infty }+2|ln{w}_0|.$$

(32)

Since $w_m\left(x\right)\to w\left(x\right)$ in $L^1[0,1]$, it follows from the continuous dependence of the eigenvalues and the pr$\ddot{\mathrm{u}}$fer argument $\theta$ on the weight functions that

$${\lambda }_{nm}\to {\lambda }_n,\theta (t;{\lambda }_{nm})\to \theta (t;{\lambda }_n),n\ge 1$$

as $m\to \infty$. Thus, letting $m\to \infty$ in (32), we obtain

$$\left|\sqrt{{\lambda }_n}{\int }_0^x(1-w\left(t\right))sin2\theta (t;{\lambda }_n)\mathrm{d}t\right|\le \frac{\underset{0}{\overset{1}{V}}\left(w\right)}{w_0}+{2\parallel w\parallel }_{\infty }+2|ln{w}_0|.$$

(33)

The proof is finished. □

Now, we prove the uniform boundedness results of normalized eigenfunctions.

Theorem 3. 

Consider the eigenvalue problems (1) and (2) with $q\left(x\right)\equiv 0$. Assume that $w\left(x\right)$ is a bounded variation function with a positive lower bound $w_0>0$, then the normalized eigenfunctions $\left\{{\varphi }_n\right\}$ of problems (1) and (2) are uniformly bounded. There exists an integer $K>0$ and $M=M(w_0,\parallel w{\parallel }_{\infty },\underset{0}{\overset{1}{V}}\left(w\right))$ such that

$$|{\varphi }_n\left(x\right)|\le M(w_0{,\parallel w\parallel }_{\infty },\underset{0}{\overset{1}{V}}\left(w\right))$$

(34)

for all $n\ge K$ and $x\in [0,1]$.

Proof. 

The normalized eigenfunction ${\varphi }_n(x;{\lambda }_n)$ of problems (1) and (2) satisfies

$$-{\varphi }_n^{″}={\lambda }_{n}w{\varphi }_n,x\in [0,1];{\mathbf{B}}_{\mathbf{1}}{\varphi }_n=0,{\mathbf{B}}_{\mathbf{2}}{\varphi }_n=0.$$

(35)

By the pr$\ddot{\mathrm{u}}$fer transformation

$$\varphi (x;{\lambda }_n)=\rho (x;{\lambda }_n)sin\theta (x;{\lambda }_n),{\varphi }_n^{{}^{\prime }}(x;{\lambda }_n)=\sqrt{{\lambda }_n}\rho (x;{\lambda }_n)cos\theta (x;{\lambda }_n),$$

(36)

and from (7) we have $x\in [0,1]$

$$\rho (x;{\lambda }_n)=\rho (0;{\lambda }_n)e^{\frac{\sqrt{{\lambda }_n}}{2}{\int }_0^x(1-w\left(t\right))sin2\theta (t;{\lambda }_n)\mathrm{d}t}.$$

(37)

Since ${\lambda }_n\to \infty$ as $n\to \infty$, there exists an integer $K\ge N$, such that

$${\lambda }_n\ge \frac{8}{w_0}{\left(|{cot}^{*}{\theta }_1|+|{cot}^{*}{\theta }_2|\right)}^2,n\ge K,$$

(38)

and, hence, Lemma 2 yields that

$${\int }_0^1{{\varphi }_n^{{}^{\prime }}(x;{\lambda }_n)}^2\mathrm{d}x\le 3{\lambda }_n,n\ge K.$$

This, together with (36), gives

$${\int }_0^1{\rho }^2(x;{\lambda }_n){cos}^2\theta (x;{\lambda }_n)\mathrm{d}x\le 3,n\ge K.$$

(39)

For simplicity, hereafter, we denote $\theta (x;{\lambda }_n)$ by ${\theta }_n\left(x\right)$, $\rho (x;{\lambda }_n)$ by ${\rho }_n\left(x\right)$. Inequality (39) combined with

$${\int }_0^{1}w\left(x\right){\varphi }_n^2\left(x\right)\mathrm{d}x={\int }_0^{1}w{\rho }_n^2{sin}^2{\theta }_n\mathrm{d}x=1$$

implies that

$$\begin{gathered} w_0{\int }_0^1{\rho }_n^2\left(x\right)\mathrm{d}x\le {\int }_0^{1}w\left(x\right){\rho }_n^2\left(x\right)\mathrm{d}x \\ ={\int }_0^{1}w{\rho }_n^2{cos}^2{\theta }_n\mathrm{d}x+{\int }_0^{1}w{\rho }_n^2{sin}^2{\theta }_n\mathrm{d}x\le 1+3{\parallel w\parallel }_{\infty }. \end{gathered}$$

(40)

Putting (37) into (40) we have $x\in [0,1]$ and $n\ge K$,

$${\rho }_n^2\left(0\right){\int }_0^1{e}^{\sqrt{{\lambda }_n}{\int }_0^x(1-w\left(t\right))sin2{\theta }_n\left(t\right)\mathrm{d}t}\mathrm{d}x\le \frac{1+{3\parallel w\parallel }_{\infty }}{w_0}.$$

(41)

It follows from (30) in Lemma 4 that there exist positive constants $A_{\pm }$

$$A_{\pm }=exp\left(\pm \left[\underset{0}{\overset{1}{V}}\left(w\right)/w_0+{2\parallel w\parallel }_{\infty }+2|ln{w}_0|\right]\right)$$

such that for ${\lambda }_n>0$ and $x\in [0,1]$,

$$A_{-}\le exp(\sqrt{{\lambda }_n}{\int }_0^x(1-w\left(t\right))sin2{\theta }_n\left(t\right)\mathrm{d}t)\le A_{+}.$$

(42)

Combined with (41), it is apparent that, for $n\ge K$

$${\rho }_n^2\left(0\right)\le \frac{1+{3\parallel w\parallel }_{\infty }}{w_0{A}_{-}}.$$

(43)

Therefore, by (37), (42), and (43), we obtain that for $x\in [0,1]$ and $n\ge K$

$${\rho }^2(x;{\lambda }_n)\le \frac{1+{3\parallel w\parallel }_{\infty }}{w_0{A}_{-}}{A}_{+},$$

(44)

Thus, for $n\ge K$ and $x\in [0,1]$, we have

$$|{\varphi }_n(x;{\lambda }_n)|=|\rho (x;{\lambda }_n)sin\theta (x;{\lambda }_n)|\le A_{+}\sqrt{\frac{1+{3\parallel w\parallel }_{\infty }}{w_0}}.$$

Set

$$M(w_0{,\parallel w\parallel }_{\infty },\underset{0}{\overset{1}{V}}\left(w\right)):=max\left\{A_{+}\sqrt{\frac{1+{3\parallel w\parallel }_{\infty }}{w_0}},\parallel {\varphi }_1{\parallel }_{\infty },\dots ,{\parallel {\varphi }_{K-1}\parallel }_{\infty }\right\}.$$

(45)

Therefore, $|{\varphi }_n\left(x\right)|\le M(w_0{,\parallel w\parallel }_{\infty },\underset{0}{\overset{1}{V}}\left(w\right))$ for $n\ge 1$ and $x\in [0,1]$. The proof is finished. □

Theorem 3 proves the uniform boundedness of normalized eigenfunctions of Sturm–Liouville problems (1) and (2) with positive-bounded variation weight. We note that the monotonicity requirement of the weight in the corresponding result in [23] is removed. Next, we show that the restriction “positive lower bound” on the weight is not necessary. For this purpose, we discuss the case where $w\left(x\right)$ is a step function, which is allowed to be zero in some subintervals.

Theorem 4. 

Consider the eigenvalue problems (1) and (2). Assume that $w\left(x\right)$ is a step function defined by

$$w\left(x\right)={\alpha }_j,x\in [x_{j-1},x_j),0=x_0<x_1<\cdots <x_m=1,$$

where ${\alpha }_j\ge 0$ and ${\Sigma }_1^m{\alpha }_j>0$. Then

(i) 

If ${\alpha }_1{\alpha }_m>0$, then ${\left\{{\varphi }_n\right\}}_{n\ge 1}$ is uniformly bounded.

(ii) 

If ${\alpha }_1{\alpha }_m=0$, then ${\left\{{\varphi }_n\right\}}_{n\ge 1}$ is bounded uniformly $\iff 1+{cot}^{*}{\theta }_1{x}_1\ne 0$ (if ${\alpha }_1=0$) and $1+{cot}^{*}{\theta }_2{x}_m\ne 0$ (if ${\alpha }_m=0$).

Proof. 

By Lemma 1, we only need to prove Theorem 4 for the case $q\left(x\right)=0$.

(i)

First, we show that ${\left\{{\varphi }_n\right\}}_{n\ge 1}$ is bounded uniformly on every subinterval $I_j=[x_{j-1},x_j)$, such that ${\alpha }_j>0$, $1\le j\le m$.

Let $I_j$ be such a subinterval. Recall that ${\lambda }_n>0$ for $n>N$. Since $w\left(x\right)\equiv {\alpha }_j>0$, $x\in I_j$ and the normalized eigenfunction ${\varphi }_n$ satisfies $-{\varphi }_n^{″}={\lambda }_{n}w{\varphi }_n$, we have

$${\varphi }_n\left(x\right)=c_{nj}sin(\sqrt{{\lambda }_n}x+{\eta }_{nj}),n>N,x\in I_j.$$

(46)

Since ${\varphi }_n$ is normalized, one sees that

$$1={\int }_0^{1}w{\varphi }_n^2\ge {\alpha }_j{c}_{nj}^2{\int }_{I_j}{sin}^2(\sqrt{{\lambda }_n}x+{\eta }_{nj})dx\ge {\alpha }_j{c}_{nj}^2{\Delta }_j/4$$

(47)

for sufficiently large ${\lambda }_n$ and, hence, $|c_{nj}|\le 2/\left(\sqrt{{\alpha }_j{\Delta }_j}\right)$, where ${\Delta }_j=|I_j|$. As a result,

$$|{\varphi }_n\left(x\right)|\le 2/\left(\sqrt{{\alpha }_j{\Delta }_j}\right),n\ge N_1,x\in I_j$$

(48)

for sufficiently large $N_1>N$. Set

$$M_1=max\left\{\frac{2}{\sqrt{{\alpha }_j{\Delta }_j}}:{\alpha }_j>0,1\le j\le m\right\}$$

(49)

$$M=max\left\{M_1,\parallel {\varphi }_1{\parallel }_{\infty },\dots ,{\parallel {\varphi }_{N_1-1}\parallel }_{\infty }\right\}$$

(50)

Therefore, $|{\varphi }_n\left(x\right)|\le M$ for $n\ge 1$ and $x\in I_j$, such that ${\alpha }_j>0$, $1\le j\le m$.

(ii)

Let $I_k$ be a subinterval of $[0,1]$, such that ${\alpha }_k=0$ and $k\ne 1,m$. Since $-{\varphi }_n^{″}\left(x\right)=0$, $x\in I_k$, it is easy to see

$$|{\varphi }_n(x,{\lambda }_n)|\le max\{|{\varphi }_n(x_{k-1},{\lambda }_n)|,|{\varphi }_n(x_k,{\lambda }_n)|\}\le M,n\ge 1.$$

For the case where $k=1$ or $k=m$, we give only the proof for $k=1$. The proof for the case $k=m$ is in the same way. From the boundary conditions, one sees that

$${\varphi }_n^{\prime }(0,{\lambda }_n)={\varphi }_n(0,{\lambda }_n){cot}^{*}{\theta }_1.$$

Hence, it follows from $-{\varphi }_n^{″}=0$ on $I_1$ that for $x\in I_1$,

$${\varphi }_n(x,{\lambda }_n)={\varphi }_n(0,{\lambda }_n)+{\varphi }_n^{\prime }(0,{\lambda }_n)x=(1+{cot}^{*}{\theta }_{1}x){\varphi }_n(0,{\lambda }_n)$$

(51)

If $1+{cot}^{*}{\theta }_1{x}_1\ne 0$, then

$$|{\varphi }_n(0,{\lambda }_n)|\le \frac{{\varphi }_n(x_1,{\lambda }_n)}{|1+{cot}^{*}\theta x_1|}\le \frac{M}{|1+{cot}^{*}\theta x_1|},$$

and, hence, for $x\in I_1$,

$$|{\varphi }_n(x,{\lambda }_n)|\le max\{|{\varphi }_n(0,{\lambda }_n)|,|{\varphi }_n(x_1,{\lambda }_n)|\}\le max\{\frac{M}{|1+{cot}^{*}\theta x_1|},M\}.$$

If $1+{cot}^{*}{\theta }_1{x}_1=0$, then (51) gives ${\varphi }_n(x_1,{\lambda }_n)=0$ for $n\ge 1$. We claim that ${\varphi }_n^{\prime }(x_1,{\lambda }_n)\to \infty$ as $n\to +\infty$.

On the contrary, and without loss of generality, we suppose that $|{\varphi }_n^{\prime }(x_1,{\lambda }_n)|\le B$ for some constant $B>0$ and all $n\ge 1$. Since $x\in I_2$,

$${\varphi }_n\left(x\right)=c_{n2}sin\left(\sqrt{{\lambda }_n}(x-x_1)\right),$$

(52)

we have $|c_{n2}\sqrt{{\lambda }_n}|=|{\varphi }_n^{\prime }(x_1,{\lambda }_n)|\le B$. If ${\alpha }_3=0$, then ${\varphi }_n^{\prime }(x_3,{\lambda }_n)={\varphi }_n^{\prime }(x_2,{\lambda }_n)$, which means

$$|{\varphi }_n^{\prime }(x_3,{\lambda }_n)|=|{\varphi }_n^{\prime }(x_2,{\lambda }_n)|=|c_{n2}\sqrt{{\lambda }_n}cos\left(\sqrt{{\lambda }_n}(x-x_1)\right)|\le B.$$

If ${\alpha }_3>0$, then ${\varphi }_n\left(x\right)=c_{n3}sin(\sqrt{{\lambda }_n}x+{\eta }_{n3})$ for $x\in I_3=[x_2,x_3]$. By the continuity and differentiability of ${\varphi }_n$ at $x=x_2$, it holds that

$$\begin{array}{l}{c}_{n2}sin\left[\sqrt{{\lambda }_n}(x_2-x_1)\right]=c_{n3}sin(\sqrt{{\lambda }_n}{x}_2+{\eta }_{n3}),\\ c_{n2}\sqrt{{\lambda }_n}cos\left[\sqrt{{\lambda }_n}(x_2-x_1)\right]=c_{n3}\sqrt{{\lambda }_n}cos(\sqrt{{\lambda }_n}{x}_2+{\eta }_{n3}),\end{array}$$

which implies that $|c_{n2}|=|c_{n3}|$ and, hence, $|c_{n3}\sqrt{{\lambda }_n}|\le B$. Inductively, we can prove that

$$|c_{nj}\sqrt{{\lambda }_n}|\le B,j\ge 2,{\alpha }_j>0.$$

Clearly,

$$1={\int }_0^{1}w\left(x\right){\varphi }_n^2\left(x\right)\mathrm{d}x\le \sum _{j=1}^m{\alpha }_j{|c_{nj}|}^2\to 0,n\to \infty ,$$

which is a contradiction. Therefore, ${\varphi }_n(0,{\lambda }_n)={\varphi }_n^{\prime }(x_1,{\lambda }_n)(0-x_1)\to +\infty$ as $n\to +\infty$. Hence ${\left\{{\varphi }_n(x,{\lambda }_n)\right\}}_{n\ge 1}$ is not uniformly bounded for $n\ge 1$ when $1+{cot}^{*}{\theta }_1{x}_1=0$. The proof of Theorem 4 is complete. □

Remark 3. 

In order to ensure the uniform boundedness of the normalized eigenfunction sequence ${\left\{{\varphi }_n\left(x\right)\right\}}_{n\ge 1}$, the restriction on $w\left(x\right)$ in [23] can be modified so that $w\left(x\right)$ is a bounded variation function with a positive bound from below or $w\left(x\right)$ satisfies the relevant conditions in Theorem 4.

On the other hand, although the restriction “positive lower bound” on the weight is not necessary for some cases, the additional restriction on boundary conditions is required for general result.

Here, we give an example of the case that, even if $infw\left(x\right)$ is not positive, conditions (i) and (ii) of Theorem 4 also ensure the uniform boundedness of normalized eigenfunctions.

Example 1. 

Consider the Sturm–Liouville problem

$$-y^{″}\left(x\right)=\lambda w\left(x\right)y\left(x\right),x\in [0,3]$$

(53)

with the Dirichlet boundary conditions

$$y\left(0\right)=0,y\left(3\right)=0,$$

(54)

where $x\in [0,1)\cup (2,3]$, $w\left(x\right)=1$, and when $x\in [1,2]$, $w\left(x\right)=0$. It is apparent that $w\left(x\right)$ satisfies condition (i) of Theorem 4. Through calculating, the normalized eigenfunction ${\psi }_n(x;{\lambda }_n)$ of (53) satisfies the equation

$${\psi }_n(x;{\lambda }_n)=\left\{\begin{array}{c}\hfill sin\sqrt{{\lambda }_n}x,x\in [0,1).\\ \hfill \sqrt{{\lambda }_n}cos\sqrt{{\lambda }_n}x+(sin\sqrt{{\lambda }_n}-\sqrt{{\lambda }_n}cos\sqrt{{\lambda }_n}),x\in [1,2].\\ \hfill sin\sqrt{{\lambda }_n}(x-3),x\in (2,3].\end{array}\right.$$

${\lambda }_n$ is an eigenvalue that satisfies the equation $tan\sqrt{{\lambda }_n}=-\frac{1}{2}\sqrt{{\lambda }_n}$.

It is apparent that ${\left\{{\psi }_n(x,{\lambda }_n)\right\}}_{n\ge 1}$ is bounded uniformly on $[0,3]$.

Next, we will provide an example to demonstrate that the normalized eigenfunction may not be uniformly bounded. This illustrates that the normalization of eigenfunctions to have unit norm does not necessarily ensure their boundedness. Such examples can provide insight into the behavior of eigenfunctions in certain scenarios.

Example 2. 

Consider the Sturm–Liouville problem

$$-y^{″}\left(x\right)=\lambda w\left(x\right)y\left(x\right),x\in [0,2]$$

(55)

with the self-adjoint separated boundary conditions

$$y^{\prime }\left(0\right)+y\left(0\right)=0,y\left(2\right)=0,$$

(56)

where $w\left(x\right)=0,x\in [0,1)$ and $w\left(x\right)=1$, $x\in [1,2]$. Calculations show that the normalized eigenfunction

$${\varphi }_n(x;{\lambda }_n)=\left\{\begin{array}{c}\hfill \sqrt{2}n\pi (x-1),x\in [0,1).\\ \hfill \sqrt{2}sinn\pi (x-1),x\in [1,2].\end{array}\right.$$

Clearly, ${\left\{{\varphi }_n(x;{\lambda }_n)\right\}}_{n\ge 1}$ is not uniformly bounded on $[0,2]$. In turn, our results were verified.

4. The Uniform Local Lipschitz Continuity of Eigenvalues with Respect to the Weights

With the help of the results in previous sections, we study the uniform local Lipschitz continuity of eigenvalues of Sturm–Liouville problems with respect to the weight functions in this section.

Theorem 5. 

The eigenvalue sequence $\{{\lambda }_n\left(w\right):n\ge 1\}$ of problems (1) and (2) is uniformly locally Lipschitz continuous with respect to weight functions in $\Omega \subset L^1$, where Ω is the subset of $L^1[0,1]$ such that every element w of Ω is a bounded variation function with a positive lower bound.

Proof. 

Set ${\breve{w}}_j=inf\{w_j\left(x\right):x\in [0,1]\}$, $j=1,2$ and

$$w_t\left(x\right)=w_1\left(x\right)+t(w_2\left(x\right)-w_1\left(x\right)),t\in [0,1].$$

Then $\underset{0}{\overset{1}{V}}\left(w_t\right)\le \underset{0}{\overset{1}{V}}\left(w_1\right)+\underset{0}{\overset{1}{V}}\left(w_2\right):=V\left(w\right)$ and

$$inf\{w_t\left(x\right):x\in [0,1]\}\ge min\{{\breve{w}}_1,{\breve{w}}_2\}:=\breve{w},\parallel w_t{\parallel }_{\infty }\le max\{\parallel w_1{\parallel }_{\infty },\parallel w_2{\parallel }_{\infty }\}:=\widehat{w}.$$

Let ${\lambda }_n\left(\widehat{w}\right)$ be the n-th eigenvalue of

$$-y^{″}={\lambda }_n\widehat{w}y,x\in [0,1];{\mathbf{B}}_{\mathbf{1}}y=0,{\mathbf{B}}_{\mathbf{2}}y=0.$$

(57)

Clearly, ${\lambda }_n\left(w_t\right)\ge {\lambda }_n\left(\widehat{w}\right)\to \infty$ as $n\to \infty$ by the monotonicity of eigenvalues with respect to weights for $t\in [0,1]$; hence, there exists $K>0$, which is independent of t, such that

$${\lambda }_n\left(w_t\right)>\frac{8}{\breve{w}}{\left(|{cot}^{*}{\theta }_1|+|{cot}^{*}{\theta }_2|\right)}^2,n\ge K,t\in [0,1].$$

With the same proof from (38) to (44) in Theorem 3, we find that

$$|{\varphi }_n(x;{\lambda }_n\left(w_t\right))|\le A_{+}\sqrt{\frac{1+3\widehat{w}}{\breve{w}}}:=M_1,n\ge K,$$

where $A_{+}=exp\left(\left[V\left(w\right)/\breve{w}+2\widehat{w}\right]\right)$. In addition, for $1<n\le K$, it follows from the continuous dependence of eigenvalues and eigenfunctions on the weights that

$$M_2:=max\left\{\parallel {\varphi }_1{\parallel }_{\infty }\left(t\right)\dots ,{\parallel {\varphi }_K\parallel }_{\infty }\left(t\right)\right\}<\infty ,t\in [0,1],$$

(58)

where $\parallel {\varphi }_n{\parallel }_{\infty }\left(t\right)=max\{|{\varphi }_n(x,\lambda \left(w_t\right))|:x\in [0,1]\}$. Therefore,

$$|{\varphi }_n(x,{\lambda }_n\left(w_t\right))|\le max\{M_1,M_2\}:=M(w_1,w_2).$$

(59)

It follows from Theorem 1 that

$$\frac{\partial {\lambda }_n\left(w\right)}{\partial w}\left(h\right)=-{\lambda }_n\left(w\right){\int }_0^1{\varphi }_n^2(x,w)h\left(x\right)\mathrm{d}x,h\in L^1([0,1],\mathbb{R}),$$

Recall the definition of N in Remark 2, we know that for $n\ge N$,

$$\frac{\partial ln{\lambda }_n\left(w\right)}{\partial w}\left(h\right)=\frac{\partial ln{\lambda }_n\left(w\right)}{\partial {\lambda }_n\left(w\right)}·\frac{\partial {\lambda }_n\left(w\right)}{\partial w}\left(h\right)=-{\int }_0^1{\varphi }_n^2(x,w)h\left(x\right)\mathrm{d}x.$$

(60)

As a result, from (59), (60), and Remark 1, for all $n\ge 1$ (except for, at most, one integer $N_0$), we have

$$\begin{array}{ll}& \left|ln|{\lambda }_n\left(w_2\right)|-ln|{\lambda }_n\left(w_1\right)|\right|=\left|ln{\lambda }_n\left(w_2\right)-ln{\lambda }_n\left(w_1\right)\right|\\ =& \left|{\int }_0^1\frac{\partial ln{\lambda }_n\left(w_t\right)}{\partial t}\mathrm{d}t\right|=\left|{\int }_0^1\frac{\partial ln{\lambda }_n\left(w_t\right)}{\partial w_t}·\frac{\mathrm{d}\left(w_t\right)}{\mathrm{d}t}\mathrm{d}t\right|\\ =& \left|{\int }_0^1{\int }_0^1{\varphi }_n^2(x,w_t)(w_2\left(x\right)-w_1\left(x\right))\mathrm{d}x\mathrm{d}t\right|\le M(w_1,w_2){\parallel w_2-w_1\parallel }_1.\end{array}$$

(61)

The proof is finished. □

5. Conclusions

In this work, we obtain the uniform boundedness of normalized eigenfunctions of the Sturm–Liouville problem and investigate the uniform local Lipschitz continuity of eigenvalues with respect to the weighted function. These properties are expected to be crucial for future research, particularly in the study of the asymptotic distribution of eigenvalues with non-constant weights. By using these results, we will be able to explore the behavior of eigenvalues as they approach infinity and gain a deeper understanding of the underlying system.