Uniform local Lipschitz continuity of eigenvalues with respect to the potential in $L^1[a,b]$

TL;DR

This paper proves that the eigenvalues of a Sturm-Liouville problem vary in a uniformly Lipschitz continuous manner with respect to the potential function in the $L^1$ space, under certain conditions.

Contribution

It establishes the uniform Lipschitz continuity of eigenvalues with respect to the potential in $L^1$, extending previous results to a broader class of problems.

Findings

Eigenvalues are uniformly Lipschitz continuous in $L^1$ on bounded sets.

The result applies to Sturm-Liouville problems with certain monotonic weights.

Continuity holds uniformly across the eigenvalue sequence.

Abstract

The present paper shows that the eigenvalue sequence $\{\lambda_n(q)\}_{n\geqslant 1}$ of regular Sturm-Liouville eigenvalue problem with certain monotonic weights is uniformly Lipschitz continuous with respect to the potential $q$ on any bounded subset of $L^1([a,b],\mathbb{R})$.

Equations

$$-(p(x)y(x{)}^{\prime }{)}^{\prime }+q(x)y(x)=\lambda \omega (x)y(x)\text{ on }[a,b]$$

$$y(a)\cos \alpha +(p{y}^{\prime })(a)\sin \alpha =0,y(b)\cos \beta +(p{y}^{\prime })(b)\sin \beta =0,$$

$$\frac{1}{p},q,\omega \in L^1([a,b],\mathbb{R}),p,\omega >0a.e.\text{ on }[a,b].$$

$$-\infty <{\lambda }_1(q)<{\lambda }_2(q)<\cdots <{\lambda }_n(q)<\cdots ,$$

$${\lambda }_n(q)\to \infty \text{ as }n\to \infty .$$

$$\left|{\lambda }_n(q_1)-{\lambda }_n(q_2)\right|⩽C(\Omega ){\Vert q_1-q_2\Vert }_{L^1}$$

$$s={\int }_a^x\frac{1}{p(t)}dt:=\mathcal{L}(x),\tilde{y}(s)=y({\mathcal{L}}^{-1}(s)),$$

$$-{\tilde{y}}^{\prime \prime }(s)+\tilde{q}(s)\tilde{y}(s)=\lambda \tilde{\omega }(s)\tilde{y}(s)\text{ on }[0,c],c={\int }_a^b\frac{1}{p(t)}dt,$$

$$\tilde{y}(0)\cos \alpha +{\tilde{y}}^{\prime }(0)\sin \alpha =0,\tilde{y}(c)\cos \beta +{\tilde{y}}^{\prime }(c)\sin \beta =0,$$

$$-y^{\prime \prime }(x)+q(x)y(x)=\lambda \omega (x)y(x)\text{ on }[0,1],$$

$$y(0)\cos \alpha +y^{\prime }(0)\sin \alpha =0,y(1)\cos \beta +y^{\prime }(1)\sin \beta =0,$$

$$\alpha ,\beta \in [0,\pi ),q,\omega \in L^1([0,1],\mathbb{R}),\omega >0a.e.\text{ on }[0,1],$$

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

$$\frac{\partial\lambda_{n}(q)}{\partial q}\Big{|}_{q=q_{0}}\cdot h=\int_{0}^{1}\varphi_{n}^{2}(x;\lambda_{n}(q_{0}))h(x)\,dx,$$

$$\rho (x;\lambda )=\sqrt{\lambda y^2(x;\lambda )+(y^{\prime }{)}^2(x;\lambda )},\theta (x;\lambda )=\arctan \frac{\sqrt{\lambda }y(x;\lambda )}{y^{\prime }(x;\lambda )},$$

$$\lambda ⩾0,\theta (0;\lambda )\in \mathbb{R},\rho (0;\lambda )>0,$$

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

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

$$-y^{\prime \prime }(x)=\lambda \omega (x)y(x)\text{ on }[0,1],y(0)=c_1,y^{\prime }(0)=c_2,$$

$${\theta }^{\prime }(x;\lambda )=\sqrt{\lambda }({\cos }^2\theta (x;\lambda )+\omega (x){\sin }^2\theta (x;\lambda ))$$

$${\rho }^{\prime }(x;\lambda )=\frac{\sqrt{\lambda }}{2}\rho (x;\lambda )(1-\omega (x))\sin 2\theta (x;\lambda ),\rho (0;\lambda )=\sqrt{\lambda c_1^2+c_2^2}\in (0,+\infty ).$$

$$y(x;\lambda )=\frac{1}{\sqrt{\lambda }}\rho (x;\lambda )\sin \theta (x;\lambda ),$$

$$\rho (x;\lambda )=\rho (0;\lambda )\cdot e^{\frac{\sqrt{\lambda }}{2}{\int }_0^x(1-\omega (t))\sin 2\theta (t;\lambda )dt}.$$

$$H(x;\lambda ):=\frac{\sqrt{\lambda }}{2}{\int }_0^{x}h(t)\sin 2\theta (t;\lambda )dt,$$

$$\underset{\lambda \to +\infty }{\lim }\theta (1;\lambda )=+\infty ,$$

$$\theta (1;\lambda )-\theta (0;\lambda )=\sqrt{\lambda }{\int }_0^1\left({\cos }^2\theta (x;\lambda )+\omega (x){\sin }^2\theta (x;\lambda )\right)dx⩾\sqrt{\lambda }{\int }_0^1\min \{\omega (x),1\}dx.$$

$${\theta }^{\prime }(x;\lambda )⩾0$$

$${\int }_0^{c}g(x)\sin 2\theta (x;\lambda )dx=O\left(\frac{1}{\sqrt{\lambda }}\right),$$

$${\int }_0^{c}g(x)\cos 2\theta (x;\lambda )dx=O\left(\frac{1}{\sqrt{\lambda }}\right),$$

$$G(c;\lambda )=\frac{\sqrt{\lambda }}{2}{\int }_0^{c}g(x)\sin 2\theta (x;\lambda )dx,c\in (0,1].$$

Full text

2000 MSC numbers: Primary 34B05, Secondary 45J05, 34L15

Keywords: Sturm-Liouville problem, eigenvalue, uniform local Lipschitz continuity

The first named author is supported by the NSF of China (Grant 11701327) and China Postdoctoral Science Foundation (Grant 2017M612252). The second named author, as the corresponding author, is supported by the NSF of China (Grant 11271229).

1. Introduction

Consider the regular Sturm-Liouville eigenvalue problem associated to the second order differential equation

[TABLE]

with the self-adjoint separated boundary conditions

[TABLE]

where $\alpha,\ \beta\in[0,\pi),$ $\lambda$ is the spectral parameter,

[TABLE]

Here $L^{1}[a,b]$ denotes the Banach space of all Lesbegue integrable, complex valued functions on the closed interval $[a,b]\subset\mathbb{R}$ equipped with the canonical $L^{1}$-norm $\left\|\cdot\right\|_{L^{1}}$. The subspace of real valued functions of $L^{1}[a,b]$ is denoted by $L^{1}([a,b],\mathbb{R})$.

Under the natural condition (1.3), the eigenvalue problem, (1.1) and (1.2), admits only countably infinite number of real eigenvalues which are isolated, bounded below and unbounded above by the spectral theory of differential operators.

Fix $p$ and $\omega$, let $\lambda_{n}(q)$ be the $n$th eigenvalue with respect to the potential function $q$. It is well known that

[TABLE]

and

[TABLE]

Moreover, $\lambda_{n}(q)$ can be viewed as a functional on $L^{1}([a,b],\mathbb{R})$ for every $n\geq 1$. It is also known that $\lambda_{n}(q)$ is continuous, and even differentiable, with respect to $q$ in $L^{1}[a,b]$ (see e.g. [9] as well as [3], [6] and [8]).

The continuity and differentiability of eigenvalues provide efficient tools in the study of properties of eigenvalues and eigenfunctions as well as in other related fields. In the recent years, Professor Meirong Zhang and his collaborators have obtained fruitful results on weak and strong continuity of eigenvalues and eigenvalue-pairs of several kinds of eigenvalue problems (see e.g. [16], [12], [2], [15], [13] and [14] as well as [4] and [5]).

The main topic of this paper is the study of a new continuity, called uniform local Lipschitz continuity, of the eigenvalue sequence $\{\lambda_{n}(q)\}_{n\geqslant 1}$ with respect to the potential function $q$ in $L^{1}([a,b],\mathbb{R})$.

Definition 1.1**.**

The eigenvalue sequence $\{\lambda_{n}(q)\}_{n\geqslant 1}$ of (1.1)-(1.2) is said to be uniformly locally Lipschitz continuous with respect to the potential $q$ in $L^{1}([a,b],\mathbb{R})$, if, for any $L^{1}$-norm bounded subset $\Omega\subset L^{1}([a,b],\mathbb{R})$, there exists a positive number $C(\Omega)$ such that

[TABLE]

for all $n\geqslant 1$, $q_{1},q_{2}\in\Omega$.

Note that $C(\Omega)$ is independent of the index $n$ of the eigenvalues $\{\lambda_{n}(q)\}_{n\geqslant 1}$, and hence this local Lipschitz continuity is uniform for all $n\geqslant 1$. This is exactly the meaning of the word “uniformly” in the definition above.

The present paper shows that, under some appropriate conditions, the eigenvalue sequence $\{\lambda_{n}(q)\}_{n\geqslant 1}$ has the desired continuity above. This result will provide a new tool or idea for the further study of Sturm-Liouville eigenvalue problem.

The paper is structured as follows. In Section 2, we present in Section 2.1 the content of the main theorem, and introduce some notations in Section 2.2 as well as recalling some known facts as preliminary which are crucial for the proof of our results. In Section 3, we conclude the proofs of some auxiliary lemmas, and further prove the main theorem.

2. The main theorem and preliminary

Throughout this paper, we denote by $\mathbb{R}$ the field of real numbers.

The symbol $L^{2}_{\omega}[a,b]$ denotes the weighted Hilbert space of all Lebesgue measurable, complex valued functions $f$ on $[a,b]$ satisfying $\int_{a}^{b}\omega|f|^{2}<\infty$ with the norm $\left\|f\right\|_{\omega}=(\int_{a}^{b}\omega|f|^{2})^{\frac{1}{2}}$ and the inner product $\langle f,g\rangle_{\omega}=\int_{a}^{b}\omega f\overline{g}$.

We denote by $L^{\infty}[a,b]$ the Banach space of all essentially bounded, complex valued functions on $[a,b]$ equipped with the canonical essential norm $\left\|\cdot\right\|_{\infty}$, and by $AC[a,b]$ the space of all absolutely continuous, complex valued functions on $[a,b]$.

2.1. The main theorem

Since $p>0$ a.e. and $1/p\in L^{1}([a,b],\mathbb{R})$, it is easily seen that, under the following transformation $\mathcal{L}$ of independent variables, called Liouville transformation (see e.g. [11, Page 2293]),

[TABLE]

the problem (1.1) and (1.2) for $y(x)$ is rewritten as the problem for $\tilde{y}(s)$ in the form

[TABLE]

[TABLE]

where $\alpha,\ \beta\in[0,\pi),$ $\tilde{q}(s)=p(\mathcal{L}^{-1}(s))q(\mathcal{L}^{-1}(s))$ and $\tilde{\omega}(s)=p(\mathcal{L}^{-1}(s))\omega(\mathcal{L}^{-1}(s))$.

It is not difficult to check that $\tilde{q}$, $\tilde{w}$ and $\tilde{y}$ satisfy the corresponding condition (1.3) with $[a,b]$ replaced by $[0,c]$. More importantly, the eigenvalues of (1.1)-(1.2) are the same as those of (2.2)-(2.3).

Furthermore, for any common eigenvalue $\lambda$ of both (1.1)-(1.2) and (2.2)-(2.3), denote by $E_{\lambda}$ and $\tilde{E}_{\lambda}$ the spaces of eigenfunctions associated to $\lambda$, respectively. Then the map $y(x)\mapsto\tilde{y}(s)$ sets up an isometry from $E_{\lambda}\subset L^{2}_{\omega}[a,b]$ onto $\tilde{E}_{\lambda}\subset L^{2}_{\tilde{\omega}}[0,c]$, and $\tilde{y}(s)$ on $[0,c]$ has the same range as that of $y(x)$ on $[a,b]$.

Hence, for simplicity, in the following theorem, we consider the equation (1.1) for the case $p\equiv 1$ on the unit interval $[0,1]$, i.e., the eigenvalue problem

[TABLE]

[TABLE]

where

[TABLE]

instead of the problem (1.1) and (1.2). Furthermore, we present two hypotheses for the weight function $\omega$ of (2.4)-(2.5) below:

H1:  $\omega(x)$ is monotonic on $[0,1]$;

H2:  $\inf_{x\in[0,1]}\omega(x)>0.$

In the present paper, we mainly prove the following result.

Theorem 2.1**.**

Suppose that the weight function $\omega$ of the eigenvalue problem (2.4) and (2.5) satisfies both of two hypothesises H1 and H2 above. Then the eigenvalue sequence $\{\lambda_{n}(q)\}_{n\geqslant 1}$ of (2.4)-(2.5) is uniformly locally Lipschitz continuous, in the sense of Definition 1.1, with respect to the potential $q$ in $L^{1}([0,1],\mathbb{R})$.

2.2. Notations and preliminary

For the benefit of the reader, we recall some well-known facts needed later.

2.2.1. Differentiability of eigenvalues with respect to potential functions

In this paper, by a normalized eigenfunction of (1.1)-(1.2) with a non-negative weight function $\omega$ we mean an eigenfunction $\varphi$ satisfying $\left\|\varphi\right\|_{\omega}=1$.

The following theorem shows the differentiability of eigenvalues of (1.1)-(1.2) with respect to the potential functions.

Theorem 2.2**.**

For any integer $n\geqslant 1$ and $q_{0}\in L^{1}([0,1],\mathbb{R})$, there exists a neighborhood $U(q_{0})$ of $q_{0}$ such that, the map

[TABLE]

is differentiable at $q_{0}$, and its Fréchet derivative is the bounded linear functional given by

[TABLE]

where $h\in L^{1}([0,1],\mathbb{R})$, and $\varphi_{n}$ is a normalized eigenfunction associated to $\lambda_{n}(q_{0})$ of (1.1)-(1.2).

Theorem 2.2 can be viewed as a special case of a well-known theorem [3, Theorem 4.2(6)] provided by Kong and Zettl. For more details about the differentiability of eigenvalues, the reader also may refer to [9, Theorem 3.6.1] and [6].

2.2.2. Prüfer transformation

Prüfer transformation is an important tool in the study of Sturm-Liouville problem, and has several variants (see e.g. [9] as well as [1], [10] and [16]). In the following, we introduce the elliptic Prüfer transformation.

Consider the problem (2.4) and (2.5). Set

[TABLE]

[TABLE]

Then

[TABLE]

is independent of $\rho$. The equation (2.8) is usually called the Prüfer equation, and $\rho$ satisfies

[TABLE]

3. The proof of Theorem 2.1

To prove our main theorem, we need to prove some lemmas and propositions as preparation. At first, consider the initial value problem

[TABLE]

where $c_{1}\geqslant 0$, $c_{2}\in\mathbb{R}$, $\omega\in L^{1}([0,1],\mathbb{R})$ and $\omega>0$ a.e. on $[0,1]$.

Applying Prüfer transformation in Section 2.2.2 to (3.1), we obtain the Prüfer equation for the case $q\equiv 0$ as follows:

[TABLE]

with the initial condition $\theta(0;\lambda)=\arctan\frac{\sqrt{\lambda}c_{1}}{c_{2}}\in[0,\pi)$, and $\rho$ satisfies

[TABLE]

Consequently, the solution $y(x;\lambda)$ of (3.1) has the following expression:

[TABLE]

where

[TABLE]

Set

[TABLE]

where $h(t)=1-\omega(t)$.

Lemma 3.1**.**

Let $\theta$ be defined as in (3.2) and $\omega(x)$ be in $L^{1}[0,1]$. If $\omega(x)\geqslant 0$ a.e. on $[0,1]$ and $\int_{0}^{1}\omega(x)\,dx>0$, then

[TABLE]

and $\theta(x;\lambda)$ is nondecreasing on $[0,1]$ for any fixed $\lambda\geqslant 0$.

Proof:  Since $\omega(x)\geqslant 0$ a.e. on $[0,1]$ and $\int_{0}^{1}\omega(x)\,dx>0$, the limit equation follows from

[TABLE]

Immediately, the remainder is proved, since the Prüfer equation (3.2), together with $\omega\geqslant 0$, shows that

[TABLE]

for any $x\in[0,1]$ and $\lambda\geqslant 0$. $\Box$

The following is the key lemma for the main theorem in this paper.

Lemma 3.2**.**

Let $\theta$ be defined as in (3.2). Assume that both of two hypotheses H1 and H2 hold. If $g(x):[0,1]\rightarrow\mathbb{R}$ is a function whose total variation on $[0,1]$ is finite, then

[TABLE]

and

[TABLE]

for any $c\in[0,1]$.

Proof:  Here we only prove this lemma when $\omega$ is increasing. For the case that $\omega$ is decreasing, by using the transform $t=1-x$, we can keep the eigenvalues invariant, and obtain the proof in the same way.

Since every function of bounded variation is the difference of two bounded monotonic functions, we may further assume that $g(x)$ is monotonic.

When $c=0$, the proof is trivial.

First, we begin to prove (3.8) for $c>0$.

Set

[TABLE]

Case 1: assume that $g(x)$ is decreasing and non-negative on $[0,1]$.

By Lemma 3.1, for any fixed $\bar{x}\in(0,1]$ and sufficiently large $\lambda>0$ , we can find two finite sequences

[TABLE]

such that

[TABLE]

satisfying for any $1\leqslant k\leqslant m,\ 1\leqslant j\leqslant m-1,$

[TABLE]

and ensuring that for any $x\in(x_{0},x_{1}),$

[TABLE]

which means that $x_{1}$ is the smallest one of those $x$ satisfying $\theta(x;\lambda)=\pi$.

Since $g$ is decreasing, we know that, for any integer $j\in\{1,2,...,m-1\}$,

[TABLE]

and

[TABLE]

Moreover, by the monotonicity of $\theta$ in Lemma 3.1, we have that

[TABLE]

and

[TABLE]

Hence,

[TABLE]

and

[TABLE]

For simplicity, hereafter we denote $\theta(x;\lambda)$ by $\theta(x)$.

Combining the inequalities (3.11)-(3.14) and nonnegativity of $h$ and $w$, we obtain that

[TABLE]

and

[TABLE]

Define an auxiliary function as follows:

[TABLE]

Then, substituting $\theta(x)$ for $u$, by the periodicity of $\sin 2u$, we have that

[TABLE]

So, it follows from (3.15)-(3.16) that

[TABLE]

and

[TABLE]

where $1\leqslant j\leqslant m-1.$

Adding together the two inequalities above, we have that

[TABLE]

where $1\leqslant j\leqslant m-1$.

For the last interval $[x_{m},\bar{x}]$, it can be known, from the similar argument as above, that

[TABLE]

From monotonicity and non-negativity of $h$ and $\omega$, it is apparent that $f(t)$ is non-negative and decreasing on $[0,1]$, and so

[TABLE]

where the finiteness of the integral in (3.23) owes to $\omega(0)>0$.

Then, it follows from (3.21)–(3.22) that, for the $\bar{x}$ arbitrarily given above,

[TABLE]

Moreover, since $\omega(0)>0$, we also have that

[TABLE]

Notice that

[TABLE]

Set

[TABLE]

Therefore, from (3.24)–(3.26) and the arbitrariness of $\bar{x}$, we can derive that, for any $c\in[0,1]$ and sufficiently large $\lambda>0$, one has

[TABLE]

which implies (3.8) in Case 1.

Case 2: assume that $g(x)$ is decreasing on $[0,1]$, but is not needed to be non-negative.

Let $u(x)=g(x)-g(1)$ and $v(x)\equiv 1$. Then $u(x)$ is non-negative and also decreasing on $[0,1]$, and $g(x)=u(x)+g(1)v(x)$. So, for any $c\in[0,1]$, we have

[TABLE]

Applying the result in Case 1 to the functions $u$ and $v$, we obtain (3.8) in Case 2.

Case 3: assume that $g(x)$ is increasing on $[0,1]$.

Set $u(x)=g(1)-g(x)$ and $v(x)\equiv 1$. So $g(x)=g(1)v(x)-u(x)$, and $u(x)$ is decreasing on $[0,1]$. Then (3.8) follows from the trick similar to that in Case 2.

From the argument above, the proof of (3.8) is done.

For (3.9), the proof is similar to that of (3.8).

Set

[TABLE]

By Lemma 3.1, for any fixed $\bar{x}\in(0,1]$ and sufficiently large $\lambda>0$ , we also can find two finite sequences

[TABLE]

such that

[TABLE]

satisfying for any $1\leqslant k\leqslant m,\ 1\leqslant j\leqslant m-1,$

[TABLE]

and ensuring that for any $x\in(x_{0},x_{1}),$

[TABLE]

which means that $x_{1}$ is the smallest one of those $x$ satisfying $\theta(x;\lambda)=\frac{5\pi}{4}$.

Then

[TABLE]

Moreover, it can be clearly seen that

[TABLE]

and

[TABLE]

Hence,

[TABLE]

and

[TABLE]

Similar to (3.18), we define the corresponding auxiliary function as follows:

[TABLE]

Note that $f(t)\in L^{\infty}[0,1]$.

Under the above setting (3.28)-(3.32), from lines of argument similar to those of (3.8), it can be shown that, when $g(x)$ is decreasing and non-negative on $[0,1]$, for any $c\in[0,1]$ and sufficiently large $\lambda>0$, one has

[TABLE]

which implies (3.9) in Case 1.

Similarly, (3.9) in both of Case 2 and Case 3 also can be obtained.

This lemma is proved.

$\Box$

The following result is a direct consequence of Lemma 3.2.

Lemma 3.3**.**

If both of H1 and H2 hold, then $H(x;\lambda)$ in (3.6) is uniformly bounded for all sufficiently large $\lambda>0$.

The next lemma can be considered as an analogue of Riemann-Lesbegue lemma.

Lemma 3.4**.**

Let $\theta$ be defined as in (3.2). Assume that H1 holds. If $g(x)$ is an arbitrary element in $L^{1}([0,1],\mathbb{R})$, then

[TABLE]

and

[TABLE]

for any $c\in[0,1]$.

Proof:  First, we claim that both of (3.34) and (3.35) hold when $g(x)\in AC[0,1]$ and $\omega$ satisfies both of two hypotheses H1 and H2. Since every absolutely continuous function has bounded variation, this claim is obviously true because of Lemma 3.2.

Next, we retain H1 but remove H2. Set $\omega_{n}=\omega+\frac{1}{n}$. Since all of absolutely continuous functions are dense in $L^{1}[0,1]$, we can deduce by the above claim that (3.34)-(3.35) hold for every $\omega_{n}$ and any $g(x)\in L^{1}[0,1]$. Consequently, (3.34)-(3.35) are true for any non-negative monotonic weight $\omega$ and integrable function $g(x)$, since $\omega_{n}$ is uniformly convergent to $\omega$ on $[0,1]$ and the Prüfer argument $\theta$ depends continuously on its weight function (see [9, Theorem 4.5.1]). $\Box$

By the above lemmas, we can establish, on any bounded subset of $L^{1}([0,1],\mathbb{R})$, the uniform boundedness of the normalized eigenfunctions of the eigenvalue problem (2.4)-(2.5).

Proposition 3.5**.**

Consider the eigenvalue problem (2.4)-(2.5), and suppose that the weight function $\omega$ satisfies H1–H2. Then, for any $L^{1}$-norm bounded subset $\Omega$ of $L^{1}([0,1],\mathbb{R})$, there exists a positive number $M(\Omega)$ such that, for any normalized eigenfunction $\varphi_{n}(x;\lambda_{n}(q))$ of (2.4)-(2.5), one has

[TABLE]

for all $n\geqslant 1$, $q\in\Omega$ and $x\in[0,1]$.

Proof:  Consider the initial value problem as follows:

[TABLE]

where $q,\ \omega\in L^{1}([0,1],\mathbb{R})$, $\omega>0$ a.e. on $[0,1]$ and $C_{1},\ C_{2}$ are two arbitrary fixed real numbers satisfying

[TABLE]

where $\alpha$ and $\beta$ are given in the boundary condition (2.5).

We may as well assume that $C_{1}\neq 0$.

Choose two linearly independent solutions $\phi$ and $\psi$ of (3.1), such that $\phi(0)=0,\ \phi^{\prime}(0)=1$ and $\psi(0)=1,\ \psi^{\prime}(0)=0$. Clearly, Wronskian determinant

[TABLE]

of $\psi$ and $\phi$ equals to 1.

We may choose $\psi$ and $\phi$ as follows:

[TABLE]

where $(r,\nu)$ and $(\mu,\sigma)$ satisfies the corresponding equation (3.5).

So, by Prüfer transformation, we obtain that

[TABLE]

For the initial condition in (3.36), using the formula of variation of constant, we can derive that the unique solution $y(x;\lambda)$ of (3.36) satisfies the integral equation

[TABLE]

Putting (3.38) into (3.40), we have

[TABLE]

where $R(x;t;\lambda)=r(x;\lambda)\sin\nu(x;\lambda)\mu(t;\lambda)\sin\sigma(t;\lambda)-r(t;\lambda)\sin\nu(t;\lambda)\mu(x;\lambda)\sin\sigma(x;\lambda)$.

Because $(r,\nu)$ and $(\mu,\sigma)$ satisfy the corresponding equation (3.5), it is easily known from Lemma 3.3 that, there exists positive numbers $M_{0}$ and $K$ such that, for any $\lambda\geqslant K$,

[TABLE]

and

[TABLE]

and then,

[TABLE]

Set

[TABLE]

By the inequalities (3.41)-(3.44)and Gronwall inequality (see e.g. [9, Theorem 1.4.1(i)] and [7, Theorem 1.3.2]), it is apparent that, for any $\lambda\geqslant\max\{1,K\}$,

[TABLE]

which implies that

[TABLE]

By (3.41), (3.42) and (3.46), it can be seen that, for any $\lambda\geqslant\max\{1,K\}$,

[TABLE]

And then, there exists a positive number $M_{1}$ such that, for any $\lambda\geqslant\max\{1,K\}$,

[TABLE]

Let $\{\lambda_{n}\}_{n\geqslant 1}$ be the eigenvalue sequence of the eigenvalue problem (2.4)-(2.5). Then the unique solution $y(x;\lambda_{n})$ of the initial value problem (3.36) is also a eigenfunction of (2.4)-(2.5) corresponding to $\lambda_{n}$. So we can find a number $\beta(\lambda_{n})$ such that

[TABLE]

satisfying

[TABLE]

Thereupon, we have

[TABLE]

Since $\lambda_{n}\rightarrow+\infty$ as $n\rightarrow+\infty$, there exists a sufficiently large positive integer $N_{0}$ such that $\lambda_{n}\geqslant\max\{1,K\}$ for any $n\geqslant N_{0}$.

Hence, by (3.48) and (3.49), we have

[TABLE]

for any $n\geqslant N_{0}$.

Since $(\mu,\sigma)$ satisfies the corresponding equation (3.5), the equation (3.50), together with $\mu(0;\lambda)=1$, yields that,

[TABLE]

for any $n\geqslant N_{0}$.

Lemma 3.3 tells us that, there exists a positive number $H_{0}$ and a sufficiently large $N_{1}\geqslant N_{0}$ such that

[TABLE]

Then, for any $n\geqslant N_{1}$, we have

[TABLE]

that is,

[TABLE]

So, by Lemma 3.4 and (1.5), for any fixed $\gamma\in(0,1)$, we can choose a sufficiently large integer $N\geqslant N_{1}$, such that, as long as $n\geqslant N$, one has

[TABLE]

and

[TABLE]

Consequently, by the inequalities (3.54)-(3.56), we have

[TABLE]

Set

[TABLE]

where $\varphi_{i}$ is the unique normalized eigenfunction corresponding to the $i$th eigenvalue $\lambda_{i}$.

Hence, it follows from (3.45) and (3.57) that, for any $n\geqslant 1$,

[TABLE]

The proof is finished. $\Box$

Now, it’s time to give the proof of Theorem 2.1.

Proof of Theorem 2.1: Set $\bar{B}_{M}:=\{q\in L^{1}[0,1]\ |\ \left\|q\right\|_{L^{1}}\leqslant M\}$, which is convex. For any $L^{1}$-norm bounded subset $\Omega$ of $L^{1}([0,1],\mathbb{R})$, set $B(\Omega):=\sup\{\left\|q\right\|_{L^{1}}|\ q\in\Omega\}$. It is easily seen that $\Omega\subset\bar{B}_{B(\Omega)}$. Hence we only need to prove our result holds for convex sets.

Let $\Omega$ be an arbitrary convex $L^{1}$-norm bounded subset of $L^{1}([0,1],\mathbb{R})$. For any two $q_{1},\ q_{2}\in\Omega$ and $\Delta q=q_{2}-q_{1}$, set

[TABLE]

and

[TABLE]

Let $\varphi_{n}(x;t)$ be the unique normalized eigenfunction of $\tilde{\lambda}_{n}(t)$. By Theorem 2.2, it is apparent that

[TABLE]

as a bounded linear functional on $L^{1}([0,1],\mathbb{R})$.

Then, by (3.58), we obtain that

[TABLE]

Finally, due to Proposition 3.5 and (3.59), the proof is done. $\Box$

Acknowledgement

The authors gratefully acknowledge the anonymous referees for their valuable comments which substantially improved the quality of this paper. It is especially helpful that a vital error about the proof of Lemma 3.4 in the original manuscript was point out by the referees. The authors also would like to thank Professor Bing Xie (Shandong University, Weihai) and Dr. Qianhong Huang (University of Alberta, Canada) for some helpful discussions and suggestions.