Bounds of Eigenvalues for Complex q-Sturm–Liouville Problem

Abstract

The eigenvalues of complex q-Sturm–Liouville boundary value problems are the focus of this paper. The coefficients of the corresponding q-Sturm–Liouville equation provide the lower bounds on the real parts of all eigenvalues, and the real part of the eigenvalue and the coefficients of this q-Sturm–Liouville equation provide the bounds on the imaginary part of each eigenvalue.

1. Introduction

Consider the q-Sturm–Liouville problem

$$-{\displaystyle \frac{1}{q}}{D}_{q^{-1}}{D}_{q}y\left(x\right)+v\left(x\right)y\left(x\right)=\lambda w\left(x\right)y\left(x\right),0\le x\le 1,$$

(1)

associated with the boundary value conditions

$${\mathcal{B}}_1\left(y\right):=a_{1}y\left(0\right)+a_2{D}_{q^{-1}}y\left(0\right)=0,{\mathcal{B}}_2\left(y\right):=b_{1}y\left(1\right)+b_2{D}_{q^{-1}}y\left(1\right)=0,$$

(2)

where the functions v and w possess appropriate q-derivatives with

$$w\in L_q^1([0,1],\mathbb{R}),w\left(x\right)>0a.e.\mathrm{on}[0,1],v\in L_q^1([0,1],\mathbb{C}\setminus \mathbb{R}),$$

where the q-difference operator $D_q$ is defined by

$$D_{q}f\left(x\right):={\displaystyle \frac{f\left(x\right)-f\left(qx\right)}{x-qx}},x\in [0,1]/\left\{0\right\},q\in \left(\mathbb{R},(0,1)\right),$$

and the q-derivative at zero is defined by $D_{q}f\left(0\right):=\underset{n\to \infty }{lim}{\displaystyle \frac{f\left(x{q}^n\right)-f\left(0\right)}{x{q}^n}},x\in (0,1),$ if the limit exists and does not depend on x. $a_i,b_i,i=1,2$ are real constants. Let $L_q^1[0,1]$ be the space of all complex-valued functions defined on $[0,1]$ such that ${\parallel y\parallel }_{1,q}:={\int }_0^1|y\left(x\right)|{\mathrm{d}}_{q}x<\infty$, where the Jackson q-integration is defined in (3) in the following. Such a problem defined by (1) and (2) is called the q-Sturm–Liouville problem, where it is proven to be well defined in [1] (Chapter 5) and [2].

Due to their numerous applications in the fields of quantum theory, orthogonal polynomials, and hypergeometric functions, q-difference equations have recently garnered a lot of attention. The q-derivatives, q-integrals, q-exponential function, q-trigonometric function, q-Taylor formula, q-Beta(Gamma) functions, and the associated difficulties are involved in a number of physical models (see [1,3,4]), and in particular, q-Sturm–Liouville problems were examined in [3,5,6,7,8].

The classic Sturm–Liouville problem $-y^{″}\left(x\right)+v\left(x\right)y\left(x\right)=\lambda w\left(x\right)y\left(x\right)$ with boundary conditions are formally non-self-adjoint if the imaginary part of v is nonzero. In contrast to self-adjoint boundary value problems, this means there may be infinitely many non-real eigenvalues when the imaginary part of v is nonzero. In classic and indefinite (w changes sign) Sturm–Liouville problems, determining a priori bounds of these non-real eigenvalues in terms of the coefficients and the boundary conditions is an intriguing and challenging topic. Qi et al. recently solved these problems in [9,10,11,12]. In particular, the author of [13] considered the complex perturbation of the Legendre eigenvalue problems with limit-circle-type non-oscillation endpoints, where the dissipative operators and the estimates of non-real eigenvalues are obtained. Moreover, the authors in [14] determined the lower bounds on the real parts and the bound on the imaginary part of each eigenvalue in terms of the coefficients and the real part of the eigenvalue for the regular complex Sturm–Liouville problem with Dirichlet boundary conditions.

The author of [1] (pp. 164–170) demonstrated that all of the eigenvalues of this system are real and that the eigenfunctions fulfill an orthogonality relation for the fundamental q-Sturm–Liouville eigenvalue problem, with v being a real-valued function and $w\left(x\right)\ge 0,x\in (0,1)$ in (1). In a Hilbert space, Annaby and Mansour developed a self-adjoint q-Sturm–Liouville operator and discussed the characteristics of the eigenvalues and eigenfunctions in [15]. The bounds of non-real eigenvalues for the indefinite q-Sturm–Liouville problem are obtained in [16].

This paper concentrates on the complex q-Sturm–Liouville eigenvalue problem (1) and (2). Since (1) is formally non-self-adjoint, we can derive lower bounds on the real parts of all the eigenvalues of (1) and (2) in terms of the coefficients of (1), as well as bounds on the imaginary part of each eigenvalue using the real part of the eigenvalue and the coefficients of (1). The approaches used in this paper were partially motivated by the non-real eigenvalue estimates for the regular indefinite and complex Sturm–Liouville problem obtained in [12,14].

The primary findings of this work are discussed in the next section.

2. A Priori Bounds of Non-Real Eigenvalues

In this section, we recall some basic concepts and useful results of quantum calculus. We refer to [3,4,17] and some references cited therein. The integral denoted by ${\int }_a^{b}f\left(t\right){\mathrm{d}}_{q}t$ is a right inverse of the q-derivative introduced in [18] by Jackson, and it is defined by

$${\int }_a^{b}f\left(t\right){\mathrm{d}}_{q}t:={\int }_0^{b}f\left(t\right){\mathrm{d}}_{q}t-{\int }_0^{a}f\left(t\right){\mathrm{d}}_{q}t,$$

(3)

where ${\int }_0^{x}f\left(t\right){\mathrm{d}}_{q}t:=(1-q){\displaystyle \sum _{n=0}^{\infty }}x{q}^{n}f\left(x{q}^n\right)$, provided that the series on the right-hand side of the above equation converges at $x=a$ and b. The rule of q-integration by parts is

$${\int }_0^1{y}_1\left(x\right)D_q{y}_2\left(x\right){\mathrm{d}}_{q}x=y_1\left(x\right)y_2\left(x\right)-\underset{n\to \infty }{lim}{y}_1\left(q^n\right)y_2\left(q^n\right)-{\int }_0^1{y}_2\left(qx\right)D_q{y}_1\left(x\right){\mathrm{d}}_{q}x,$$

(4)

and the non-symmetric q-product rule is

$$D_q\left(fg\right)\left(x\right)=f\left(qx\right)D_{q}g\left(x\right)+g\left(x\right)D_{q}f\left(x\right).$$

(5)

Let us consider the following q-Sturm–Liouville problem:

$$\left\{\begin{array}{l}-{\displaystyle \frac{1}{q}}{D}_{q^{-1}}{D}_{q}y\left(x\right)+v\left(x\right)y\left(x\right)=\lambda w\left(x\right)y\left(x\right),0\le x\le 1,\\ {\mathcal{B}}_1\left(y\right):=a_{1}y\left(0\right)+a_2{D}_{q^{-1}}y\left(0\right)=0,\\ {\mathcal{B}}_2\left(y\right):=b_{1}y\left(1\right)+b_2{D}_{q^{-1}}y\left(1\right)=0,\end{array}\right.$$

(6)

where $a_i,b_i,i=1,2$ are real constants and the functions v and w satisfy the conditions

$$\begin{array}{l}w\in L_q^1([0,1],\mathbb{R}),w\left(x\right)>0a.e.\mathrm{on}[0,1],v\in L_q^1([0,1],\mathbb{C}\setminus \mathbb{R}),\\ v=v_1+i{v}_2,v_k^{-}=-min\{v_k,0\},v_k^{+}=max\{v_k,0\}k=1,2.\end{array}$$

(7)

A complex number $\lambda$ is called an eigenvalue of boundary value problem (6) if there is a nontrivial solution y that satisfies the boundary conditions. Such a solution y is called an eigenfunction of $\lambda$. It is no restriction to assume that ${\int }_0^1{|f\left(x\right)|}^2{\mathrm{d}}_{q}x=1$ in the following discussion. To state the main results, we need the following notations to simplify our statements. Let ${\parallel ·\parallel }_c$ be the maximum norm of $C[0,1]$, which is the set of the continuous functions on $[0,1]$, and assume that $\widehat{w}\ge 1/w\left(qx\right)$ for $x\in [0,1]$, $\mathcal{W}={\left({\int }_0^1{|D_{q}w\left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}$,

$$\begin{array}{l}{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}=|\alpha |+|\beta |+\parallel v_1^{-}{\parallel }_{1,q},{\mathcal{V}}_{\alpha ,\beta ,v_1^{+}}=|\alpha |+|\beta |+\parallel v_1^{+}{\parallel }_{1,q},\alpha ={\displaystyle \frac{a_1}{a_2}},\beta ={\displaystyle \frac{b_1}{b_2}},\\ \mathcal{V}=\sqrt{{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}\left({\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}+1\right)}+{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}},{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}={\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}+\mathrm{Re}\lambda {\parallel w\parallel }_{1,q},\\ \mathsf{\Gamma }=\sqrt{{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}\left(1+{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}\right)}+{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w},\Delta =\sqrt{\widehat{\Delta }\left(\widehat{\Delta }+1\right)}+\widehat{\Delta },\\ \widehat{\Delta }=\widehat{w}\left({\parallel w\parallel }_c{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}+{\displaystyle \frac{\mathcal{W}}{2}}\right)/\left(1-{\displaystyle \frac{\widehat{w}\mathcal{W}}{2}}\right).\end{array}$$

(8)

Therefore, we give the estimated results on the non-real eigenvalues of problem (6) in the following.

Theorem 1.

Let (7) and (8) hold. If λ is a non-real eigenvalue of problem (6) with $\mathrm{Re}\lambda \le 0$, then the following holds:

$$\mathrm{Re}\lambda \ge -{\displaystyle \frac{2}{{\epsilon }_1}}\left[{\mathcal{V}}_{\alpha ,\beta ,v_1^{+}}\left(2\mathcal{V}+1\right)+{\mathcal{V}}^2\right],|\mathrm{Im}\lambda |\le {\displaystyle \frac{2}{{\epsilon }_1}}\left(2\mathcal{V}+1\right){\parallel v_2^{+}\parallel }_{1,q},$$

where ${\epsilon }_1$ satisfies $\mathrm{mes}\mathsf{\Omega }\left({\epsilon }_1\right)\le 1/\left[2\left(2\mathcal{V}+1\right)\right]$.

Theorem 2.

Let (7) and (8) hold. If λ is a non-real eigenvalue of problem (6) with $\mathrm{Re}\lambda >0$, then the following estimates hold:

$$|\mathrm{Im}\lambda |\le {\displaystyle \frac{2}{{\epsilon }_1}}{\parallel v_2^{+}\parallel }_{1,q}\left(2\mathsf{\Gamma }+1\right),$$

where ${\epsilon }_1$ satisfies $\mathrm{mes}\mathsf{\Omega }\left({\epsilon }_1\right)\le 1/\left[2\left(2\mathsf{\Gamma }+1\right)\right]$.

Theorem 3.

Let (7) and (8) holds. Assume that $D_{q}w\in L_q^2[0,1]$. If λ is a non-real eigenvalue of problem (6) with $\mathrm{Re}\lambda \le 0$, then

$$\begin{array}{l}\mathrm{Re}\lambda \ge -{\displaystyle \frac{2}{{\epsilon }_2}}\left\{{\parallel w\parallel }_c\left[{\mathcal{V}}_{\alpha ,\beta ,v_1^{+}}\left(2\mathcal{V}+1\right)+{\mathcal{V}}^2\right]+\mathcal{W}\mathcal{V}\sqrt{2\mathcal{V}+1}\right\},\\ |\mathrm{Im}\lambda |\le {\displaystyle \frac{2}{{\epsilon }_2}}\left[{\parallel w\parallel }_c\left(2\mathcal{V}+1\right){\parallel v_2^{+}\parallel }_{1,q}+\mathcal{W}\mathcal{V}\sqrt{2\mathcal{V}+1}\right],\end{array}$$

where ${\epsilon }_2$ satisfies $\mathrm{mes}\mathsf{\Omega }\left({\epsilon }_2\right)\le 1/\left[2\left(2\mathcal{V}+1\right)\right]$ and $\mathcal{W}={\left({\int }_0^1{|D_{q}w\left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}$.

Theorem 4.

Let (7) and (8) hold. Suppose that $\widehat{w}\ge 1/w\left(qx\right)$ for $x\in [0,1]$ and $D_{q}w\in L_q^2[0,1]$. If λ is a non-real eigenvalue of problem (6) with $\mathrm{Re}\lambda >0$, then the following holds:

$$|\mathrm{Im}\lambda |\le {\displaystyle \frac{2}{{\epsilon }_2}}\left({\parallel w\parallel }_c\left(2\Delta +1\right){\parallel v_2^{+}\parallel }_{1,q}+\mathcal{W}\Delta \sqrt{2\Delta +1}\right),$$

where ${\epsilon }_2$ satisfies $\mathrm{mes}\mathsf{\Omega }\left({\epsilon }_2\right)\le 1/\left[2\left(2\Delta +1\right)\right]$ and $\mathcal{W}={\left({\int }_0^1{|D_{q}w\left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}$.

3. The Proofs of Theorems

The following lemma is the estimate of ${\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x$, where $\varphi$ is an eigenfunction of the q-Sturm–Liouville problem (6) corresponding to the eigenvalue $\lambda$; that is, ${\mathcal{B}}_1\varphi ={\mathcal{B}}_2\varphi =0$ and

$$-{\displaystyle \frac{1}{q}}{D}_{q^{-1}}{D}_q\varphi \left(x\right)+v\left(x\right)\varphi \left(x\right)=\lambda w\left(x\right)\varphi \left(x\right).$$

(9)

Since problem (6) is a linear system and $\varphi$ is continuous, we choose ${\int }_0^1{|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x=1$ in the following discussion.

Lemma 1.

Let λ and ϕ be defined as above, with $\mathrm{Re}\lambda \le 0$. Then,

$${\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\mathcal{V}}^2,{\parallel \varphi \parallel }_{\infty }^2\le 2\mathcal{V}+1,$$

where $\mathcal{V}$ is defined in (8).

Proof. 

Multiply both sides of (9) by $\overline{\varphi }$ and integrate over the interval $[0,1]$. Then, by ${\mathcal{B}}_1\varphi ={\mathcal{B}}_2\varphi =0$ and the q-integration by parts in (4), we have

$$\lambda {\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x={\beta |\varphi \left(1\right)|}^2-{\alpha |\varphi \left(0\right)|}^2+{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+{\int }_0^{1}v\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x,$$

(10)

where $\alpha =a_1/a_2$ and $\beta =b_1/b_2$. Separating the real parts yields

$$\mathrm{Re}\lambda {\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x={\beta |\varphi \left(1\right)|}^2-{\alpha |\varphi \left(0\right)|}^2+{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+{\int }_0^1{v}_1\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x.$$

(11)

This, together with $w\left(x\right)>0$ and $\mathrm{Re}\lambda \le 0$, yields the following:

$${\beta |\varphi \left(1\right)|}^2-{\alpha |\varphi \left(0\right)|}^2+{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+{\int }_0^1{v}_1\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le 0.$$

This, together with $v_1^{-}=max\{-v_1,0\}$, yields the following:

$$\begin{array}{ll}{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x& \le {\alpha |\varphi \left(0\right)|}^2-{\beta |\varphi \left(1\right)|}^2-{\int }_0^1{v}_1{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ & \le {\alpha |\varphi \left(0\right)|}^2-{\beta |\varphi \left(1\right)|}^2+{\int }_0^1{v}_1^{-}\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x.\end{array}$$

(12)

For every $x,y\in [0,1],y<x$, from the non-symmetric q-product rule in (5), we obtain

$$\begin{array}{l}{|\varphi \left(x\right)|}^2-{|\varphi \left(y\right)|}^2={\int }_y^x{D}_q{(|\varphi \left(t\right)|}^2){\mathrm{d}}_{q}t={\int }_y^x\overline{\varphi \left(t\right)}{D}_q\varphi \left(t\right){\mathrm{d}}_{q}t+{\int }_y^x\varphi \left(qt\right)\overline{D_q\varphi \left(t\right)}{\mathrm{d}}_{q}t\\ \le {\left({\int }_0^1{|\varphi \left(t\right)|}^2{\mathrm{d}}_{q}t\right)}^{1/2}{\left({\int }_0^1{|D_q\varphi \left(t\right)|}^2{\mathrm{d}}_{q}t\right)}^{1/2}+{\left({\int }_0^1{|\varphi \left(qt\right)|}^2{\mathrm{d}}_{q}t\right)}^{1/2}{\left({\int }_0^1{|\overline{D_q\varphi \left(t\right)}|}^2{\mathrm{d}}_{q}t\right)}^{1/2}\\ \le 2{\left({\int }_0^1{|\varphi \left(t\right)|}^2{\mathrm{d}}_{q}t\right)}^{1/2}{\left({\int }_0^1{|D_q\varphi \left(t\right)|}^2{\mathrm{d}}_{q}t\right)}^{1/2}=2{\left({\int }_0^1{|D_q\varphi \left(t\right)|}^2{\mathrm{d}}_{q}t\right)}^{1/2}.\end{array}$$

Integrating the above inequality over $[0,1]$ with respect to y gives

$${\int }_0^1{|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}y-{\int }_0^1{|\varphi \left(y\right)|}^2{\mathrm{d}}_{q}y\le 2{\int }_0^1{\left({\int }_0^1{|D_q\varphi \left(t\right)|}^2{\mathrm{d}}_{q}t\right)}^{1/2}{\mathrm{d}}_{q}y.$$

This implies that

$$\begin{array}{ll}{|\varphi \left(x\right)|}^2& \le 2{\left({\int }_0^1{|D_q\varphi \left(t\right)|}^2{\mathrm{d}}_{q}t\right)}^{1/2}{\int }_0^1{\mathrm{d}}_{q}y+{\int }_0^1{|\varphi \left(y\right)|}^2{\mathrm{d}}_{q}y\\ & =2{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+1.\end{array}$$

(13)

Hence, we obtain the following from (12):

$$\begin{array}{ll}{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x& \le {\alpha |\varphi \left(0\right)|}^2-{\beta |\varphi \left(1\right)|}^2+{\int }_0^1{v}_1^{-}\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ & \le \left(|\alpha |+|\beta |+\parallel v_1^{-}{\parallel }_{1,q}\right)\left[2{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+1\right]\\ & ={\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}\left[2{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+1\right].\end{array}$$

This yields

$${\left[{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}-{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}\right]}^2\le {\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}\left({\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}+1\right).$$

And hence,

$${\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}\le \sqrt{{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}\left({\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}+1\right)}+{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}},$$

such that the first estimate in this lemma is proved. Hence, it follows from (13) that

$${|\varphi \left(x\right)|}^2\le 2{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+1\le 2\mathcal{V}+1,$$

which completes the proof of Lemma 1. □

According to $w\left(x\right)>0$ a.e. on $[0,1]$, we choose ${\epsilon }_1>0$ to be so small that

$$\mathsf{\Omega }\left({\epsilon }_1\right)=\{x\in [0,1]:w\left(x\right)\le {\epsilon }_1\}.$$

(14)

Lemma 2.

Let λ and ϕ be defined as above, with $\mathrm{Re}\lambda \le 0$. Then,

$${\int }_0^{1}w\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\ge {\epsilon }_1/2,$$

where ${\epsilon }_1$ is defined in (14).

Proof. 

In fact, since the measure of $\mathsf{\Omega }\left({\epsilon }_1\right)$, i.e., the $\mathrm{mes}\left(\mathsf{\Omega }\left({\epsilon }_1\right)\right)\to 0$ as ${\epsilon }_1\to 0$, the $\mathrm{mes}\mathsf{\Omega }\left({\epsilon }_1\right)\le 1/\left[2\left(2\mathcal{V}+1\right)\right]$ holds. Hence,

$$\begin{array}{l}{\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\ge {\int }_{[0,1]\setminus \mathsf{\Omega }\left({\epsilon }_1\right)}w\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ \ge {\epsilon }_1\left({\int }_0^1{|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x-{\int }_{\mathsf{\Omega }\left({\epsilon }_1\right)}{|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)\\ \ge {\epsilon }_1\left(1-{\parallel \varphi \parallel }_{\infty }^2{\int }_{\mathsf{\Omega }\left({\epsilon }_1\right)}{\mathrm{d}}_{q}x\right)\ge {\epsilon }_1/2.\end{array}$$

□

By applying the above lemmas, we can prove Theorem 1 in the following:

Proof of Theorem 1.

It follows from (11) in Lemma 1 that

$$\begin{array}{l}|\mathrm{Re}\lambda |{\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x=\left|{\beta |\varphi \left(1\right)|}^2-\alpha {|\varphi \left(0\right)|}^2+{\int }_0^1\left(|D_q{\varphi \left(x\right)|}^2+v_1\left(x\right){|\varphi \left(x\right)|}^2\right){\mathrm{d}}_{q}x\right|\\ \le {|\alpha ||\varphi \left(0\right)|}^2+{|\beta ||\varphi \left(1\right)|}^2+{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+{\int }_0^1{v}_1^{+}\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ \le \left(|\alpha |+|\beta |+\parallel v_1^{+}{\parallel }_{1,q}\right)\left(2\mathcal{V}+1\right)+{\mathcal{V}}^2\\ ={\mathcal{V}}_{\alpha ,\beta ,v_1^{+}}\left(2\mathcal{V}+1\right)+{\mathcal{V}}^2.\end{array}$$

This, together with Lemma 2, yields the following:

$${\displaystyle \frac{{\epsilon }_1}{2}}|\mathrm{Re}\lambda |\le |\mathrm{Re}\lambda |{\int }_0^{1}w\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\mathcal{V}}_{\alpha ,\beta ,v_1^{+}}\left(2\mathcal{V}+1\right)+{\mathcal{V}}^2.$$

(15)

Separating the imaginary parts of (10) in Lemma 1 yields

$$\mathrm{Im}\lambda {\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x={\int }_0^1{v}_2\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x.$$

It follows from $v_2^{+}=max\{v_2,0\}$ and Lemma 2 that

$${\displaystyle \frac{{\epsilon }_1}{2}}|\mathrm{Im}\lambda |\le |\mathrm{Im}\lambda |{\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\int }_0^1{v}_2^{+}{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le \left(2\mathcal{V}+1\right){\parallel v_2^{+}\parallel }_{1,q}.$$

(16)

Therefore, the inequalities in Theorem 1 follow from (15) and (16) hold immediately. □

Now, we consider another situation. If $\lambda$ is the eigenvalue of the q-Sturm–Liouville problem (6) with $\mathrm{Re}\lambda >0$, then we consider the following eigenvalue problem:

$$-{\displaystyle \frac{1}{q}}{D}_{q^{-1}}{D}_{q}y\left(x\right)+\left[v\left(x\right)-\mathrm{Re}\lambda w\left(x\right)\right]y\left(x\right)=\lambda w\left(x\right)y\left(x\right),{\mathcal{B}}_{1}y=0={\mathcal{B}}_{2}y.$$

(17)

It is easy to see that $\lambda -\mathrm{Re}\lambda$ is the eigenvalue of (17) and $\mathrm{Re}(\lambda -\mathrm{Re}\lambda )=0$.

Lemma 3.

Let λ be an eigenvalue of the q-Sturm–Liouville problem (6), with $\mathrm{Re}\lambda >0$, and let ϕ be the corresponding eigenfunction. Then,

$${\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\mathsf{\Gamma }}^2,{\parallel \varphi \parallel }_{\infty }^2\le 2\mathsf{\Gamma }+1,$$

where Γ is defined in (8).

Proof. 

By multiplying both sides of equation of (17) by $\overline{\varphi }$ and integrating over the interval $[0,1]$, similar to the proof of Lemma 1, we have

$$\begin{array}{r}(\lambda -\mathrm{Re}\lambda ){\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x={\beta |\varphi \left(1\right)|}^2-{\alpha |\varphi \left(0\right)|}^2+{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ +{\int }_0^1\left[v\left(x\right)-\mathrm{Re}\lambda w\left(x\right)\right]{|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x,\end{array}$$

(18)

where $\alpha =a_1/a_2$ and $\beta =b_1/b_2$. Separating the real parts yields

$$\begin{array}{ll}0& =\mathrm{Re}(\lambda -\mathrm{Re}\lambda ){\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ & ={\beta |\varphi \left(1\right)|}^2-{\alpha |\varphi \left(0\right)|}^2+{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+{\int }_0^1[v_1\left(x\right)-\mathrm{Re}\lambda w\left(x\right)]{|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x.\end{array}$$

According to ${|\varphi \left(x\right)|}^2\le 2{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+1$ of (13) in Lemma 1 and $v_1^{-}=max\{-v_1,0\}$, we have

$$\begin{array}{l}{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x={\alpha |\varphi \left(0\right)|}^2-{\beta |\varphi \left(1\right)|}^2-{\int }_0^1[v_1\left(x\right)-\mathrm{Re}\lambda w\left(x\right)]{|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ \le {\alpha |\varphi \left(0\right)|}^2+{\beta |\varphi \left(1\right)|}^2+{\int }_0^1{v}_1^{-}{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+\mathrm{Re}\lambda {\int }_0^{1}w\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ \le \left(|\alpha |+|\beta |+\parallel v_1^{-}{\parallel }_{1,q}+\mathrm{Re}\lambda {\parallel w\parallel }_{1,q}\right)\left[2{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+1\right]\\ ={\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}\left[2{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+1\right].\end{array}$$

Hence,

$${\left[{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}-{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}\right]}^2\le {\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}\left(1+{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}\right).$$

This yields

$${\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}\le \mathsf{\Gamma }\mathrm{and}{|\varphi \left(x\right)|}^2\le 2\mathsf{\Gamma }+1,$$

which completes the proof of Lemma 3. □

It follows from (14) that the $\mathrm{mes}\left(\mathsf{\Omega }\left({\epsilon }_1\right)\right)\to 0$ as ${\epsilon }_1\to 0$, and thus, we can set $\mathrm{mes}\mathsf{\Omega }\left({\epsilon }_1\right)\le 1/\left[2\left(2\mathsf{\Gamma }+1\right)\right]$. Similar to the proof of Lemma 2, we have the following.

Lemma 4.

Let λ and ϕ be defined as above, with $\mathrm{Re}\lambda >0$. Then,

$${\int }_0^{1}w\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\ge {\epsilon }_1/2,$$

where ${\epsilon }_1$ satisfies $\mathrm{mes}\mathsf{\Omega }\left({\epsilon }_1\right)\le 1/\left[2\left(2\mathsf{\Gamma }+1\right)\right]$.

By applying Lemmas 3 and 4, we now prove Theorem 2.

Proof of Theorem 2.

Separating the imaginary parts of (18) in Lemma 3 yields

$$\mathrm{Im}(\lambda -\mathrm{Re}\lambda ){\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x={\int }_0^1{v}_2\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x.$$

It follows from Lemma 3, Lemma 4, and $v_2^{+}=max\{v_2,0\}$ that

$$\begin{array}{l}{\displaystyle \frac{{\epsilon }_1}{2}}|\mathrm{Im}\lambda |={\displaystyle \frac{{\epsilon }_1}{2}}|\mathrm{Im}(\lambda -\mathrm{Re}\lambda )|\le |\mathrm{Im}(\lambda -\mathrm{Re}\lambda )|{\int }_0^1{w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ ={\int }_0^1{v}_2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\int }_0^1{v}_2^{+}{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\parallel v_2^{+}\parallel }_{1,q}\left(2\mathsf{\Gamma }+1\right).\end{array}$$

Thus, the inequality in Theorem 2 is established and the proof is complete. □

In the following, we prove Theorems 3 and 4. Since $w\left(x\right)>0$ a.e. on $[0,1]$, we choose ${\epsilon }_2>0$ to be so small that

$$\mathsf{\Omega }\left({\epsilon }_2\right)=\{x\in [0,1]:w^2\left(x\right)\le {\epsilon }_2\}.$$

(19)

Lemma 5.

Let λ and ϕ be defined as above, with $\mathrm{Re}\lambda \le 0$. Then,

$${\int }_0^1{w}^2\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\ge {\epsilon }_2/2,$$

where ${\epsilon }_2$ satisfies the $\mathrm{mes}\mathsf{\Omega }\left({\epsilon }_2\right)\le 1/\left[2\left(2\mathcal{V}+1\right)\right]$.

Proof. 

In fact, since $\mathrm{mes}\left(\mathsf{\Omega }\left({\epsilon }_2\right)\right)\to 0$ as ${\epsilon }_2\to 0$, the $\mathrm{mes}\left(\mathsf{\Omega }\left({\epsilon }_2\right)\right)\le 1/\left[2\left(2\mathcal{V}+1\right)\right]$ holds. Similar to Lemma 2, we have

$${\int }_0^1{w}^2\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\ge {\epsilon }_2\left({\int }_0^1{|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x-{\int }_{\mathsf{\Omega }\left({\epsilon }_2\right)}{|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)\ge {\epsilon }_2/2.$$

□

With the aid of Lemma 5, we have the following proof of Theorem 3.

Proof of Theorem 3.

Multiply both sides of (9) by $w\left(x\right)\overline{\varphi \left(x\right)}$ and integrate by parts on $[0,1]$. Then, from ${\mathcal{B}}_1\varphi ={\mathcal{B}}_2\varphi =0$ and the q-integration by parts in (4), we have

$$\begin{array}{ll}\lambda {\int }_0^1{w}^2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x=& {\beta w\left(1\right)|\varphi \left(1\right)|}^2-{\alpha w\left(0\right)|\varphi \left(0\right)|}^2+{\int }_0^{1}w\left(qx\right)|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ & +{\int }_0^{1}w\left(x\right)v\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+{\int }_0^1{D}_{q}w\left(x\right)D_q\varphi \left(x\right)\overline{\varphi \left(x\right)}{\mathrm{d}}_{q}x.\end{array}$$

(20)

Separating the real parts yields

$$\begin{array}{ll}\mathrm{Re}\lambda {\int }_0^1{w}^2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x& ={\beta w\left(1\right)|\varphi \left(1\right)|}^2-{\alpha w\left(0\right)|\varphi \left(0\right)|}^2+{\int }_0^{1}w\left(qx\right)|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ & +{\int }_0^{1}w\left(x\right)v_1\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+\mathrm{Re}\left({\int }_0^1{D}_{q}w\left(x\right)D_q\varphi \left(x\right)\overline{\varphi \left(x\right)}{\mathrm{d}}_{q}x\right).\end{array}$$

(21)

According to ${\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\mathcal{V}}^2,{\parallel \varphi \parallel }_{\infty }^2\le 2\mathcal{V}+1,$ in Lemma 1,

$${\beta w\left(1\right)|\varphi \left(1\right)|}^2-{\alpha w\left(0\right)|\varphi \left(0\right)|}^2+{\int }_0^{1}w\left(x\right)v_1{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\parallel w\parallel }_c\left(|\alpha |+|\beta |+\parallel v_1^{+}{\parallel }_{1,q}\right)\left(2\mathcal{V}+1\right).$$

(22)

$${\int }_0^{1}w\left(qx\right)|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\parallel w\parallel }_c{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\parallel w\parallel }_c{\mathcal{V}}^2.$$

(23)

$$\mathrm{Re}\left({\int }_0^1{D}_{q}w\left(x\right)D_q\varphi \left(x\right)\overline{\varphi \left(x\right)}{\mathrm{d}}_{q}x\right)\le {\int }_0^1\left|D_{q}w\left(x\right)D_q\varphi \left(x\right)\overline{\varphi \left(x\right)}\right|{\mathrm{d}}_{q}x\le \mathcal{W}\mathcal{V}\sqrt{2\mathcal{V}+1},$$

(24)

where $\mathcal{W}={\left({\int }_0^1{|D_{q}w\left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}$. Now, by using inequations (22)–(24), together with Lemma 5, the integration of (21) yields

$$\begin{array}{l}{\displaystyle \frac{{\epsilon }_2}{2}}|\mathrm{Re}\lambda |\le |\mathrm{Re}\lambda |{\int }_0^1{w}^2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ \le {\parallel w\parallel }_c\left[\left(2\mathcal{V}+1\right)\left(|\alpha |+|\beta |+\parallel v_1^{+}{\parallel }_{1,q}\right)+{\mathcal{V}}^2\right]+\mathcal{W}\mathcal{V}\sqrt{2\mathcal{V}+1}\\ \\ \le {\parallel w\parallel }_c\left[{\mathcal{V}}_{\alpha ,\beta ,v_1^{+}}\left(2\mathcal{V}+1\right)+{\mathcal{V}}^2\right]+\mathcal{W}\mathcal{V}\sqrt{2\mathcal{V}+1}.\end{array}$$

From $\mathrm{Re}\lambda \le 0$, one sees that

$$\mathrm{Re}\lambda \ge -{\displaystyle \frac{2}{{\epsilon }_2}}\left\{{\parallel w\parallel }_c\left[{\mathcal{V}}_{\alpha ,\beta ,v_1^{+}}\left(2\mathcal{V}+1\right)+{\mathcal{V}}^2\right]+\mathcal{W}\mathcal{V}\sqrt{2\mathcal{V}+1}\right\}.$$

Separating the imaginary parts of (20) yields

$$\mathrm{Im}\lambda {\int }_0^1{w}^2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x={\int }_0^{1}w\left(x\right)v_2\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+\mathrm{Im}\left({\int }_0^1{D}_{q}w\left(x\right)D_q\varphi \left(x\right)\overline{\varphi \left(x\right)}{\mathrm{d}}_{q}x\right).$$

It follows from $v_2^{+}=max\{v_2,0\}$, (24), and Lemma 5 that

$$\begin{array}{ll}|\mathrm{Im}\lambda |{\displaystyle \frac{{\epsilon }_2}{2}}& \le |\mathrm{Im}\lambda |{\int }_0^1{w}^2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ & \le {\int }_0^{1}w\left(x\right)v_2^{+}{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+{\int }_0^1|D_{q}w\left(x\right)D_q\varphi \left(x\right)\overline{\varphi \left(x\right)}|{\mathrm{d}}_{q}x\\ & \le {\parallel w\parallel }_c\left(2\mathcal{V}+1\right){\parallel v_2^{+}\parallel }_{1,q}+\mathcal{W}\mathcal{V}\sqrt{2\mathcal{V}+1}.\end{array}$$

Therefore, the inequality of imaginary parts of $\lambda$ immediately holds. □

Now, we consider the eigenvalue problem (17). Similar to Lemma 3, we have the following.

Lemma 6.

Let λ and ϕ be defined as above, with $\mathrm{Re}\lambda >0$. Assume that $\widehat{w}\ge 1/w\left(qx\right)$ for $x\in [0,1]$. Then,

$${\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\Delta }^2,{\parallel \varphi \parallel }_{\infty }^2\le 2\Delta +1,$$

where Δ is defined in (8).

Proof. 

By multiplying both sides of Equation (9) by $w\left(x\right)\overline{\varphi \left(x\right)}$ and integrating over the interval $[0,1]$, similar to the proofs of Lemma 3 and Theorem 3, we have

$$\begin{array}{l}(\lambda -\mathrm{Re}\lambda ){\int }_0^1{w}^2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ ={\beta w\left(1\right)|\varphi \left(1\right)|}^2-{\alpha w\left(0\right)|\varphi \left(0\right)|}^2+{\int }_0^{1}w\left(qx\right)|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ +{\int }_0^1{D}_{q}w\left(x\right)D_q\varphi \left(x\right)\overline{\varphi \left(x\right)}{\mathrm{d}}_{q}x+{\int }_0^1[v\left(x\right)-\mathrm{Re}\lambda w\left(x\right)]w\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x.\end{array}$$

(25)

It follows from $\mathrm{Re}(\lambda -\mathrm{Re}\lambda )=0$ that

$$\begin{array}{ll}0& =\mathrm{Re}(\lambda -\mathrm{Re}\lambda ){\int }_0^1{w}^2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ & ={\beta w\left(1\right)|\varphi \left(1\right)|}^2-{\alpha w\left(0\right)|\varphi \left(0\right)|}^2+{\int }_0^{1}w\left(qx\right)|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ & +\mathrm{Re}\left({\int }_0^1{D}_{q}w\left(x\right)D_q\varphi \left(x\right)\overline{\varphi \left(x\right)}{\mathrm{d}}_{q}x\right)+{\int }_0^1[v_1\left(x\right)-\mathrm{Re}\lambda w\left(x\right)]w\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x.\end{array}$$

This, together with ${|\varphi \left(x\right)|}^2\le 2{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+1$ in (13) of Lemma 1 and ${\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}={\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}+\mathrm{Re}\lambda {\parallel w\parallel }_{1,q}$, similar to the proof of Lemma 3, we have

$$\begin{array}{r}{\int }_0^{1}w\left(qx\right)|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le \left({2\parallel w\parallel }_c{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}+\mathcal{W}\right){\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}\\ +{\displaystyle \frac{\mathcal{W}}{2}}{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+{\parallel w\parallel }_c{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}+{\displaystyle \frac{\mathcal{W}}{2}}.\end{array}$$

From $\widehat{w}\ge 1/w\left(qx\right)$, one sees that

$$\begin{array}{l}{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x={\int }_0^1{\displaystyle \frac{1}{w\left(qx\right)}}w\left(qx\right)|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le \widehat{w}{\int }_0^{1}w\left(qx\right){|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ \le \widehat{w}\left[\left({2\parallel w\parallel }_c{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}+\mathcal{W}\right){\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+{\displaystyle \frac{\mathcal{W}}{2}}{\int }_0^1|D_q{\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+{\parallel w\parallel }_c{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}+{\displaystyle \frac{\mathcal{W}}{2}}\right].\end{array}$$

Using $\widehat{\Delta }=\widehat{w}\left({\parallel w\parallel }_c{\mathcal{V}}_{\alpha ,\beta ,v_1^{-}}^{\lambda ,w}+{\displaystyle \frac{\mathcal{W}}{2}}\right)/\left(1-{\displaystyle \frac{\widehat{w}\mathcal{W}}{2}}\right)$ in (8), one sees that

$${\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le 2\widehat{\Delta }{\left({\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\right)}^{1/2}+\widehat{\Delta },$$

and then, ${\int }_0^1{|D_q\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\le {\Delta }^2.$ Therefore, ${\parallel \varphi \parallel }_{\infty }^2\le 2\Delta +1$. The proof is completed. □

It follows from (19) that the $\mathrm{mes}\left(\mathsf{\Omega }\left({\epsilon }_2\right)\right)\to 0$ as ${\epsilon }_2\to 0$, and thus, we can set $\mathrm{mes}\mathsf{\Omega }\left({\epsilon }_2\right)\le 1/\left[2\left(2\Delta +1\right)\right]$. Similar to the proof of Lemma 5, we have the following.

Lemma 7.

Let λ and ϕ be defined as above, with $\mathrm{Re}\lambda >0$. Then,

$${\int }_0^1{w}^2\left(x\right){|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\ge {\epsilon }_2/2,$$

where ${\epsilon }_2$ satisfies $\mathrm{mes}\left(\mathsf{\Omega }\left({\epsilon }_2\right)\right)\le 1/\left[2\left(2\Delta +1\right)\right]$.

By applying Lemmas 6 and 7, we can prove Theorem 4 in the following.

Proof of Theorem 4.

Separating the imaginary parts of (25) in Lemma 6 yields

$$\begin{array}{ll}|\mathrm{Im}\lambda |{\displaystyle \frac{{\epsilon }_2}{2}}& \le |\mathrm{Im}\lambda |{\int }_0^1{w}^2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x=|\mathrm{Im}(\lambda -\mathrm{Re}\lambda )|{\int }_0^1{w}^2{\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x\\ & \le {\int }_0^1{v}_2^{+}{\left(x\right)w\left(x\right)|\varphi \left(x\right)|}^2{\mathrm{d}}_{q}x+\mathrm{Im}\left({\int }_0^1{D}_{q}w\left(x\right)D_q\varphi \left(x\right)\overline{\varphi \left(x\right)}{\mathrm{d}}_{q}x\right)\\ & \le {\parallel w\parallel }_c\left(2\Delta +1\right){\parallel v_2^{+}\parallel }_{1,q}+\mathcal{W}\Delta \sqrt{2\Delta +1}.\end{array}$$

As a result, the above inequality yields the inequalities in Theorem 4. □

4. Conclusions

This paper considers the eigenvalue problems of complex q-Sturm–Liouville boundary value problems (1) and (2). We obtained the bounds of the non-real eigenvalues of these problems, including the lower bounds on the real parts and the upper bounds on the imaginary parts and the real part of the eigenvalue of complex q-Sturm–Liouville boundary value problems in terms of the coefficients of (1) and the real part of the eigenvalue.