Classification and criteria of the limit cases for singular second-order linear equations on time scales

Abstract

This paper is concerned with classification and criteria of the limit cases for singular second-order linear equations on time scales. By the different cases of the limiting set, the equations are divided into two cases: the limit-point and limit-circle cases just like the continuous and discrete cases. Several sufficient conditions for the limit-point cases are established. It is shown that the limit cases are invariant under a bounded perturbation. These results unify the existing ones of second-order singular differential and difference equations.

1 Introduction

In this paper, we consider classification and criteria of the limit cases for the following singular second-order linear equation:

$$-{(p(t)y^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+q(t)y^{\sigma }(t)=\lambda w(t)y^{\sigma }(t),t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T},$$

(1.1)

where $p^{\mathrm{\Delta }}$, q, and w are real and piecewise continuous functions on $[\rho (0),\mathrm{\infty })\cap \mathbb{T}$, $p(t)\ne 0$ and $w(t)>0$ for all $t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$; $\lambda \in \mathbb{C}$ is the spectral parameter; $\mathbb{T}$ is a time scale with $\rho (0)\in \mathbb{T}$ and $sup\mathbb{T}=\mathrm{\infty }$; $\sigma (t)$ and $\rho (t)$ are the forward and backward jump operators in $\mathbb{T}$; $y^{\mathrm{\Delta }}$ is the Δ-derivative of y; and $y^{\sigma }(t):=y(\sigma (t))$.

The spectral problems of symmetric linear differential operators and difference operators can both be divided into two cases. Those defined over finite closed intervals with well-behaved coefficients are called regular. Otherwise, they are called singular. In 1910, Weyl [1] gave a dichotomy of the limit-point and limit-circle cases for the following singular second-order linear differential equation:

$$-y^{″}(t)+q(t)y(t)=\lambda y(t),t\in [0,\mathrm{\infty }),$$

(1.2)

where q is a real and continuous function on $[0,\mathrm{\infty })$, $\lambda \in \mathbb{C}$ is the spectral parameter. Later, Titchmarsh, Coddington, Levinson etc. developed his results and established the Weyl-Titchmarsh theory [2, 3]. Their work has been greatly developed and generalized to higher-order differential equations and Hamiltonian systems, and a classification and some criteria of limit cases were formulated [4–9]. Singular spectral problems of self-adjoint scalar second-order difference equations over infinite intervals were firstly studied by Atkinson [10]. His work was followed by Hinton, Jirari etc. [11, 12]. In 2001, some sufficient and necessary conditions and several criteria of the limit-point and limit-circle cases were obtained for the following formally self-adjoint second-order linear difference equations with real coefficients [13]:

$$-\mathrm{\nabla }(p(n)\mathrm{\Delta }y(n))+q(n)y(n)=\lambda w(n)y(n),n\in {\{n\}}_{n=0}^{\mathrm{\infty }},$$

(1.3)

where ∇ and Δ are the backward and forward difference operators respectively, namely $\mathrm{\nabla }y(n):=y(n)-y(n-1)$ and $\mathrm{\Delta }y(n):=y(n+1)-y(n)$; $p(n)$, $q(n)$, and $w(n)$ are real numbers with $w(n)>0$ for $n\in [0,\mathrm{\infty })$ and $p(n)\ne 0$ for $n\in [-1,\mathrm{\infty })$; λ is a complex spectral parameter. In 2006, Shi [14] established the Weyl-Titchmarsh theory of discrete linear Hamiltonian systems. Later, several sufficient conditions and sufficient and necessary conditions for the limit-point and limit-circle cases were established for the singular second-order linear difference equation with complex coefficients (see [15]).

In the past twenty years, a lot of effort has been put into the study of regular spectral problems on time scales (see [16–23]). But singular spectral problems have started to be considered only quite recently. In 2007, we employed Weyl’s method to divide the following singular second-order linear equations on time scales into two cases: limit-point and limit-circle cases [24]:

$$-y^{\mathrm{\Delta }\mathrm{\Delta }}(t)+q(t)y^{\sigma }(t)=\lambda y^{\sigma }(t),t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T},$$

where q is real and continuous on $[\rho (0),\mathrm{\infty })\cap \mathbb{T}$, $\lambda \in \mathbb{C}$ is the spectral parameter. By using the similar method, Huseynov [25] studied the classification of limit cases for the following singular second-order linear equations on time scales:

$$-{(p(t)y^{\mathrm{\Delta }}(t))}^{\mathrm{\nabla }}+q(t)y(t)=\lambda y(t),t\in (a,\mathrm{\infty })\cap \mathbb{T},$$

as well as of the form

$$-{(p(t)y^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+q(t)y^{\sigma }(t)=\lambda y^{\sigma }(t),t\in [a,\mathrm{\infty })\cap \mathbb{T},$$

(1.4)

where $p^{\mathrm{\nabla }}$ (or $p^{\mathrm{\Delta }}$) and q are real and piecewise continuous functions in $(a,\mathrm{\infty })\cap \mathbb{T}$ (or $[a,\mathrm{\infty })\cap \mathbb{T}$), $p(t)\ne 0$ for all t, and $\lambda \in \mathbb{C}$ is the spectral parameter. Obviously, let $w(t)\equiv 1$ and $\rho (0)=a$, then (1.1) is the same as (1.4). In 2010, by using the properties of the Weyl matrix disks, Sun [26] established the Weyl-Titchmarsh theory of Hamiltonian systems on time scales. It has been found that the second-order singular differential and difference equations can be divided into limit-point and limit-circle cases. We wonder whether the classification of the limit cases holds on time scales. In the present paper, we extend these results obtained in [24] to Eq. (1.1) and establish several sufficient conditions and sufficient and necessary conditions for the limit-point and limit-circle cases for Eq. (1.1).

This paper is organized as follows. In Section 2, some basic concepts and a fundamental theory about time scales are introduced. In Section 3, a family of nested circles which converge to a limiting set is constructed. The dichotomy of the limit-point and limit-circle cases for singular second-order linear equations on time scales is given by the geometric properties of the limiting set. Finally, several criteria of the limit-point case are established, and the invariance of the limit cases is shown under a bounded perturbation for the potential function q in Section 4.

2 Preliminaries

In this section, some basic concepts and fundamental results on time scales are introduced.

Let $\mathbb{T}\subset \mathbb{R}$ be a non-empty closed set. The forward and backward jump operators $\sigma ,\rho :\mathbb{T}\to \mathbb{T}$ are defined by

$$\sigma (t):=inf\{s\in \mathbb{T}:s>t\},\rho (t):=sup\{s\in \mathbb{T}:s<t\},$$

respectively, where $inf\mathrm{\varnothing }=sup\mathbb{T}$, $sup\mathrm{\varnothing }=inf\mathbb{T}$. A point $t\in \mathbb{T}$ is called right-scattered, right-dense, left-scattered, and left-dense if $\sigma (t)>t,\sigma (t)=t,\rho (t)<t$, and $\rho (t)=t$ separately. Denote ${\mathbb{T}}^k:=\mathbb{T}$ if $\mathbb{T}$ is unbounded above and ${\mathbb{T}}^k:=\mathbb{T}\setminus (\rho (max\mathbb{T}),max\mathbb{T}]$ otherwise. The graininess $\mu :\mathbb{T}\to [0,\mathrm{\infty })$ is defined by

$$\mu (t):=\sigma (t)-t.$$

Let f be a function defined on $\mathbb{T}$. f is said to be Δ-differentiable at $t\in {\mathbb{T}}^k$ provided there exists a constant a such that, for any $\epsilon >0$, there is a neighborhood U of t (i.e., $U=(t-\delta ,t+\delta )\cap \mathbb{T}$ for some $\delta >0$) with

$$|f(\sigma (t))-f(s)-a(\sigma (t)-s)|\le \epsilon |\sigma (t)-s|\text{for all }s\in U.$$

In this case, denote $f^{\mathrm{\Delta }}(t):=a$. If f is Δ-differentiable for every $t\in {\mathbb{T}}^k$, then f is said to be Δ-differentiable on $\mathbb{T}$. If f is Δ-differentiable at $t\in {\mathbb{T}}^k$, then

$$f^{\mathrm{\Delta }}(t)=\{\begin{array}{cc}{lim}_{\stackrel{s\to t}{s\in \mathbb{T}}}\frac{f(t)-f(s)}{t-s},\hfill & \text{if }\mu (t)=0,\hfill \\ \frac{f(\sigma (t))-f(t)}{\mu (t)},\hfill & \text{if }\mu (t)>0.\hfill \end{array}$$

(2.1)

If $F^{\mathrm{\Delta }}(t)=f(t)$ for all $t\in {\mathbb{T}}^k$, then $F(t)$ is called an anti-derivative of f on $\mathbb{T}$. In this case, define the Δ-integral by

$${\int }_s^{t}f(\tau )\mathrm{\Delta }\tau =F(t)-F(s)\text{for all }s,t\in \mathbb{T}.$$

For convenience, we introduce the following results ([[27], Chapter 1] and [[28], Chapter 1]), which are useful in this paper.

Lemma 2.1 Let $f,g:\mathbb{T}\to \mathbb{R}$ and $t\in {\mathbb{T}}^k$.

• (i) If f is Δ-differentiable at t, then f is continuous at t.

• (ii) If f and g are Δ-differentiable at t, then fg is Δ-differentiable at t and

  $${(fg)}^{\mathrm{\Delta }}(t)=f^{\sigma }(t)g^{\mathrm{\Delta }}(t)+f^{\mathrm{\Delta }}(t)g(t)=f^{\mathrm{\Delta }}(t)g^{\sigma }(t)+f(t)g^{\mathrm{\Delta }}(t).$$

• (iii) If f and g are Δ-differentiable at t, and $f(t)f^{\sigma }(t)\ne 0$, then $f^{-1}g$ is Δ-differentiable at t and

  $${\left(g{f}^{-1}\right)}^{\mathrm{\Delta }}(t)=(g^{\mathrm{\Delta }}(t)f(t)-g(t)f^{\mathrm{\Delta }}(t)){(f^{\sigma }(t)f(t))}^{-1}.$$

A function f defined on $\mathbb{T}$ is said to be rd-continuous if it is continuous at every right-dense point in $\mathbb{T}$ and its left-sided limit exists at every left-dense point in $\mathbb{T}$. The set of rd-continuous functions $f:\mathbb{T}\to \mathbb{R}$ is denoted by $C_{rd}(\mathbb{T})=C_{rd}(\mathbb{T},\mathbb{R})$. The set of k th Δ-differentiable functions with rd-continuous k th derivative is denoted by $C_{rd}^k(\mathbb{T})=C_{rd}^k(\mathbb{T},\mathbb{R})$.

Lemma 2.2 If f, g are rd-continuous functions on $\mathbb{T}$, then

• (i) $f^{\sigma }$ is rd-continuous and f has an anti-derivative on $\mathbb{T}$;

• (ii) ${\int }_t^{\sigma (t)}f(\tau )\mathrm{\Delta }\tau =\mu (t)f(t)$ for all $t\in \mathbb{T}$.

• (iii) (Integration by parts) ${\int }_a^b{f}^{\sigma }(\tau )g^{\mathrm{\Delta }}(\tau )\mathrm{\Delta }\tau =f(b)g(b)-f(a)g(a)-{\int }_a^b{f}^{\mathrm{\Delta }}(\tau )g(\tau )\mathrm{\Delta }\tau$.

• (iv) (Hölder’s inequality [[29], Lemma 2.2(iv)]) Let $r,s\in \mathbb{T}$ with $r\le s$, then

  $${\int }_r^s|f(\tau )g(\tau )|\mathrm{\Delta }\tau \le {\left\{{\int }_r^s{|f(\tau )|}^p\mathrm{\Delta }\tau \right\}}^{\frac{1}{p}}{\left\{{\int }_r^s{|g(\tau )|}^q\mathrm{\Delta }\tau \right\}}^{\frac{1}{q}},$$

where $p>1$ and $q=p/(p-1)$.

Let

$$L_w^2(\rho (0),\mathrm{\infty }):=\{y^{\sigma }:[\rho (0),\mathrm{\infty })\to \mathbb{C}|{\int }_{\rho (0)}^{\mathrm{\infty }}w(t){|y^{\sigma }(t)|}^2\mathrm{\Delta }t<\mathrm{\infty }\}.$$

A function $g:\mathbb{T}\to \mathbb{R}$ is called regressive if

$$1+\mu (t)g(t)\ne 0\text{for all }t\in \mathbb{T}.$$

Higer [30] showed that for any given $t_0\in \mathbb{T}$ and for any given rd-continuous and regressive g, the initial value problem

$$y^{\mathrm{\Delta }}(t)=g(t)y(t),y(t_0)=1$$

has a unique solution

$$\begin{array}{r}{e}_g(t,t_0)=exp\left\{{\int }_{t_0}^t{\xi }_{\mu (\tau )}(g(\tau ))\mathrm{\Delta }\tau \right\},\\ {\xi }_h(z)=\{\begin{array}{cc}\frac{Log(1+hz)}{h},\hfill & \text{if }h\ne 0,\hfill \\ z,\hfill & \text{if }h=0.\hfill \end{array}\end{array}$$

(2.2)

Lemma 2.3 ([[27], Theorem 6.1])

Let $y,f\in C_{rd}(\mathbb{T})$ and $g\in {\mathcal{R}}^{+}:=\{g\in C_{rd}(\mathbb{T}):1+\mu (t)g(t)>0,t\in \mathbb{T}\}$. Then

$$y^{\mathrm{\Delta }}(t)\le g(t)y(t)+f(t),\mathrm{\forall }t\in \mathbb{T},$$

implies

$$y(t)\le y(t_0)e_g(t,t_0)+{\int }_{t_0}^t{e}_g(t,\sigma (\tau ))f(\tau )\mathrm{\Delta }\tau ,\mathrm{\forall }t\in \mathbb{T}.$$

We define the Wronskian by

$$W[x,y](t)=p(t)[x(t)y^{\mathrm{\Delta }}(t)-x^{\mathrm{\Delta }}(t)y(t)],x,y\in C_{rd}^2(\mathbb{T}).$$

(2.3)

The following result is a direct consequence of the Lagrange identity [[27], Theorem 4.30].

Lemma 2.4 Let x and y be any two solutions of (1.1). Then $W[x,y](t)$ is a constant in $[\rho (0),\mathrm{\infty })\cap \mathbb{T}$.

3 Classification

In this section, we focus on the classification of the limit cases for singular second-order linear equations on time scales.

Let $y_1(t,\lambda )$ and $y_2(t,\lambda )$ be the two solutions of (1.1) satisfying the following initial conditions:

respectively. Since their Wronskian is identically equal to 1, these two solutions form a fundamental solution system of (1.1). We form a linear combination of $y_1(t,\lambda )$ and $y_2(t,\lambda )$

$$y(t,\lambda ,m):=y_1(t,\lambda )+m{y}_2(t,\lambda ).$$

(3.1)

Let $b\in (\rho (0),\mathrm{\infty })\cap \mathbb{T}$, $k\in \mathbb{R}$, $\lambda =\mu +i\nu$ with $\nu \ne 0$, and let (3.1) satisfy

$$p(b)y^{\mathrm{\Delta }}(b,\lambda ,m)+ky(b,\lambda ,m)=0.$$

(3.2)

Then

$$m=-\frac{p(b)y_1^{\mathrm{\Delta }}(b,\lambda )+k{y}_1(b,\lambda )}{p(b)y_2^{\mathrm{\Delta }}(b,\lambda )+k{y}_2(b,\lambda )}.$$

(3.3)

It can be verified that the integral identity

$$[\overline{y(t,\lambda )}p(t)y^{\mathrm{\Delta }}(t,\lambda )]{|}_{t_1}^{t_2}-{\int }_{t_1}^{t_2}p(t){|y^{\mathrm{\Delta }}(t,\lambda )|}^2\mathrm{\Delta }t+{\int }_{t_1}^{t_2}[\lambda w(t)-q(t)]{|y^{\sigma }(t,\lambda )|}^2\mathrm{\Delta }t=0$$

(3.4)

holds for any solution $y(t,\lambda )$ of (1.1) and for any $t_1,t_2\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$. Setting $y(t,\lambda )=y_2(t,\lambda )$, $t_1=\rho (0)$, $t_2=b$ in (3.4) and taking its imaginary part, we obtain

$$\mathrm{\Im }[\overline{y_2(b,\lambda )}p(b)y_2^{\mathrm{\Delta }}(b,\lambda )]=-\nu {\int }_{\rho (0)}^{b}w(t){|y_2^{\sigma }(t,\lambda )|}^2\mathrm{\Delta }t.$$

(3.5)

So

$$\mathrm{\Im }\left(\frac{p(b)y_2^{\mathrm{\Delta }}(b,\lambda )}{y_2(b,\lambda )}\right)=\frac{\mathrm{\Im }[\overline{y_2(b,\lambda )}p(b)y_2^{\mathrm{\Delta }}(b,\lambda )]}{{|y_2(b,\lambda )|}^2}\ne 0.$$

It follows from (3.2) and $k\in \mathbb{R}$ that $k\ne -\frac{p(b)y_2^{\mathrm{\Delta }}(b,\lambda )}{y_2(b,\lambda )}$. Hence, the denominator in (3.3) is not equal to zero, and consequently, m is well defined.

Next, we will show that (3.3) describes a circle for any fixed b. It follows from (3.1) and (3.2) that

$$\overline{y(\rho (0),\lambda ,m)}p(\rho (0))y^{\mathrm{\Delta }}(\rho (0),\lambda ,m)=m$$

and

$$\overline{y(b,\lambda ,m)}p(b)y^{\mathrm{\Delta }}(b,\lambda ,m)=-k{|y(b,\lambda ,m)|}^2\in \mathbb{R}.$$

By (3.4) and the above two relations, we have

$$\mathrm{\Im }(m)=\nu {\int }_{\rho (0)}^{b}w(t){|y^{\sigma }(t,\lambda ,m)|}^2\mathrm{\Delta }t,$$

(3.6)

which implies that m lies in the upper half-plane if $\nu >0$. It follows from (3.2) that

$$k=-\frac{p(b)y^{\mathrm{\Delta }}(b,\lambda ,m)}{y(b,\lambda ,m)},$$

which, together with $k\in \mathbb{R}$, yields that

$$p(b)[y^{\mathrm{\Delta }}(b,\lambda ,m)\overline{y(b,\lambda ,m)}-\overline{y^{\mathrm{\Delta }}(b,\lambda ,m)}y(b,\lambda ,m)]=0.$$

It is equivalent to

$$W[y,\overline{y}](b,\lambda ,m)=0.$$

(3.7)

By using (3.1), (3.7) can be expanded as

$${|m|}^{2}W[y_2,\overline{y_2}](b,\lambda )+mW[y_2,\overline{y_1}](b,\lambda )+\overline{m}W[y_1,\overline{y_2}](b,\lambda )+W[y_1,\overline{y_1}](b,\lambda )=0.$$

(3.8)

Moreover, setting $m=u+iv$, we have

$$W[y_2,\overline{y_2}](b,\lambda )=2iA,W[y_1,\overline{y_1}](b,\lambda )=2iD,-W[y_2,\overline{y_1}](b,\lambda )=B+iC.$$

(3.9)

It follows from the last relation in (3.9) that we have $W[y_1,\overline{y_2}](b,\lambda )=B-iC$. By using (2.3) and (3.9), it can be verified that

(3.10)

It follows from the first relation in (3.9) and (3.5) that we have $A=\nu {\int }_{\rho (0)}^{b}w(t){|y_2^{\sigma }(t,\lambda )|}^2\mathrm{\Delta }t\ne 0$. Then (3.8) becomes

$${(u-\frac{C}{2A})}^2+{(v-\frac{B}{2A})}^2=\frac{B^2+C^2-4AD}{4A^2},$$

(3.11)

which implies that (3.3) forms a circle $C_b$ as k varies. It is evident that the center of $C_b$ is

$$z_0=\frac{C+iB}{2A}=-\frac{B-iC}{2iA}=-\frac{W[y_1,\overline{y_2}](b,\lambda )}{W[y_2,\overline{y_2}](b,\lambda )}.$$

It follows from Lemma 2.4 and (3.10) that

$$B^2+C^2-4AD={|W[y_1,y_2](b,\lambda )|}^2={|W[y_1,y_2](\rho (0),\lambda )|}^2=1.$$

From (3.11), (3.9), (2.3), and (3.5) we have that the radius of $C_b$ is

$$\begin{array}{rcl}{r}_b& =& |\frac{B^2+C^2-4AD}{4A^2}{|}^{\frac{1}{2}}\\ =& {|2iA|}^{-1}\\ =& {|W[y_2,\overline{y_2}](b,\lambda )|}^{-1}\\ =& {[2|\nu |{\int }_{\rho (0)}^{b}w(t){|y_2^{\sigma }(t,\lambda )|}^2\mathrm{\Delta }t]}^{-1}.\end{array}$$

(3.12)

Let $\overline{C_b}$ denote the closed disk bounded by $C_b$. We are going to show that the circle sequence $\{\overline{C_b}\}(\rho (0)<b<\mathrm{\infty })$ is nested.

Set

$$U+iV=\nu {\int }_{\rho (0)}^{b}w(t)y_1^{\sigma }(t,\lambda )\overline{y_2^{\sigma }(t,\lambda )}\mathrm{\Delta }t.$$

From the first relation in (3.9), we have

$$A=\nu {\int }_{\rho (0)}^{b}w(t){|y_2^{\sigma }(t,\lambda )|}^2\mathrm{\Delta }t.$$

Similarly,

$$D=\nu {\int }_{\rho (0)}^{b}w(t){|y_1^{\sigma }(t,\lambda )|}^2\mathrm{\Delta }t.$$

So, it follows from (3.6) that

$$v=A(u^2+v^2)+2Uu+2Vv+D.$$

(3.13)

In the case of $\nu >0$, the point $m=u+iv$ is interior to the circle if $v>A(u^2+v^2)+2Uu+2Vv+D$. This shows that $m\in \overline{C_b}$ if and only if

$$\mathrm{\Im }(m)\ge \nu {\int }_{\rho (0)}^{b}w(t){|y^{\sigma }(t,\lambda ,m)|}^2\mathrm{\Delta }t.$$

Let $b_1,b_2\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$ with $b_1<b_2$ and consider the corresponding disks $\overline{C_{b_1}}$ and $\overline{C_{b_2}}$. For any $m\in \overline{C_{b_2}}$, we have

$$\mathrm{\Im }(m)\ge \nu {\int }_{\rho (0)}^{b_2}w(t){|y^{\sigma }(t,\lambda ,m)|}^2\mathrm{\Delta }t\ge \nu {\int }_{\rho (0)}^{b_1}w(t){|y^{\sigma }(t,\lambda ,m)|}^2\mathrm{\Delta }t.$$

Hence, $m\in \overline{C_{b_1}}$. This yields that $\overline{C_{b_2}}\subset \overline{C_{b_1}}$. Therefore, $\{\overline{C_b}\}$ is nested. Consequently, there are the following two alternatives:

• (1) $r_b\to 0$ as $b\to \mathrm{\infty }$. In this case there is one point $m=m(\lambda )$ which is common to all the disks $\overline{C_b}$, $b\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$. This is called the limit-point case. It follows from (3.12) that this case occurs if and only if

  $${\int }_{\rho (0)}^{\mathrm{\infty }}w(t){|y_2^{\sigma }(t,\lambda )|}^2\mathrm{\Delta }t=\mathrm{\infty }.$$

  (3.14)

• (2) $r_b\to r_{\mathrm{\infty }}>0$ as $b\to \mathrm{\infty }$. In this case there is a disk $\overline{C_{\mathrm{\infty }}}$ contained in all the disks $\overline{C_b}$, $b\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$. This is called the limit-circle case. It follows from (3.12) that this case occurs if and only if the integral in (3.14) is convergent, i.e., $y_2(\cdot ,\lambda )\in L_w^2(\rho (0),\mathrm{\infty })$.

Theorem 3.1 For every non-real $\lambda \in \mathbb{C}$, Eq. (1.1) has at least one non-trivial solution in $L_w^2(\rho (0),\mathrm{\infty })$.

Proof In the limit-circle case, it follows from the above discussion that $y_2(\cdot ,\lambda )\in L_w^2(\rho (0),\mathrm{\infty })$.

Next, we will show that $y_1(\cdot ,\lambda )+m(\lambda )y_2(\cdot ,\lambda )\in L_w^2(\rho (0),\mathrm{\infty })$ in the limit-point case. Let $\{b_n\}\subset \mathbb{T}$ with $0<b_n<b_{n+1}\to \mathrm{\infty }$ and choose any $m_n\in C_{b_n}$. Then $m_n\to m(\lambda )$ as $n\to \mathrm{\infty }$ and $y^{\sigma }(t,\lambda ,m_n)$ uniformly converges to $y^{\sigma }(t,\lambda ,m(\lambda ))$ on any finite interval $[\rho (0),\omega ]\cap \mathbb{T}$, $\omega \in \mathbb{T}$. Since the sequence $\{\mathrm{\Im }(m_n)\}$ is bounded from above and its upper bound is denoted by $y_0$, then for $b_n>\omega$,

$$y_0\ge \mathrm{\Im }(m_n)=\nu {\int }_{\rho (0)}^{b_n}w(t){|y^{\sigma }(t,\lambda ,m_n)|}^2\mathrm{\Delta }t\ge \nu {\int }_{\rho (0)}^{\omega }w(t){|y^{\sigma }(t,\lambda ,m_n)|}^2\mathrm{\Delta }t.$$

Hence, by the uniform convergence of $y^{\sigma }(t,\lambda ,m_n)$, we have

$$y_0\ge \nu {\int }_{\rho (0)}^{\omega }w(t){\left|y^{\sigma }(t,\lambda ,m(\lambda ))\right|}^2\mathrm{\Delta }t$$

for all ω. Therefore, $y(\cdot ,\lambda ,m(\lambda ))=y_1(\cdot ,\lambda )+m(\lambda )y_2(\cdot ,\lambda )\in L_w^2(\rho (0),\mathrm{\infty })$. This completes the proof. □

Remark 3.1 Similar to the proof of Theorem 3.1, it can be easily verified that $y(\cdot ,\lambda ,m)\in L_w^2(\rho (0),\mathrm{\infty })$ for any $m\in C_{\mathrm{\infty }}$ with $\mathrm{\Im }(\lambda )\ne 0$ in the limit-circle case. Clearly, $y(t,\lambda ,m)$ and $y_2(t,\lambda )$ are linearly independent. Hence, all the solutions of Eq. (1.1) belong to $L_w^2(\rho (0),\mathrm{\infty })$ for any $\lambda \in \mathbb{C}$ with $\mathrm{\Im }(\lambda )\ne 0$ in the limit-circle case.

Remark 3.2 It follows from (3.14) and Theorem 3.1 that Eq. (1.1) has exactly one linearly independent solution in $L_w^2(\rho (0),\mathrm{\infty })$ in the limit point case for any $\lambda \in \mathbb{C}$ with $\mathrm{\Im }(\lambda )\ne 0$.

Theorem 3.2 If Eq. (1.1) has two linearly independent solutions in $L_w^2(\rho (0),\mathrm{\infty })$ for some ${\lambda }_0\in \mathbb{C}$, then this property holds for all $\lambda \in \mathbb{C}$.

Proof Suppose that Eq. (1.1) has two linearly independent solutions in $L_w^2(\rho (0),\mathrm{\infty })$ for $\lambda ={\lambda }_0\in \mathbb{C}$. Then $y_1(t,{\lambda }_0)$ and $y_2(t,{\lambda }_0)$ are in $L_w^2(\rho (0),\mathrm{\infty })$. For briefness, denote

$$u_1(t)=y_1(t,{\lambda }_0),u_2(t)=y_2(t,{\lambda }_0).$$

For any $\lambda \in \mathbb{C}$, let $v(t)$ be an arbitrary non-trivial solution of (1.1), and let $u(t)$ be the solution of (1.1) with $\lambda ={\lambda }_0$ and with the initial values

$$u(a)=v(a),u^{\mathrm{\Delta }}(a)=v^{\mathrm{\Delta }}(a),a\in (0,\mathrm{\infty })\cap \mathbb{T}.$$

From the variation of constants [[27], Theorem 3.73], we have

$$v(t)=u(t)+(\lambda -{\lambda }_0){\int }_a^t[u_1(t)u_2^{\sigma }(s)-u_2(t)u_1^{\sigma }(s)]w(s)v^{\sigma }(s)\mathrm{\Delta }s,t\in [a,\mathrm{\infty })\cap \mathbb{T}.$$

(3.15)

Replacing t with $\sigma (t)$ in (3.15) and using (ii) of Lemma 2.2, we obtain

which implies by the Hölder inequality in Lemma 2.2 that

$$\begin{array}{rcl}|w^{\frac{1}{2}}(t)v^{\sigma }(t)|& \le & |w^{\frac{1}{2}}(t)u^{\sigma }(t)|+|\lambda -{\lambda }_0||w^{\frac{1}{2}}(t)u_1^{\sigma }(t)|\\ \times {[{\int }_a^{t}w(s){|u_2^{\sigma }(s)|}^2\mathrm{\Delta }s{\int }_a^{t}w(s){|v^{\sigma }(s)|}^2\mathrm{\Delta }s]}^{\frac{1}{2}}\\ +|\lambda -{\lambda }_0||w^{\frac{1}{2}}(t)u_2^{\sigma }(t)|{[{\int }_a^{t}w(s){|u_1^{\sigma }(s)|}^2\mathrm{\Delta }s{\int }_a^{t}w(s){|v^{\sigma }(s)|}^2\mathrm{\Delta }s]}^{\frac{1}{2}}.\end{array}$$

It follows from the inequality

$${(A+B+C)}^2\le 3(A^2+B^2+C^2),$$

where A, B, C are non-negative numbers, that

$$\begin{array}{rcl}\frac{1}{3}w(t){|v^{\sigma }(t)|}^2& \le & w(t){|u^{\sigma }(t)|}^2+{|\lambda -{\lambda }_0|}^2[w(t){|u_1^{\sigma }(t)|}^2{\int }_a^{t}w(s){|u_2^{\sigma }(s)|}^2\mathrm{\Delta }s\\ +w(t){|u_2^{\sigma }(t)|}^2{\int }_a^{t}w(s){|u_1^{\sigma }(s)|}^2\mathrm{\Delta }s]{\int }_a^{t}w(s){|v^{\sigma }(s)|}^2\mathrm{\Delta }s.\end{array}$$

Integrating the two sides of the above inequality with respect to t from a to $\tau \in (a,\mathrm{\infty })\cap \mathbb{T}$, we get

$$\begin{array}{rcl}\frac{1}{3}{\int }_a^{\tau }w(t){|v^{\sigma }(t)|}^2\mathrm{\Delta }t& \le & {\int }_a^{\tau }w(t){|u^{\sigma }(t)|}^2\mathrm{\Delta }t\\ +{|\lambda -{\lambda }_0|}^2{\int }_a^{\tau }[w(t){|u_1^{\sigma }(t)|}^2{\int }_a^{t}w(s){|u_2^{\sigma }(s)|}^2\mathrm{\Delta }s\\ +w(t){|u_2^{\sigma }(t)|}^2{\int }_a^{t}w(s){|u_1^{\sigma }(s)|}^2\mathrm{\Delta }s]{\int }_a^{t}w(s){|v^{\sigma }(s)|}^2\mathrm{\Delta }s\mathrm{\Delta }t,\end{array}$$

which yields that

Hence,

(3.16)

The constant a can be chosen in advance so large that

$$6{|\lambda -{\lambda }_0|}^2{\int }_a^{\mathrm{\infty }}w(t){|u_1^{\sigma }(t)|}^2\mathrm{\Delta }t{\int }_a^{\mathrm{\infty }}w(t){|u_2^{\sigma }(t)|}^2\mathrm{\Delta }t<1.$$

It follows from (3.16) that $v\in L_w^2(a,\mathrm{\infty })$ and hence $v\in L_w^2(\rho (0),\mathrm{\infty })$. Therefore, all the solutions of Eq. (1.1) are in $L_w^2(\rho (0),\mathrm{\infty })$. The proof is complete. □

At the end of this section, from the above discussions we present the classification of the limit cases for singular second-order linear equations over the infinite interval $[\rho (0),\mathrm{\infty })\cap \mathbb{T}$ on time scales.

Definition 3.1 If Eq. (1.1) has only one linear independent solution in $L_w^2(\rho (0),\mathrm{\infty })$ for some $\lambda \in \mathbb{C}$, then Eq. (1.1) is said to be in the limit-point case at $t=\mathrm{\infty }$. If Eq. (1.1) has two linear independent solutions in $L_w^2(\rho (0),\mathrm{\infty })$ for some $\lambda \in \mathbb{C}$, then Eq. (1.1) is said to be in the limit-circle case at $t=\mathrm{\infty }$.

4 Several criteria of the limit-point and limit-circle cases

In this section, we establish several criteria of the limit-point and limit-circle cases for Eq. (1.1).

We first give two criteria of the limit-point case.

Theorem 4.1 Let $w(t)\equiv 1$ and $p(t)>0$ for all $t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$. If there exists a positive Δ-differentiable function $M(t)$ on $[a,\mathrm{\infty })\cap \mathbb{T}$ for some $a\ge \rho (0)$ and two positive constants $k_1$ and $k_2$ such that for all $t\in [a,\mathrm{\infty })\cap \mathbb{T}$,

• (i) $q(t)\ge -k_1{M}^{\sigma }(t)$,

• (ii) $p^{\frac{1}{2}}(t)|M^{\mathrm{\Delta }}(t)|{(M(t))}^{-1}{(M^{\sigma }(t))}^{-\frac{1}{2}}\le k_2$,

• (iii) ${\int }_a^{\mathrm{\infty }}{(p(t)M^{\sigma }(t))}^{-\frac{1}{2}}\mathrm{\Delta }t=\mathrm{\infty }$,

then Eq. (1.1) is in the limit-point case at $t=\mathrm{\infty }$.

Proof Suppose that Eq. (1.1) is in the limit-circle case at $t=\mathrm{\infty }$. By Theorem 3.2, all the solutions of

$$-{(p(t)y^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+q(t)y^{\sigma }(t)=0,t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$$

(4.1)

are in $L_w^2(\rho (0),\mathrm{\infty })$. Let $y_1(t)$ and $y_2(t)$ be the solutions of (4.1) satisfying the following initial conditions:

$$y_1(\rho (0))=p(\rho (0))y_2^{\mathrm{\Delta }}(\rho (0))=0,p(\rho (0))y_1^{\mathrm{\Delta }}(\rho (0))=y_2(\rho (0))=1.$$

(4.2)

It is evident that $y_1(t)$ and $y_2(t)$ are two linearly independent solutions of (4.1) in $L_w^2(\rho (0),\mathrm{\infty })$. By Lemma 2.4, $W[y_1,y_2](t)\equiv 1$ for all $t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$. Hence, we have

It follows from the Hölder inequality and assumption (iii) that

$${\int }_a^{\mathrm{\infty }}\frac{p(\tau ){(y_1^{\mathrm{\Delta }}(\tau ))}^2}{M^{\sigma }(\tau )}\mathrm{\Delta }\tau \text{or}{\int }_a^{\mathrm{\infty }}\frac{p(\tau ){(y_2^{\mathrm{\Delta }}(\tau ))}^2}{M^{\sigma }(\tau )}\mathrm{\Delta }\tau$$

are divergent. Suppose

$${\int }_a^{\mathrm{\infty }}\frac{p(\tau ){(y_1^{\mathrm{\Delta }}(\tau ))}^2}{M^{\sigma }(\tau )}\mathrm{\Delta }\tau =\mathrm{\infty }.$$

From (4.1) and assumption (i), we have

$$\begin{array}{rcl}{\int }_a^t\frac{y_1^{\sigma }(\tau ){[p(\tau )y_1^{\mathrm{\Delta }}(\tau )]}^{\mathrm{\Delta }}}{M^{\sigma }(\tau )}\mathrm{\Delta }\tau & =& {\int }_a^t\frac{q(\tau ){(y_1^{\sigma }(\tau ))}^2}{M^{\sigma }(\tau )}\mathrm{\Delta }\tau \\ \ge & -k_1{\int }_a^t{(y_1^{\sigma }(\tau ))}^2\mathrm{\Delta }\tau .\end{array}$$

(4.3)

Applying integration by parts in Lemma 2.2, by (iii) in Lemma 2.1, we get

(4.4)

Again applying the Hölder inequality, from condition (ii), we have

(4.5)

where

$$H(t):={\int }_a^t\frac{p(\tau ){(y_1^{\mathrm{\Delta }}(\tau ))}^2}{M^{\sigma }(\tau )}\mathrm{\Delta }\tau .$$

Since

$${\int }_a^{\mathrm{\infty }}{(y_1^{\sigma }(\tau ))}^2\mathrm{\Delta }\tau >{\int }_a^t{(y_1^{\sigma }(\tau ))}^2\mathrm{\Delta }\tau ,$$

it follows from (4.3)-(4.5) that

It follows from the assumption that $H(t)\to \mathrm{\infty }$ as $t\to \mathrm{\infty }$. From the above relation and $p(t)>0$ for all $t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$, we have that $y_1(t)y_1^{\mathrm{\Delta }}(t)$ is ultimately positive. Therefore, $y_1(t)\nrightarrow 0$ as $t\to \mathrm{\infty }$; and consequently, $y_1(t)$ does not belong to $L_w^2(\rho (0),\mathrm{\infty })$. This contradicts the assumption that all the solutions of (4.1) are in $L_w^2(\rho (0),\mathrm{\infty })$. Then Eq. (4.1) has at least one non-trivial solution outside of $L_w^2(\rho (0),\mathrm{\infty })$. It follows from Theorem 3.2 that Eq. (1.1) is in the limit-point case at $t=\mathrm{\infty }$. This completes the proof. □

Remark 4.1 Since $\mathbb{R}$ and $\mathbb{N}$ are two special time scales, Theorem 4.1 not only contains the criterion of the limit-point case for second-order differential equations [[5], Chapter 9, Theorem 2.4], but also the criterion of the limit-point case for second-order difference equation (1.3) [[15], Theorem 3.3].

The following corollary is a direct consequence of Theorem 4.1 by setting $M(t)\equiv 1$ for $t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$.

Corollary 4.1 If $w(t)\equiv 1$, $p(t)>0$, $q(t)$ is bounded below in $[\rho (0),\mathrm{\infty })\cap \mathbb{T}$, and ${\int }_{\rho (0)}^{\mathrm{\infty }}{(p(t))}^{-\frac{1}{2}}\mathrm{\Delta }t=\mathrm{\infty }$, then Eq. (1.1) is in the limit-point case at $t=\mathrm{\infty }$.

Theorem 4.2 If

$${\int }_{\rho (0)}^{\mathrm{\infty }}\frac{{\mu }^{\sigma }(t){[w(t)w^{\sigma }(t)]}^{\frac{1}{2}}}{|p^{\sigma }(t)|}\mathrm{\Delta }t=\mathrm{\infty },$$

(4.6)

then Eq. (1.1) is in the limit-point case at $t=\mathrm{\infty }$.

Proof On the contrary, suppose that Eq. (1.1) is in the limit-circle case at $t=\mathrm{\infty }$. Let $y_1(t)$ and $y_2(t)$ be two linearly independent solutions of (1.1) in $L_w^2(\rho (0),\mathrm{\infty })$ satisfying the initial conditions (4.2). By Lemma 2.4, we have

$$W[y_1,y_2](t)=W[y_1,y_2](\rho (0))\equiv 1,t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T},$$

which, together with (2.1), implies that

So, we get

$$|y_1(t)||y_2^{\sigma }(t)|+|y_2(t)||y_1^{\sigma }(t)|\ge \frac{\mu (t)}{|p(t)|},t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T},$$

which implies

(4.7)

where $y^{{\sigma }^2}(t)=y^{\sigma }(\sigma (t))$. By the Hölder inequality and the assumption that $y_1,y_2\in L_w^2(\rho (0),\mathrm{\infty })$, one has

Hence, it follows from (4.7) that

$${\int }_{\rho (0)}^{\mathrm{\infty }}\frac{{\mu }^{\sigma }(t){[w(t)w^{\sigma }(t)]}^{\frac{1}{2}}}{|p^{\sigma }(t)|}\mathrm{\Delta }t<\mathrm{\infty },$$

which is a contradiction to the assumption (4.6). Therefore, Eq. (1.1) is in the limit-point case at $t=\mathrm{\infty }$. This completes the proof. □

Remark 4.2 Let $\mathbb{T}=\mathbb{N}$, Theorem 4.2 is the same as that obtained by Chen and Shi for second-order difference equations [[13], Corollary 3.1].

Next, we study the invariance of the limit cases under a bounded perturbation for the potential function q. Let $f(t)=M$ and $p(t)=f(t)$ in [[27], Theorem 2.4(i)]. It follows from [[27], Theorem 2.36(i)], [[27], Theorem 2.39(i)], and [[27], Theorem 2.4(i)] that we have the following lemma, which is useful in the subsequent discussion.

Lemma 4.1 (Gronwall’s inequality)

Let $y,f\in C_{rd}(\mathbb{T})$ be two non-negative functions on $[\rho (0),\mathrm{\infty })\cap \mathbb{T}$ and M be a non-negative constant. If

$$y(t)\le M+{\int }_{\rho (0)}^{t}f(\tau )y(\tau )\mathrm{\Delta }\tau \mathit{\text{for all}}t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T},$$

(4.8)

then

$$y(t)\le M{e}_f(t,\rho (0))\mathit{\text{for all}}t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T},$$

where $e_f(t,s)$ is defined as in (2.2).

The following result shows that if Eq. (1.1) is in the limit-circle case, so is it under a bounded perturbation for the potential function q.

Lemma 4.2 Let $q(t)=d(t)+e(t)$ for all $t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$ and $e(t)$ be bounded with respect to $w(t)$ on $[\rho (0),\mathrm{\infty })\cap \mathbb{T}$; that is, there exists a positive constant M such that

$$|e(t)|\le Mw(t),t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}.$$

(4.9)

Then Eq. (1.1) is in the limit-circle case at $t=\mathrm{\infty }$ if and only if the equation

$$-{(p(t)y^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+d(t)y^{\sigma }(t)=\lambda w(t)y^{\sigma }(t)$$

(4.10)

is in the limit-circle case at $t=\mathrm{\infty }$.

Proof Suppose that (4.10) is in the limit-circle case at $t=\mathrm{\infty }$. To show that Eq. (1.1) is in the limit-circle case, it suffices to show that each solution (4.1) is in $L_w^2(\rho (0),\mathrm{\infty })$ by Theorem 3.2.

Let $y_1(t)$ and $y_2(t)$ be two solutions of the equation

$$-{(p(t)y^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+d(t)y^{\sigma }(t)=0$$

(4.11)

satisfying the initial conditions (4.2). Then $y_1(t)$, $y_2(t)$ are two linearly independent solutions in $L_w^2(\rho (0),\mathrm{\infty })$ by Theorem 3.2.

Let $y(t)$ be any solution of (4.1). Then

$$-{(p(t)y^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+d(t)y^{\sigma }(t)=r(t)\text{for all }t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T},$$

where $r(t):=-e(t)y^{\sigma }(t)$. By the variation of constants [[27], Theorem 3.73] there exist two constants α and β such that

$$y(t)=\alpha y_1(t)+\beta y_2(t)+{\int }_{\rho (0)}^{t}r(\tau )(y_1^{\sigma }(\tau )y_2(t)-y_2^{\sigma }(\tau )y_1(t))\mathrm{\Delta }\tau \text{for all }t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}.$$

Hence, replacing t by $\sigma (t)$ and by (ii) in Lemma 2.2, we get

$$y^{\sigma }(t)=\alpha y_1^{\sigma }(t)+\beta y_2^{\sigma }(t)+{\int }_{\rho (0)}^{t}r(\tau )(y_1^{\sigma }(\tau )y_2^{\sigma }(t)-y_2^{\sigma }(\tau )y_1^{\sigma }(t))\mathrm{\Delta }\tau .$$

(4.12)

From (4.9) and (4.12), we have

$$\begin{array}{rcl}|y^{\sigma }(t)|& \le & |\alpha ||y_1^{\sigma }(t)|+|\beta ||y_2^{\sigma }(t)|\\ +M{\int }_{\rho (0)}^t(|y_1^{\sigma }(\tau )||y_2^{\sigma }(t)|+|y_2^{\sigma }(\tau )||y_1^{\sigma }(t)|)w(\tau )|y^{\sigma }(\tau )|\mathrm{\Delta }\tau .\end{array}$$

(4.13)

Since $y_1(t)$, $y_2(t)$ are solutions of Eq. (4.11), which satisfy the initial conditions (4.2), it follows from the existence-uniqueness theorem that $|y_1^{\sigma }(t)|+|y_2^{\sigma }(t)|\ne 0$ for all $t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$. Let

$$y_0^{\sigma }(t):=\frac{|y^{\sigma }(t)|}{|y_1^{\sigma }(t)|+|y_2^{\sigma }(t)|}\text{for all }t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}.$$

From (4.13), we have

$$\begin{array}{rcl}{y}_0^{\sigma }(t)& \le & \frac{|\alpha ||y_1^{\sigma }(t)|+|\beta ||y_2^{\sigma }(t)|}{|y_1^{\sigma }(t)|+|y_2^{\sigma }(t)|}\\ +M{\int }_{\rho (0)}^t\frac{(|y_1^{\sigma }(\tau )||y_2^{\sigma }(t)|+|y_2^{\sigma }(\tau )||y_1^{\sigma }(t)|)w(\tau )|y^{\sigma }(\tau )|}{|y_1^{\sigma }(t)|+|y_2^{\sigma }(t)|}\mathrm{\Delta }\tau \\ \le & |\alpha |+|\beta |+M{\int }_{\rho (0)}^t(|y_1^{\sigma }(\tau )|+|y_2^{\sigma }(\tau )|)w(\tau )|y^{\sigma }(\tau )|\mathrm{\Delta }\tau \\ =& |\alpha |+|\beta |+M{\int }_{\rho (0)}^t{(|y_1^{\sigma }(\tau )|+|y_2^{\sigma }(\tau )|)}^{2}w(\tau )|y_0^{\sigma }(\tau )|\mathrm{\Delta }\tau \\ \le & |\alpha |+|\beta |+2M{\int }_{\rho (0)}^t({|y_1^{\sigma }(\tau )|}^2+{|y_2^{\sigma }(\tau )|}^2)w(\tau )|y_0^{\sigma }(\tau )|\mathrm{\Delta }\tau .\end{array}$$

It follows from (i) of Lemma 2.2 that $y_0^{\sigma }(\cdot )\in C_{rd}(\mathbb{T})$. By Lemma 4.1, we have

$$\begin{array}{rcl}{y}_0^{\sigma }(t)& \le & (|\alpha |+|\beta |)e_{[2Mw({|y_1^{\sigma }|}^2+{|y_2^{\sigma }|}^2)]}(t,\rho (0))\\ =& (|\alpha |+|\beta |)exp\left[{\int }_{\rho (0)}^t{\xi }_{\mu (\tau )}(2Mw(\tau )({|y_1^{\sigma }(\tau )|}^2+{|y_2^{\sigma }(\tau )|}^2))\mathrm{\Delta }\tau \right]\\ =& (|\alpha |+|\beta |)exp[{\int }_{\rho (0)}^t\frac{1}{\mu (\tau )}Log(1+\mu (\tau )2Mw(\tau )({|y_1^{\sigma }(\tau )|}^2+{|y_2^{\sigma }(\tau )|}^2))\mathrm{\Delta }\tau ]\\ \le & (|\alpha |+|\beta |)exp[{\int }_{\rho (0)}^{t}2Mw(\tau )({|y_1^{\sigma }(\tau )|}^2+{|y_2^{\sigma }(\tau )|}^2)\mathrm{\Delta }\tau ]\\ \le & (|\alpha |+|\beta |)exp[{\int }_{\rho (0)}^{\mathrm{\infty }}2Mw(\tau )({|y_1^{\sigma }(\tau )|}^2+{|y_2^{\sigma }(\tau )|}^2)\mathrm{\Delta }\tau ]=:C<\mathrm{\infty },\end{array}$$

which implies that $|y^{\sigma }(t)|\le C(|y_1^{\sigma }(t)|+|y_2^{\sigma }(t)|)$. Hence, $y(\cdot )\in L_w^2(\rho (0),\mathrm{\infty })$; and consequently, Eq. (1.1) is in the limit-circle case at $t=\mathrm{\infty }$.

On the other hand, using

$$-{(p(t)y^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+d(t)y^{\sigma }(t)=-{(p(t)y^{\mathrm{\Delta }}(t))}^{\mathrm{\Delta }}+(q(t)-e(t))y^{\sigma }(t)$$

one can easily conclude that if Eq. (1.1) is in the limit-circle case, then Eq. (4.10) is in the limit-circle case. This completes the proof. □

Theorem 4.3 Let $q(t)=d(t)+e(t)$ for all $t\in [\rho (0),\mathrm{\infty })\cap \mathbb{T}$ and $e(t)$ be bounded with respect to $w(t)$ on $[\rho (0),\mathrm{\infty })\cap \mathbb{T}$. Then the limit cases for Eq. (1.1) are invariant.

Remark 4.3 Lemma 4.2 extends the related result [[13], Lemma 2.4] for the singular second-order difference equation to the time scales. In addition, let $\mathbb{T}=\mathbb{R}$ in Lemma 4.2, then we can directly prove [[31], Theorem 6.1] with the similar method.