Zhang and Sun Advances in Difference Equations (2016) 2016:119
DOI 10.1186/s13662-016-0836-0
RESEARCH
Open Access
Classification and criteria of limit cases for
singular second-order linear equations with
complex coefficients on time scales
Chao Zhang* and Shurong Sun
*
Correspondence:
[email protected]
School of Mathematical Sciences,
University of Jinan, Jinan, Shandong
250022, P.R. China
Abstract
The main object of this paper is to establish the classification and some criteria of the
limit cases for singular second-order linear equations with complex coefficients on
time scales. According to the number of linearly independent solutions in suitable
weighted square integrable spaces, this class of equations is classified into cases I, II,
and III. Moreover, the exact dependence of cases II and III on the corresponding
half-planes is given and some criteria of the limit cases are established.
Keywords: singular second-order linear equation; complex coefficient; time scales;
limit case
1 Introduction
In this paper, we consider the classification and criteria of limit cases for the following
singular second-order linear equations with complex coefficients:
– p(t)y (t) + q(t)yσ (t) = λw(t)yσ (t),
t ∈ ρ(), +∞ ∩ T,
(.)
where p and q are complex-valued rd-continuous functions, w is a real rd-continuous function; p(t) =  and w(t) >  for all t ∈ [ρ(), +∞) ∩ T; p– is -integrable on [ρ(), +∞) ∩ T;
λ ∈ C is the spectral parameter; T is a time scale with ρ() ∈ T and sup T = +∞; σ (t)
and ρ(t) are the forward and backward jump operators in T; y is the -derivative; and
yσ (t) := y(σ (t)). In general, equation (.) is formally self-adjoint if and only if p(t) and q(t)
are real, equation (.) is called formally non-self-adjoint when p(t) =  or q(t) = .
In , Weyl gave a dichotomy of the limit-point and limit-circle cases for singular formally self-adjoint second-order linear differential equation []. Later, Titchmarsh,
Coddington, Levinson et al. developed his results and established the theory of WeylTitchmarsh [, ]. Their work was further developed to higher-order differential equations and continuous Hamiltonian systems [–]. Singular spectral problems of selfadjoint scalar second-order difference equations over infinite intervals were first studied
by Atkinson []. His work was followed by Hinton, Jirari et al. [, ]. Further, their work
has been developed to formally self-adjoint Hamiltonian difference systems [, ]. In the
past few years, the theory of Weyl-Titchmarsh has been greatly developed and generalized
to the discrete symplectic systems. Many important results have been established [–].
© 2016 Zhang and Sun. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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Zhang and Sun Advances in Difference Equations (2016) 2016:119
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In , Sims obtained an extension of the Weyl classification for formally non-selfadjoint second-order linear differential equations, which the leading coefficients are identical to  and the potential functions are complex []. Later, Brown et al. in  []
extended this work to a more general case:
– p(t)y (t) + q(t)y(t) = λw(t)y(t),
t ∈ [a, b),
(.)
where –∞ < a < b ≤ +∞, p and q are complex-valued functions, w is a weight function,
p(t) =  and w(t) >  a.e. t ∈ [a, b), p– (t), q(t), and w(t) are locally integrable on [a, b), λ is a
spectral parameter. They divided the equations into three cases by using the m-function,
which was defined on a collection of rotated half-planes. Recently, non-self-adjoint SturmLiouville difference equations and Hamiltonian difference systems have been discussed
[, ]. Especially, Wilson in [] gave a discrete analog of the work of Brown et al. []
on the following non-self-adjoint second-order difference equations:
– p(n – )x(n – ) + q(n)x(n) = λw(n)x(n),
n ∈ N ,
(.)
where N = {, , , . . .}, is the forward operator, i.e., x(n) = x(n + ) – x(n); p(n) and
q(n) are complex numbers, and w(n) is a real number; p(n) =  for n ∈ {–} ∪ N and
w(n) >  for n ∈ N ; λ is a spectral parameter. He also divided the equations into three
cases, by using the m-function. The classification for formally non-self-adjoint secondorder differential or difference equations is related to the corresponding half-planes. But
in [, ], the authors did not discuss whether there existed the case where the equation
was in the case II with respect to a rotated half-plane and in case III with respect to another
one. More recently, Qi, Zheng, and Sun [–] proved that case II and III depend on the
corresponding half-planes by illustrating two examples and gave two propositions to show
how case II and III depend on the corresponding half-planes.
In the past  years, a lot of efforts have been made in the study of regular spectral problems on time scales [–]. But singular spectral problems have started to be considered
only quite recently [–]. In , we employed Weyl’s method to divide the following formally self-adjoint second-order linear equations on time scales into limit-point and
limit-circle cases []:
– p(t)y (t) + q(t)yσ (t) = λw(t)yσ (t),
t ∈ ρ(), +∞ ∩ T,
where p , q, and w are real and piecewise continuous functions on [ρ(), +∞) ∩ T, p(t) = 
and w(t) >  for all t ∈ [ρ(), +∞) ∩ T, λ ∈ C is the spectral parameter. It has been found
that the formally non-self-adjoint second-order linear differential equations (.) and difference equations (.) can be both divided into three cases, by using the Weyl method.
We wonder whether it holds on time scales. The main purpose of this paper is to extend
the pioneering work of classification of (.) and (.) to equation (.), present the exact dependence of cases II and III on the corresponding half-planes, and establish several
criteria of the limit cases for equation (.).
The rest of this paper is organized as follows. In Section , some basic concepts, fundamental theories, and propositions are introduced. In Section , a family of nested circles
which converge to a limiting set is constructed. The classification of the limit cases and
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the exact dependence of limit cases on the corresponding half-planes are given. Finally,
several criteria of the limit cases are established in Section .
2 Preliminaries
In this section, first, we introduce some basic concepts and fundamental results on time
scales.
Let T ⊂ R be a nonempty closed set. The forward and backward jump operators σ , ρ :
T → T are defined by
σ (t) := inf{s ∈ T : s > t},
ρ(t) := sup{s ∈ T : s < t},
respectively, where inf ∅ = sup T, sup ∅ = inf T. A point t ∈ T is called right-scattered, rightdense, left-scattered, and left-dense if σ (t) > t, σ (t) = t, ρ(t) < t, and ρ(t) = t, separately.
Denote Tk := T if T is unbounded above and Tk := T \ (ρ(max T), max T] otherwise. The
graininess μ : T → [, +∞) is defined by
μ(t) := σ (t) – t.
Let f be a function defined on T. f is said to be -differentiable at t ∈ Tk provided
there exists a constant a such that for any ε > , there is a neighborhood U of t (i.e., U =
(t – δ, t + δ) ∩ T for some δ > ) with
f σ (t) – f (s) – a σ (t) – s ≤ εσ (t) – s
for all s ∈ U.
In this case, denote f (t) := a. If f is -differentiable for every t ∈ Tk , then f is said to be
-differentiable on T. If f is -differentiable at t ∈ Tk , then
⎧
⎨ lim s→t f (t)–f (s) , if μ(t) = ,
t–s
s∈T
(.)
f (t) =
⎩ f (σ (t))–f (t) ,
if
μ(t)
>
.
μ(t)
If F (t) = f (t) for all t ∈ Tk , then F(t) is called an anti-derivative of f on T. In this case,
define the -integral by
t
f (τ )τ = F(t) – F(s) for all s, t ∈ T.
s
For convenience, we introduce the following results ([], Chapter  and [], Chapter ),
which are useful in this paper.
Lemma . Let f , g : T → R and t ∈ Tk .
(i) If f is -differentiable at t, then f is continuous at t.
(ii) If f and g are -differentiable at t, then fg is -differentiable at t and
(fg) (t) = f σ (t)g (t) + f (t)g(t) = f (t)g σ (t) + f (t)g (t).
(iii) If f and g are -differentiable at t, and f (t)f σ (t) = , then f – g is -differentiable at
t and
gf –
–
(t) = g (t)f (t) – g(t)f (t) f σ (t)f (t) .
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A function f defined on T is said to be rd-continuous if it is continuous at every rightdense point in T and its left-sided limit exists at every left-dense point in T. The set of
rd-continuous functions f : T → R is denoted by Crd (T) = Crd (T, R). The set of kth -difk
k
(T) = Crd
(T, R).
ferentiable functions with rd-continuous kth derivative is denote by Crd
Lemma . If f , g are rd-continuous functions on T, then
(i) f σ is rd-continuous and f has an anti-derivative on T.
σ (t)
(ii) t f (τ )τ = μ(t)f (t) for all t ∈ T.
b
b
(iii) (Integration by parts) a f σ (τ )g (τ )τ = f (b)g(b) – f (a)g(a) – a f (τ )g(τ )τ .
(iv) (Hölder inequality [], Lemma .(iv)) Let r, s ∈ T with r ≤ s, then
s
f (τ )g(τ )τ ≤
r
s
f (τ )p τ
p r
s
g(τ )q τ
q
,
r
where p >  and q = p/(p – ).
We define the Wronskian by
W [x, y](t) = p(t) x(t)y (t) – x (t)y(t) .
The following result is a direct consequence of the Lagrange identity [], Theorem ..
Lemma . Let x and y be any two solutions of equation (.). Then W [x, y](t) is a constant
on [ρ(), +∞) ∩ T.
Now, it is assumed throughout the present paper that
Lw ρ(), +∞ := y : ρ(), +∞ → C +∞

w(t)yσ (t) t < +∞
ρ()
and
q(t)
+ rp(t), t ∈ ρ(), +∞ ∩ T,  < r < +∞ = C,
Q := co
w(t)
where co denotes the closed convex hull. For λ ∈ C \ Q, denote by K = K(λ ) its nearest
point in Q and denote by L = L(λ ) the tangent to Q at K if it exists, and otherwise any
line touching Q at K . We then perform a transformation of the complex plane z → z – K
and a rotation through an angle η = η(λ ) ∈ (–π, π], so that the image of L coincides with
the imaginary axis. Furthermore, the images of λ and the set Q lie in the negative and
non-negative half-planes, respectively. In other words, for all t ∈ [ρ(), +∞) ∩ T and r ∈
(, +∞),
q(t)
+ rp(t) – K eiη ≥ 
w(t)
and
(λ – K)eiη < .
(.)
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For such admissible K and η, the negative rotated half-plane can be expressed as
η,K := λ ∈ C, (λ – K)eiη <  .
Clearly, for all λ ∈ η,K , we have
(λ – K)eiη = –δ < ,
(.)
where δ = δη,K (λ) is the distance from λ to the boundary ∂η,K . Set
S := (η, K), K ∈ ∂Q, (z – K)eiη ≥  for all z ∈ Q .
Then S consists of all the admissible value of K and η.
We shall initially establish the analog of the theory of Weyl-Titchmarsh on the halfplanes η,K , but subject to the condition
eiη cos αsin α ≤ 
(.)
for some fixed α ∈ C. Denote by S(α) the set {(η, K) ∈ S, (.) is satisfied}.
Proposition . For each pair of (η, K) ∈ S,
q(t) – Kw(t) eiη ≥  and
p(t)eiη ≥  for all t ∈ ρ(), +∞ ∩ T.
(.)
Proof Note that (.) holds for all  < r < +∞ for each pair of (η, K) ∈ S. Then, by setting
r → , it can be seen from (.) that
q(t) – Kw(t) eiη ≥  for all t ∈ ρ(), +∞ ∩ T.
Further, we get from (.), for  < r < +∞,
K iη
q(t)
+ p(t) –
e ≥
rw(t)
r
for all t ∈ ρ(), +∞ ∩ T
and hence by letting r → +∞, it follows that
p(t)eiη ≥  for all t ∈ ρ(), +∞ ∩ T.
This completes the proof.
3 Classification
In this section, we focus on the classification of equation (.) and show how cases II and
III of this classification depend on the corresponding half-planes.
Let y (t, λ) and y (t, λ) be the two solutions of equation (.) satisfying the following
initial conditions:
y ρ(), λ = cos α,
y ρ(), λ = sin α,
p ρ() y
 ρ(), λ = sin α,
p ρ() y
 ρ(), λ = – cos α,
(.)
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where α ∈ C. Since their Wronskian is identically equal to –, these two solutions form a
fundamental solution system of (.). We form a linear combination of y (t, λ) and y (t, λ)
y(t, λ, m) := y (t, λ) + my (t, λ).
(.)
Let b ∈ (ρ(), +∞) ∩ T, z ∈ C, and let (.) satisfy
y(b, λ, m)z + p(b)y (b, λ, m) = .
Then
m ≡ mb (λ, z) = –
y (b, λ)z + p(b)y
 (b, λ)
.
y (b, λ)z + p(b)y
 (b, λ)
(.)
This has inverse
z = zb (λ, m) = –
p(b)y
 (b, λ)m + p(b)y (b, λ)
.
y (b, λ)m + y (b, λ)
(.)
Theorem . For η satisfying (.) and λ ∈ η,K , the transformation (.) maps the halfplane [zeiη ] ≥  onto a closed disc Db (λ) in C, which has radius

– eiη cos αsin α
rb (λ) =

–
b
  iη σ
t
e p(t) y (t, λ) + q(t) – λw(t) y (t, λ)
+
(.)
ρ()
and center
ab (λ) = –
–iη
eiη p(b)y
p(b)y (b, λ)y
 (b, λ)y (b, λ) + e
 (b, λ)
–iη p(b)y (b, λ)y (b, λ)
eiη p(b)y

 (b, λ)y (b, λ) + e

.
Proof First, we set z̃ = zeiη , so that the function (.) becomes
m̃b (λ, z̃) ≡ mb (λ, z) = –
iη
y (b, λ)z̃ + p(b)y
 (b, λ)e
.
y (b, λ)z̃ + p(b)y (b, λ)eiη
(.)
eiη p(b)y (b,λ)

, and we require this point to be such that [z̃]
The critical point of (.) is z̃ = – y (b,λ)
is negative. Upon calculation, we find that
[z̃] = –
eiη p(b)y
 (b, λ)y (b, λ)
.
|y (b, λ)|
By Lemma .(iii) and (.), it follows that
b
ρ()
yσ (t, λ) – p(t)y
+ q(t)yσ (t, λ) t
 (t, λ)
b
= –p(t)y
 (t, λ)y (t, λ)|ρ() +
b
ρ()

p(t)y
 (t, λ) t +
b
ρ()

q(t)yσ (t, λ) t
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= –p(b)y
 (b, λ)y (b, λ) – cos αsin α
b

σ
 t.
p(t)y
+
 (t, λ) + q(t) y (t, λ)
ρ()
This yields
iη
y (b, λ) e p(b)y (b, λ)y (b, λ)
|y (b, λ)|
= – eiη cos αsin α
b 

p(t) q(t)
+
y (t, λ) +
– λ yσ (t, λ) w(t) t.
eiη
w(t)
w(t)
ρ()
(.)
Hence, by (.) and (.), [z̃] <  as required. Therefore, when (.) is satisfied, z →
mb (λ, z) maps [zeiη ] ≥  onto a closed disc Db (λ). By the properties of this transformation, the center ab (λ) of the disc Db (λ) corresponding to the reflection of the critical point
in the imaginary axis. Therefore,
–iη
e p(b)y
 (b, λ)
ab (λ) = m̃b λ,
y (b, λ)
=–
–iη
p(b)y (b, λ)y
eiη p(b)y
 (b, λ)y (b, λ) + e
 (b, λ)
–iη p(b)y (b, λ)y (b, λ)
eiη p(b)y

 (b, λ)y (b, λ) + e

.
Furthermore, z̃ =  is mapped onto a point on the circle Cb (λ) bounding Db (λ). That is,
m̃b (λ, ) = –
y
 (b, λ)
.
y
 (b, λ)
By using the Wronskian of y (t, λ), y (t, λ), and (.), we find that the radius rb (λ) of Db (λ)
is given by
–iη
iη
e p(b)y (b, λ)y
y (b, λ)  (b, λ) + e p(b)y (b, λ)y (b, λ)
– – 
rb (λ) = –
iη
y (b, λ) e–iη p(b)y (b, λ)y
 (b, λ) + e p(b)y (b, λ)y (b, λ)
–
=  eiη p(b)y
 (b, λ)y (b, λ)

=
– eiη cos αsin α

–
b
  iη σ
t .
+
e p(t) y (t, λ) + q(t) – λw(t) y (t, λ)
ρ()
This completes the proof.
Theorem . If b < b , then Db (λ) ⊂ Db (λ). That is, the discs Db (λ) are nested as b →
+∞.
Proof By (.), (.) can be written into
z = zb (λ, m) = –
p(b)y (b, λ, m)
.
y(b, λ, m)
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It follows from Theorem . that m = mb (λ, z) ∈ Db (λ) if and only if [eiη zb (λ, m)] ≥ , that
is, [eiη p(b)y (b, λ, m)y(b, λ, m)] ≤ . As in (.), [eiη p(b)y (b, λ, m)y(b, λ, m)] ≤  can be
written as
eiη p ρ() y ρ(), λ, m y ρ(), λ, m
b
  eiη p(t)y (t, λ, m) + q(t) – λw(t) yσ (t, λ, m) t ≤ .
+
ρ()
On substituting (.), this gives m ∈ Db (λ) if and only if
b
  eiη p(t)y (t, λ, m) + q(t) – λw(t) yσ (t, λ, m) t
ρ()
≤ – eiη (sin α – m cos α)(cos α + m sin α)
=: A α, η; m(λ) .
(.)
By (.) and (.), the integrand on the left side of (.) is positive. Therefore, if b < b ,
b
ρ()
  eiη p(t)y (t, λ, m) + q(t) – λw(t) yσ (t, λ, m) t
b
≤
  eiη p(t)y (t, λ, m) + q(t) – λw(t) yσ (t, λ, m) t
ρ()
≤ A α, η; m(λ) .
Hence, Db (λ) ⊂ Db (λ). That is, the discs Db (λ) are nested as b → +∞. This completes
the proof.
Corollary . For λ ∈ η,K , as b → +∞, the discs Db (λ) contract either to a disc D∞ (λ) or
to a point m(λ). These represent limit-circle and limit-point cases, respectively.
In the limit-point case, it follows from (.) that
+∞
ρ()
  σ
t = +∞
eiη p(t)y
 (t, λ) + q(t) – λw(t) y (t, λ)
(.)
whereas in the limit-circle case the left side of (.) is finite. Also note that, by (.), a
solution y of (.) for λ ∈ η,K satisfies
+∞
  eiη p(t)y (t, λ) + q(t) – λw(t) yσ (t, λ) t < +∞
ρ()
if and only if
+∞
ρ()
  eiη p(t)y (t, λ) + q(t) – Kw(t) yσ (t, λ) t
+∞
+
ρ()

w(t)yσ (t, λ) t < +∞;
(.)
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in particular this yields
y(·, λ) ∈ Lw ρ(), +∞ .
This result enables us to give the following full characterization of equation (.).
Definition . Let (η, K) ∈ S(α). Then for λ ∈ η,K ,
(i) if equation (.) has exactly one linearly independent solution satisfying (.) and
this is the only linearly independent solution of equation (.) in Lw (ρ(), +∞), then
equation (.) is called case I;
(ii) if equation (.) has exactly one linearly independent solution satisfying (.), but
all the solutions of equation (.) are in Lw (ρ(), +∞), then equation (.) is called
case II;
(iii) if all the solutions of equation (.) satisfy (.) and hence are in Lw (ρ(), +∞),
then equation (.) is said to be in the case III.
Remark . It follows from Corollary . and (.) that case I and case II are the sub-cases
of the limit-point case and case III is the limit-circle case.
The next theorem in this section shows that the classification of equation (.) into
cases I, II, and III is independent of the choice of λ.
Theorem .
(i) If all the solutions of equation (.) are in Lw (ρ(), +∞) for some λ ∈ C, then the
same is true for all λ ∈ C.
(ii) If all the solutions of equation (.) satisfy (.) for some λ ∈ η,K , then the same is
true for all λ ∈ C.
Proof (i) Suppose that equation (.) has two linearly independent solutions in Lw (ρ(),
+∞) for λ = λ ∈ C. Then y (t, λ ) and y (t, λ ) are both in Lw (ρ(), +∞). For briefness,
denote
u (t) = y (t, λ ),
u (t) = y (t, λ ).
For any λ ∈ C, let v(t) be an arbitrary non-trivial solution of (.), and let u(t) be the solution of (.) with λ = λ and with the initial values
u(a) = v(a),
u (a) = v (a),
a ∈ [, +∞) ∩ T.
From the variation of constants [], Theorem ., we have
t
v(t) = u(t) + (λ – λ )
a
u (t)uσ (s) – u (t)uσ (s) w(s)vσ (s)s,
t ∈ [a, +∞) ∩ T. (.)
Replacing t with σ (t) in (.) and using (ii) of Lemma ., we obtain

w  (t)vσ (t)

σ (t) = w  (t)uσ (t) + (λ – λ )
a




w  (t)uσ (t)w  (s)uσ (s) – w  (t)uσ (t)w  (s)uσ (s)
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
× w  (s)vσ (s)s

t
= w  (t)uσ (t) + (λ – λ )




w  (t)uσ (t)w  (s)uσ (s) – w  (t)uσ (t)w  (s)uσ (s)
a


× w (s)vσ (s)s,
which implies by the Hölder inequality in Lemma . that

w  (t)vσ (t) ≤ w  (t)uσ (t)
t

t

σ 
σ 
σ

w(s) u (s) s
w(s) v (s) s
+ |λ – λ | w (t)u (t)
a
a
t

t



+ |λ – λ |w  (t)uσ (t)
w(s)uσ (s) s
w(s)vσ (s) s .
a
a
It follows from the inequality (A+B +C) ≤ (A +B +C  ), where A, B, C are non-negative
numbers, that
σ 

σ  t
σ 


w(s)uσ (s) s
w(t) v (t) ≤ w(t) u (t) + |λ – λ | w(t) u (t)

a
t
σ 

σ  t
w(s)u (s) s
w(s)vσ (s) s.
+ w(t)u (t)
a
a
Integrating the two sides of the above inequality with respect to t from a to τ ∈ (a, +∞)∩T,
we get


τ

w(t)vσ (t) t ≤
a
τ

w(t)uσ (t) t
a
+ |λ – λ |
τ
 t

w(t)uσ (t)
w(s)uσ (s) s
a
a

+ w(t)uσ (t)
t
a

w(s)uσ (s) s
t

w(s)vσ (s) st,
τ

w(t)vσ (t) t
a
which yields


τ
a

w(t)vσ (t) t
+∞
≤
a
×

w(t)uσ (t) t + |λ – λ |
+∞
a

w(t)uσ (t) t
+∞
a
τ

w(t)uσ (t) t

w(t)vσ (t) t.
a
Hence,
 – |λ – λ |
a
+∞
≤
a
+∞

w(t)uσ (t) t

w(t)uσ (t) t.
a
+∞

w(t)uσ (t) t
a
(.)
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The constant a can be chosen in advance so large that
+∞
|λ – λ |
a

w(t)uσ (t) t
+∞
a

w(t)uσ (t) t < .
It follows from (.) that v(·) ∈ Lw (a, +∞) and hence v(·) ∈ Lw (ρ(), +∞). Therefore, all
the solutions of (.) are in Lw (ρ(), +∞) for all λ ∈ C.
(ii) Suppose that equation (.) has two linearly independent solutions satisfying (.)
for λ = λ ∈ η,K . Then u (t) = y (t, λ ) and u (t) = y (t, λ ) also satisfy (.). For any
λ ∈ C, let v(t) be an arbitrary non-trivial solution of (.), and u(t) be the solution of (.)
with λ = λ , which has the initial values u(a) = v(a), u (a) = v (a), a ∈ (, +∞) ∩ T. It
follows from the variation of constants, u(t) and v(t) also satisfy (.). Differentiating both
sides of (.), we get
t
σ
σ
σ
σ
σ
u (s)w(s)v (s)s + u (t)u (t)w(t)v (t)
v (t) = u (t) + (λ – λ ) u (t)
a
t
σ
σ
σ
σ
σ
– (λ – λ ) u
(t)
u
(s)w(s)v
(s)s
+
u
(t)u
(t)w(t)v
(t)




= u (t) + (λ – λ )
a
a
t
σ
σ
σ
u
 (t)u (s) – u (t)u (s) w(s)v (s)s.
It follows from (.) and a similar method in the proof of part (i) that we have


+∞
a

eiη p(t) v (t) t
+∞
≤

eiη p(t) u (t) t
a
+∞
+ |λ – λ |

a
+∞
+
a

eiη p(t) u
 (t) t

eiη p(t) u
 (t) t
+∞
a
+∞
a

w(t)uσ (t) t

w(t)uσ (t) t
+∞

w(t)vσ (t) t
a
and


+∞
a

eiη q(t) – Kw(t) vσ (t) t
+∞
≤
a

eiη q(t) – Kw(t) uσ (t) t + |λ – λ |
+∞
×
a
+∞
+
a
×

eiη q(t) – Kw(t) uσ (t) t

e q(t) – Kw(t) uσ (t) t
+∞
iη
+∞
a
+∞

w(t)uσ (t) t

w(t)uσ (t) t

a

w(t)vσ (t) t.
a
Since u(t), u (t), u (t) satisfy (.) and v(·) ∈ Lw (ρ(), +∞),
a
+∞
  eiη p(t)v (t) + q(t) – λw(t) vσ (t) t +
+∞
a

w(t)vσ (t) t < +∞,
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and hence v(t) satisfy (.). Therefore, all solutions of equation (.) satisfy (.) for all
λ ∈ C. This completes the proof.
Remark .
(i) Equation (.) is in the case I if it has a solution y not to be in Lw (ρ(), +∞). In fact,
it can be concluded from y(·) ∈/ Lw (ρ(), +∞) and (.) that y does not satisfy (.).
(ii) By (i) of this remark and (i) of Theorem ., if equation (.) is in the case I with
respect to some η ,K , then the same is true for all η,K , that is, case I is
independent of η,K .
Remark .
(i) In the continuous case: μ(t) ≡ . Theorem .-., Corollary ., and Definition .
are the same as those obtained by Brown et al. for formally non-self-adjoint
second-order differential equations [], Section .
(ii) In the discrete case: All the points of [ρ(), +∞) ∩ T are isolated. Let
[ρ(), +∞) ∩ T = {t– , t , t , . . .}, where t– < t < t < · · · . In this case, equation (.)
can be written as
– p̃(tn )y(tn ) + q̃(tn )y(tn+ ) = λw̃(nj )y(tn+ ),
n ∈ {–, , , . . .},
where p̃(tn ) = p(tn )/μ(tn ), q̃(tn ) = q(tn )μ(tn ), and w̃(tn ) = w(tn )μ(tn ). By setting
x(n) = y(tn ), p̂(n) = p̃(tn ), q̂(n) = q̃(tn ), and ŵ(n) = w̃(tn ), the above problems can be
rewritten as
– p̂(n)x(n) + q̂(n)x(n + ) = λŵ(n)x(n + ),
n ∈ {–, , , . . .}.
It is evident that the above equation is of a form similar to (.). Hence,
Theorems .-., Corollary ., and Definition . in this special case are the same
as Theorems .-. in [].
It has been known that case I is independent of η,K by (ii) of Remark .. Time scales
contain two special cases: continuous case and discrete case. It follows from the former
examples [], Examples ., . and [], Examples ., ., that cases II and III are dependent on η,K , that is, there exist η ,K and η ,K with η ,K ∩ η ,K = ∅ such that
equation (.) is in the case II with respect to η ,K and case III with respect to η ,K .
Now, we give the exact dependent of case II and case III on the corresponding half-planes
by the following results.
Theorem . Let
B := η, there exists K ∈ ∂Q such that (η, K) ∈ S(α) .
Assume that η , η ∈ B and η = η (mod π ). If equation (.) is in the case III with respect
to η ,K and η ,K , respectively, then equation (.) is in the case III with respect to all
η,K .
Proof Let p(t) = |p(t)|eiφ(t) . Then we have
eiηj p(t) = p(t) cos ηj + φ(t) ,
j = , .
(.)
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Using sin(η – η ) = sin η cos η – cos η sin η and cos(ηj + φ(t)) = cos ηj cos φ(t) – sin ηj ×
sin φ(t), j = , , and noting that sin(η – η ) =  by η = η (mod π ), we have
sin η
sin η
cos η + φ(t) –
cos η + φ(t) ,
sin(η – η )
sin(η – η )
cos η
cos η
sin φ(t) =
cos η + φ(t) –
cos η + φ(t) .
sin(η – η )
sin(η – η )
cos φ(t) =
(.)
It follows from (.) and (.) that
sin η
sin η
eiη p(t) –
eiη p(t) ,
sin(η – η )
sin(η – η )
iη
cos η
cos η
p(t) =
e  p(t) –
eiη p(t) .
sin(η – η )
sin(η – η )
p(t) =
(.)
Hence, it follows from (.) and (.) that there exists a positive constant M such that
p(t) ≤ M eiη p(t) + eiη p(t) .
(.)
With a similar argument, we can prove that there exists a positive constant M̃ such that
for λ ∈ C,
q(t) – λw(t) ≤ M̃ eiη q(t) – λw(t) + eiη q(t) – λw(t) .
(.)
Now, let (η, K) ∈ S(α) and λ ∈ η,K . Let u(t) be a solution of (.). Suppose that equation (.) is in the case III with respect to η ,K and η ,K , respectively. Then, it follows
from (ii) of Theorem . that u(t) satisfies (.) with η, K replaced by η , K and η , K ,
respectively. Then, we can get from (.), (.), and u(·) ∈ Lw (ρ(), +∞),
+∞

eiηj p(t) u (t) t < +∞,
ρ()
+∞

eiηj q(t) – λw(t) uσ (t) t < +∞,
(.)
j = , .
ρ()
It follows from (.)-(.) that
+∞ p(t)u (t) t < +∞ and
ρ()
+∞ q(t) – λw(t)uσ (t) t < +∞.
(.)
ρ()
Clearly, (.) gives
+∞

eiη p(t) u (t) t < +∞
and
ρ()
+∞

eiη q(t) – λw(t) uσ (t) t < +∞.
(.)
ρ()
Note that λ ∈ η,K . We then see from (.) and (.) that (.) holds for each solution
u(t) of (.). Hence, equation (.) is in the case III with respect to η,K . This completes
the proof.
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The following result is a direct consequence of Theorem ..
Corollary . If equation (.) is in the case II with respect to an η,K , then there exists at
most one η ∈ B (mod π ) such that equation (.) is in the case III with respect to η ,K .
4 Several criteria of the limit cases
σ

Denote y (t) := y (σ (t)), yσ (t) := (yσ (t)) , and yσ (t) := yσ (σ (t)). In this section, we
σ
assume that y (t) = yσ (t) for all t ∈ [ρ(), +∞) ∩ T in order to establish several criteria for equation (.) to be in the case I on the special time scales. It follows from
σ
σ
yσ (t) = ( + μ (t))y (t) and the assumption y (t) = yσ (t) that μ (t) ≡ , that is, the
graininess μ(t) is constant, which implies T = R or T = hZ. Since μ(t) is a constant, the
+∞ σ
+∞
f (t)t and
f (t)t are convergent or divergent simultaneously.
integrals
Theorem . Assume that B contains at least two different elements η and η (mod π ).
If
+∞

(w(t)wσ (t)) 
t = +∞,
|pσ (t)|
ρ()
then equation (.) is in the case I.
Proof Since η , η ∈ B are different (mod π ), the imaginary axes of the two half-planes
η ,K and η ,K intersect. Therefore, we have η ,K ∩ η ,K = ∅. Choose λ ∈ η ,K ∩
η ,K . Let u(t) be the solution of equation (.) satisfying the initial value conditions
u ρ() = ,
u ρ() = .
(.)
Then we get from (.)
p(t)u (t)uσ (t)

σ 
= q(t) – λw(t) uσ (t) + pσ (t)u (t) ,
which together with (.), (.), and (.) yields, for j = , ,
eiηj p(t)u (t)uσ (t)
t

eiηj q(s) – λw(s) uσ (s) s +
=
ρ()
≥ δj
t
σ 
eiηj pσ (s) u (s) s
ρ()
t

w(s)uσ (s) s,
(.)
ρ()
where δj is the distance from λ to ∂ηj ,Kj . It is clear that, for t ∈ [ρ(), +∞) ∩ T,
p(t)u (t)uσ (t) = eiηj p(t)u (t)uσ (t) ≥ eiηj p(t)u (t)uσ (t) ,
Choose t ∈ (ρ(), +∞) ∩ T and let h := min{δ
Then we get from (.) and (.)
j = , .
(.)
t
t
σ

σ

ρ() w(s)|u (s)| s, δ ρ() w(s)|u (s)| s}.
p(t)u (t)uσ (t) ≥ h for all t ∈ [t , +∞) ∩ T.
(.)
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Note that [eiηj (q(t) – λw(t))] > , j = , . It follows from (.) and (.) that
p(t)u (t)uσ (t) ≥
σ 
eiηj pσ (s) u (s) s,
t
j = , .
(.)
ρ()
In addition, since η = η (mod π ), it can be seen from the proof of Theorem . that (.)
holds. Hence, we get from (.) and (.)
p(t)u (t)uσ (t) ≥

V (t),
M
(.)
where
pσ (s)uσ (s) s.
t
V (t) =
t
Now, we show that u(·) ∈/ Lw (ρ(), +∞). Suppose on the contrary that u(·) ∈ Lw (ρ(),
+∞). Then it follows from (.), (.), (.), (.), (.), and the Schwarz inequality that
t
pσ (s)uσ (s) s
V (t) =
t
μ(s) q(s) – λw(s) uσ (s) + p(s)u (s)uσ (s)s
t
=
t
t
=
t
t
=
t
t
≥
t
t
=
σ
|μ(s)(q(s) – λw(s))uσ (s)uσ (s) + p(s)u (s)uσ (s)||u (s)|
s
|uσ (s)|
σ
|μ(s)eiηj (q(s) – λw(s))uσ (s)uσ (s) + eiηj p(s)u (s)uσ (s)||u (s)|
s
|uσ (s)|
σ
|p(s)u (s)uσ (s)||pσ (s)u (s)uσ  (s)|
s
|pσ (s)||uσ (s)||uσ  (s)|

σ
(w(s)wσ (s))  |p(s)u (s)uσ (s)||pσ (s)u (s)uσ  (s)|
s

(w(s)wσ (s))  |pσ (s)||uσ (s)||uσ  (s)|
t

t
(w(s)wσ (s)) 


s


σ (s)|
σ
σ
|p


t
w (s)|u (s)| (w (s)) |uσ  (s)|
 t
– 
t

σ 
(w(s)wσ (s)) 

s
w(s) u (s) s
≥h
|pσ (s)|
t
t
t
– 
σ  
σ
×
w (s) u (s) s
,
≥ h
t
which yields limt→+∞ V (t) = +∞ by
On the other hand, we get from (.)
+∞
ρ()

(w(t)wσ (t)) 
|pσ (t)|
t = +∞ and u(·) ∈ Lw (ρ(), +∞).
σ 
V (t) = pσ (t)u (t)
σ = μ(t) p(t)u (t) + p(t)u (t)u (t)
σ (t)
=
|μ(t)(q(t) – λw(t))uσ (t)uσ (t) + p(t)u (t)uσ (t)||u
|uσ (t)|
|
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σ
|p(t)u (t)uσ (t)||pσ (t)u (t)uσ  (t)|
≥
|pσ (t)||uσ (t)||uσ  (t)|
≥

V (t)V σ (t)
.

σ
M |p (t)||uσ (t)||uσ  (t)|
So, it follows that


V (t)
(w(t)wσ (t)) 
 ≥
=
.
–
V (t)
V (t)V σ (t) M (w(t)wσ (t))  |pσ (t)||uσ (t)||uσ  (t)|
(.)
Integrating (.) from t to t and using the Schwarz inequality, we get


–
V (t ) V (t)
t

(w(s)wσ (s)) 

s
≥
M t (w(s)wσ (s))  |pσ (s)||uσ (s)||uσ  (s)|
t
 t
– 

σ 

(w(s)wσ (s)) 
≥
s
w(s) u (s) s
|pσ (s)|
M t
t
t
– 
 
×
wσ (s)uσ (s) s
,
t
which implies that limt→+∞ V(t) = –∞. Then we have a contradiction with limt→+∞ V (t) =
+∞. So, u(·) ∈/ Lw (ρ(), +∞) and equation (.) is in the case I by (i) of Remark .. This
completes the proof.
Example . Consider the equation
–y (t) + –t  – it  yσ (t) = λyσ (t),
t ∈ ρ(), +∞ ∩ T.
(.)
In this case, p(t) = w(t) ≡  and q(t) = –t  – it  . Then
Q := co –t  – it  + r, t ∈ ρ(), +∞ ∩ T,  < r < +∞ = C.
Clearly, Bcontains at least two different elements η and η (mod π ). Further, it is clear

+∞
σ (t)) 
t = +∞. So, (.) is in the case I by Theorem ..
that ρ() (w(t)w
|pσ (t)|
+∞

Corollary . Suppose that p(t) ≡ , q(t) is real, ρ() (w(t)wσ (t))  t = +∞, and
bounded from below. Then equation (.) is in the case I.
q(t)
w(t)
is
Proof From the assumptions, we have
q(t)
Q = co
+ r, t ∈ ρ(), +∞ ∩ T,  < r < +∞ = C.
w(t)
Clearly, B contains at least two different elements η and η (mod π ) since
all the points

σ (t)) 
+∞
q(t)
} form a half line in the complex plane xoy. In addition, ρ() (w(t)w
t =
x ≥ inf{ w(t)
σ
|p (t)|
Zhang and Sun Advances in Difference Equations (2016) 2016:119
+∞
Page 17 of 21

(w(t)wσ (t))  t = +∞. So, equation (.) is in the case I by Theorem .. This completes the proof.
ρ()
In the following, denote p (t) := [p(t)], q (t) := [q(t)], p (t) := [p(t)], and q (t) :=
[q(t)]. It is noted that Theorem . cannot be used for equation (.) for which there
is only one element in B . So, we establish the following criterion which can be used in
this case.
Theorem . Let p (t) >  for all t ∈ [ρ(), +∞) ∩ T. If there exists a positive -differentiable function M(t) on [t , +∞) ∩ T for some t ∈ [ρ(), +∞) ∩ T and four positive
constants k , k , k , k , such that for all t ∈ [t , +∞) ∩ T:
(i) Mσ (t) ≥ k ,
(ii) p (t) ≥ k |p (t)|,
(iii) q (t) ≥ –k Mσ (t)w(t),



(iv) |p(t)|  M (t)M–  (t)(Mσ (t))– ≤ k w  (t),
(v)

pσ (s)(w(s)wσ (s)) 
+∞

|pσ (s)|(Mσ (s)) 
t
s = +∞,
(.)
then equation (.) is in the case I.
Proof Let (η, K) ∈ S(α) and η,K be the corresponding half-plane. Choose λ ∈ η,K . Let
u(t) be the solution of (.) satisfying the initial value conditions (.). Then (.) and (.)
hold with ηj and δj replaced by η and δ, where δ is the distance from λ to ∂η,K . Then it
follows that there exist t̃ ∈ (ρ(), +∞) ∩ T and a positive constant h̃ such that
p(t)u (t)uσ (t) ≥ h̃ for all t ∈ [t̃ , +∞) ∩ T.
(.)
On the other hand, from the fact that u(t) is the solution of (.), we get
–
(p(t)u (t)) uσ (t) q(t)|uσ (t)|
w(t)|uσ (t)|
+
=λ
.
M(t)
M(t)
M(t)
It follows that
p(t)u (t)uσ (t)
M(t)
=
(p(t)u (t)uσ (t)) M(t) – (p(t)u (t)uσ (t))M (t)
M(t)Mσ (t)
=
(q(t)|uσ (t)| – λw(t)|uσ (t)| )M(t) + pσ (t)|u (t)| M(t) – M (t)p(t)u (t)uσ (t)
M(t)Mσ (t)
=
pσ (t)|u (t)| M (t)p(t)u (t)uσ (t)
–
Mσ (t)
M(t)Mσ (t)
σ
σ
+
q(t)|uσ (t)|
w(t)|uσ (t)|
–
λ
.
Mσ (t)
Mσ (t)
(.)
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Integrating both sides of (.) from t := max{t , t̃ } to t and then taking the real part, we
can get
t p(t)u (t)uσ (t)
M (s)p(s)u (s)uσ (s)
= W (t) –
s
M(t)
M(s)Mσ (s)
t
t
t
q (s)|uσ (s)|
w(s)|uσ (s)|
+
s
–
(λ)
s + C ,
Mσ (s)
Mσ (s)
t
t
(.)
t pσ (s)|uσ (s)|
(t )uσ (t )


where W (t) = t  Mσ (s) s and C = [ p(t )uM(t
].
)

Next, we show that u(·) ∈/ Lw (ρ(), +∞). Assume the contrary. Suppose that u(·) ∈
Lw (ρ(), +∞). Then, using the assumptions (ii), (iv), and the Schwarz inequality, we have
M (s)p(s)u (s)uσ (s)
s
M(s)Mσ (s)
t
t
 σ 
≤ k
w(s) u (s) s
t

|p(s)||u (s)|
s
M(s)
t
t
t

 t
σ 

p (s)|u (s)|
s .
≤ k  + 
w(s) u (s) s
M(s)
k t 
t
t
Hence, we get from the assumptions (i), (iii), and (.)
p(t)u (t)uσ (t)
M(t)

 t
t
σ 

p (s)|u (s)|
s
w(s) u (s) s
≥ W (t) – k  + 
M(s)
k t 
t
t

|(λ)|
– k +
w(s)uσ (s) s + C .
k
t
(.)
Furthermore, it can be seen from (.), p (t) > , the Schwarz inequality, and a similar
argument to V (t) in the proof of Theorem . that
t
W (t) =
t
σ
pσ (s)|pσ (s)||u (s) |
s
|pσ (s)|Mσ (s)
σ
pσ (s)|p(s)u (s)uσ (s)||pσ (s)u (s) uσ  (s)|
≥
s
|pσ (s)| Mσ (s)|uσ (s)||uσ  (s)|
t

t
 t
– 
pσ (s)(w(s)wσ (s)) 
σ 
u (s) s
s
w(s)
≥ h̃

t
t
|pσ (s)|(Mσ (s)) 
t
– 
 
×
wσ (s)uσ (s) s
,
t
t
which, together with (.) and u(·) ∈ Lw (ρ(), +∞), implies that limt→+∞ W (t) = +∞.
It can be seen from Mσ (t) > k , limt→+∞ W (t) = +∞, and (.) that there exists t̃ ∈
Zhang and Sun Advances in Difference Equations (2016) 2016:119
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(t , +∞) ∩ T such that
k
p(t)u (t)uσ (t)
≥ W (t) for all t ∈ [t̃ , +∞) ∩ T.
p(t)u (t)uσ (t) ≥ k M(t)

It follows from a similar argument to V (t) in the proof of Theorem . that
σ
W (t) =
pσ (t)|u (t)|
Mσ (t)
σ
pσ (t)|pσ (t)||u (t)|
= 
|pσ (t)|Mσ (t)
σ
≥
pσ (t)|p(t)u (t)uσ (t)||pσ (t)u (t)uσ  (t)|
Mσ (t)|pσ (t)| |uσ (t)||uσ  (t)|
≥
k
pσ (t)
W (t)W σ (t) σ
.

M (t)|pσ (t)| |uσ (t)||uσ  (t)|
So, we have
k
pσ (t)
W (t)

.
≥  σ
=
–
σ
W (t)
W (t)W (t)
 M (t)|pσ (t)| |uσ (t)||uσ  (t)|
(.)
Integrating (.) from t̃ to t and using the Schwarz inequality, we get


–
W
(t)
W (t̃ )
k t
pσ (s)
≥ 
s
 t̃ Mσ (s)|pσ (s)| |uσ (s)||uσ  (s)|

t
 t
– 
pσ (s)(w(s)wσ (s)) 
σ 
k
u (s) s
s
w(s)
≥


t̃
t̃
|pσ (s)|(Mσ (s)) 
t
– 
 
×
wσ (s)uσ (s) s
,
t̃
which implies that limt→+∞ W(t) = –∞ by (.). Then, we get a contradiction with
limt→+∞ W (t) = +∞. So, u(·) ∈/ Lw (ρ(), +∞) and equation (.) is in the case I by (i) of
Remark .. This completes the proof.
Remark . Theorem . extended the related result Theorem . of [] for secondorder differential equation with complex coefficients to the time scales. In addition, let
p , q be real functions on [ρ(), +∞) ∩ T, then Theorem . contains the criterion of
the limit-point case for the formally self-adjoint systems, which is similar to Theorem .
of [].
It is noted that more limitations are imposed on [p(t)] and [q(t)] in Theorem ..
Integrating both sides of (.) and taking the imaginary part, we can get the following
criterion.
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Corollary . Assume that there exist t ∈ [ρ(), +∞) ∩ T, four positive constants k , k ,
k , k , a positive -differentiable function M(t) such that for t ∈ [t , +∞) ∩ T, p (t) >


, Mσ (t) ≥ k , p (t)
≥ k |p (t)|, q (t) ≥ –k Mσ (t)w(t), |p(t)|  M (t)M–  (t)(Mσ (t))– ≤


k w (t), and
+∞
t

pσ (s)(w(s)wσ (s)) 

|pσ (s)|(Mσ (s)) 
s = +∞, then equation (.) is in the case I.
Example . Consider the equation
– ty (t) + (–t + i)yσ (t) = λyσ (t),
t ∈ ρ(), +∞ ∩ T.
(.)
In this case, p(t) = t, q(t) = –t + i, and w(t) ≡ . Then
Q := co –t + i + rt, t ∈ ρ(), +∞ ∩ T,  < r < +∞ = C.
By choosing M(t) = t, it can be verified that (.) is in the case I by Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
SS supervised the study and helped the revision. CZ carried out the main results of this article and drafted the
manuscript. All the authors have read and approved the final manuscript.
Acknowledgements
Many thanks are due to Roman Šimon Hilscher (the editor) and the anonymous reviewers for helpful comments and
suggestions. This research was supported by the NNSF of China (Grant 11571202) and the NSF of University of Jinan
(XKY1511).
Received: 29 May 2015 Accepted: 11 April 2016
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