R E S E A R C H
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, (.) wherepandqare complex-valued rd-continuous functions,wis a real rd-continuous func-tion;p(t)= andw(t) > for allt∈[ρ(), +∞)∩T;p–is-integrable on [ρ(), +∞)∩T; λ∈Cis the spectral parameter;Tis a time scale withρ()∈TandsupT= +∞;σ(t) andρ(t) are the forward and backward jump operators inT;yis the-derivative; and
yσ(t) :=y(σ(t)). In general, equation (.) is formally self-adjoint if and only ifp(t) andq(t) are real, equation (.) is called formally non-self-adjoint whenp(t)= orq(t)= .
In , Weyl gave a dichotomy of the limit-point and limit-circle cases for singu-lar formally self-adjoint second-order linear differential equation []. Later, Titchmarsh, Coddington, Levinsonet al. developed his results and established the theory of Weyl-Titchmarsh [, ]. Their work was further developed to higher-order differential equa-tions and continuous Hamiltonian systems [–]. Singular spectral problems of self-adjoint scalar second-order difference equations over infinite intervals were first studied by Atkinson []. His work was followed by Hinton, Jirariet 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 [–].
In , Sims obtained an extension of the Weyl classification for formally non-self-adjoint second-order linear differential equations, which the leading coefficients are iden-tical to and the potential functions are complex []. Later, Brownet 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≤+∞,pandqare complex-valued functions,wis a weight function,
p(t)= andw(t) > a.e.t∈[a,b),p–(t),q(t), andw(t) are locally integrable on [a,b),λis a spectral parameter. They divided the equations into three cases by using them-function, which was defined on a collection of rotated half-planes. Recently, non-self-adjoint Sturm-Liouville difference equations and Hamiltonian difference systems have been discussed [, ]. Especially, Wilson in [] gave a discrete analog of the work of Brownet 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, (.)
whereN={, , , . . .},is the forward operator,i.e.,x(n) =x(n+ ) –x(n);p(n) and q(n) are complex numbers, andw(n) is a real number;p(n)= forn∈ {–} ∪N and w(n) > forn∈N;λis a spectral parameter. He also divided the equations into three
cases, by using them-function. The classification for formally non-self-adjoint second-order 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 prob-lems on time scales [–]. But singular spectral probprob-lems have started to be considered only quite recently [–]. In , we employed Weyl’s method to divide the follow-ing 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,
wherep,q, andware real and piecewise continuous functions on [ρ(), +∞)∩T,p(t)= andw(t) > for allt∈[ρ(), +∞)∩T,λ∈Cis the spectral parameter. It has been found that the formally non-self-adjoint second-order linear differential equations (.) and dif-ference 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 ex-act dependence of cases II and III on the corresponding half-planes, and establish several criteria of the limit cases for equation (.).
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.
LetT⊂Rbe a nonempty closed set. The forward and backward jump operatorsσ,ρ:
T→Tare defined by
σ(t) :=inf{s∈T:s>t}, ρ(t) :=sup{s∈T:s<t},
respectively, whereinf∅=supT,sup∅=infT. A pointt∈Tis called scattered, right-dense, left-scattered, and left-dense ifσ(t) >t,σ(t) =t,ρ(t) <t, andρ(t) =t, separately. DenoteTk:=TifTis unbounded above andTk:=T\(ρ(maxT),maxT] otherwise. The graininessμ:T→[, +∞) is defined by
μ(t) :=σ(t) –t.
Letf be a function defined on T.f is said to be-differentiable att∈Tk provided there exists a constantasuch that for anyε> , there is a neighborhoodUoft(i.e.,U= (t–δ,t+δ)∩Tfor someδ> ) with
fσ(t)–f(s) –aσ(t) –s≤εσ(t) –s for alls∈U.
In this case, denotef(t) :=a. Iff is-differentiable for everyt∈Tk, thenf is said to be
-differentiable onT. Iff is-differentiable att∈Tk, then
f(t) = ⎧ ⎨ ⎩
lims→t
s∈T
f(t)–f(s)
t–s , ifμ(t) = , f(σ(t))–f(t)
μ(t) , ifμ(t) > .
(.)
IfF(t) =f(t) for allt∈Tk, thenF(t) is called an anti-derivative off onT. In this case, define the-integral by
t
s
f(τ)τ=F(t) –F(s) for alls,t∈T.
For convenience, we introduce the following results ([], Chapter and [], Chapter ), which are useful in this paper.
Lemma . Let f,g:T→Rand t∈Tk.
(i) Iff is-differentiable att,thenf is continuous att.
(ii) Iff andgare-differentiable att,thenfgis-differentiable attand
(fg)(t) =fσ(t)g(t) +f(t)g(t) =f(t)gσ(t) +f(t)g(t).
(iii) Iff andgare-differentiable att,andf(t)fσ(t)= ,thenf–gis-differentiable at tand
A functionf defined onTis said to be rd-continuous if it is continuous at every right-dense point inTand its left-sided limit exists at every left-dense point inT. The set of rd-continuous functionsf :T→Ris denoted byCrd(T) =Crd(T,R). The set ofkth -dif-ferentiable functions with rd-continuouskth derivative is denote byCk
rd(T) =Ckrd(T,R).
Lemma . If f,g are rd-continuous functions onT,then
(i) fσ is rd-continuous andf has an anti-derivative onT.
(ii) tσ(t)f(τ)τ=μ(t)f(t)for allt∈T.
(iii) (Integration by parts)abfσ(τ)g(τ)τ=f(b)g(b) –f(a)g(a) –b
a f
(τ)g(τ)τ.
(iv) (Hölder inequality [], Lemma .(iv))Letr,s∈Twithr≤s,then
s
r
f(τ)g(τ)τ≤
s
r
f(τ)pτ
p s
r
g(τ)qτ
q ,
wherep> andq=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
Lwρ(), +∞:=
y:ρ(), +∞→C
+∞
ρ()
w(t)yσ(t)t< +∞
and
Q:=co
q(t)
w(t)+rp(t),t∈
ρ(), +∞∩T, <r< +∞
=C,
wherecodenotes the closed convex hull. Forλ∈C\Q, denote byK=K(λ) its nearest point inQand denote byL=L(λ) the tangent toQatKif it exists, and otherwise any
line touchingQatK. We then perform a transformation of the complex planez→z–K
and a rotation through an angleη=η(λ)∈(–π,π], so that the image ofLcoincides with
the imaginary axis. Furthermore, the images ofλand the setQlie in the negative and
non-negative half-planes, respectively. In other words, for allt∈[ρ(), +∞)∩Tandr∈
(, +∞),
q(t)
w(t)+rp(t) –K
eiη
≥ (.)
and
(λ–K)eiη
For such admissibleKandη, the negative rotated half-plane can be expressed as
η,K:=
λ∈C,(λ–K)eiη< . Clearly, for allλ∈η,K, we have
(λ–K)eiη= –δ< , (.)
whereδ=δη,K(λ) is the distance fromλto the boundary∂η,K. Set
S:=(η,K),K∈∂Q,(z–K)eiη≥ for allz∈Q.
ThenSconsists of all the admissible value ofKandη.
We shall initially establish the analog of the theory of Weyl-Titchmarsh on the half-planesη,K, but subject to the condition
eiηcosαsinα≤ (.)
for some fixedα∈C. Denote byS(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 allt∈ρ(), +∞∩T.
Further, we get from (.), for <r< +∞,
q(t)
rw(t)+p(t) –
K r
eiη
≥ for allt∈ρ(), +∞∩T
and hence by lettingr→+∞, it follows that
p(t)eiη≥ for allt∈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.
Lety(t,λ) andy(t,λ) be the two solutions of equation (.) satisfying the following initial conditions:
yρ(),λ=cosα, pρ()yρ(),λ=sinα,
yρ(),λ=sinα, pρ()yρ(),λ= –cosα,
whereα∈C. Since their Wronskian is identically equal to –, these two solutions form a fundamental solution system of (.). We form a linear combination ofy(t,λ) andy(t,λ)
y(t,λ,m) :=y(t,λ) +my(t,λ). (.)
Letb∈(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 . Forηsatisfying(.)andλ∈η,K,the transformation(.)maps the
half-plane[zeiη]≥onto a closed disc D
b(λ)inC,which has radius
rb(λ) =
–eiηcosαsinα
+ b
ρ()
eiηp(t)y(t,λ)+q(t) –λw(t)yσ(t,λ)t
–
(.)
and center
ab(λ) = –
eiηp(b)y
(b,λ)y(b,λ) +e–iηp(b)y(b,λ)y(b,λ) eiηp(b)y
(b,λ)y(b,λ) +e–iηp(b)y(b,λ)y(b,λ)
.
Proof First, we setz˜=zeiη, so that the function (.) becomes
˜
mb(λ,z˜)≡mb(λ,z) = –
y(b,λ)z˜+p(b)y
(b,λ)eiη y(b,λ)z˜+p(b)y
(b,λ)eiη
. (.)
The critical point of (.) isz˜= –eiηp(b)y(b,λ)
y(b,λ) , and we require this point to be such that[˜z]
is negative. Upon calculation, we find that
[z˜] = –
eiηp(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(t,λ)+q(t)yσ(t,λ)t
= –p(t)y(t,λ)y(t,λ)|bρ()+ b
ρ()
p(t)y(t,λ)t+ b
ρ()
It follows from Theorem . thatm=mb(λ,z)∈Db(λ) if and only if[eiηzb(λ,m)]≥, that is,[eiηp(b)y(b,λ,m)y(b,λ,m)]≤. As in (.),[eiηp(b)y(b,λ,m)y(b,λ,m)]≤ can be written as
eiηpρ()yρ(),λ,myρ(),λ,m
+ b
ρ()
eiηp(t)y(t,λ,m)+q(t) –λw(t)yσ(t,λ,m)t≤.
On substituting (.), this givesm∈Db(λ) if and only if
b
ρ()
eiηp(t)y(t,λ,m)+q(t) –λw(t)yσ(t,λ,m)t
≤–eiη(sinα–mcosα)(cosα+msinα)
=:Aα,η;m(λ). (.)
By (.) and (.), the integrand on the left side of (.) is positive. Therefore, ifb<b,
b
ρ()
eiηp(t)y(t,λ,m)+q(t) –λw(t)yσ(t,λ,m)t
≤ b
ρ()
eiηp(t)y(t,λ,m)+q(t) –λw(t)yσ(t,λ,m)t
≤Aα,η;m(λ).
Hence,Db(λ)⊂Db(λ). That is, the discsDb(λ) are nested asb→+∞. This completes
the proof.
Corollary . Forλ∈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()
eiηp(t)y(t,λ)+q(t) –λw(t)yσ(t,λ)t= +∞ (.)
whereas in the limit-circle case the left side of (.) is finite. Also note that, by (.), a solutionyof (.) forλ∈η,Ksatisfies
+∞
ρ()
eiηp(t)y(t,λ)+q(t) –λw(t)yσ(t,λ)t< +∞
if and only if
+∞
ρ()
eiηp(t)y(t,λ)+q(t) –Kw(t)yσ(t,λ)t
+
+∞
ρ()
in particular this yields
y(·,λ)∈Lwρ(), +∞.
This result enables us to give the following full characterization of equation (.).
Definition . Let (η,K)∈S(α). Then forλ∈η,K,
(i) if equation (.) has exactly one linearly independent solution satisfying (.) and this is the only linearly independent solution of equation (.) inL
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 inL
w(ρ(), +∞), then equation (.) is called
case II;
(iii) if all the solutions of equation (.) satisfy (.) and hence are inL
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 ofλ.
Theorem .
(i) If all the solutions of equation(.)are inLw(ρ(), +∞)for someλ∈C,then the same is true for allλ∈C.
(ii) If all the solutions of equation(.)satisfy(.)for someλ∈η,K,then the same is
true for allλ∈C.
Proof (i) Suppose that equation (.) has two linearly independent solutions inL
w(ρ(), +∞) forλ=λ∈C. Theny(t,λ) andy(t,λ) are both inL
w(ρ(), +∞). For briefness, denote
u(t) =y(t,λ), u(t) =y(t,λ).
For anyλ∈C, letv(t) be an arbitrary non-trivial solution of (.), and letu(t) be the solu-tion of (.) withλ=λand with the initial values
u(a) =v(a), u(a) =v(a), a∈[, +∞)∩T. From the variation of constants [], Theorem ., we have
v(t) =u(t) + (λ–λ)
t
a
u(t)uσ(s) –u(t)uσ(s)w(s)vσ(s)s, t∈[a, +∞)∩T. (.)
Replacingtwithσ(t) in (.) and using (ii) of Lemma ., we obtain
w(t)vσ(t)
=w(t)uσ(t) + (λ–λ)
σ(t)
a
w(t)uσ
(t)w
(s)uσ
(s) –w
(t)uσ
(t)w
(s)uσ
(s)
×w(s)vσ(s)s
The constantacan be chosen in advance so large that
(ii) Suppose that equation (.) has two linearly independent solutions satisfying (.) forλ=λ∈η,K. Then u(t) =y(t,λ) andu(t) =y(t,λ) also satisfy (.). For any λ∈C, letv(t) be an arbitrary non-trivial solution of (.), andu(t) be the solution of (.) with λ=λ, which has the initial valuesu(a) =v(a),u(a) =v(a),a∈(, +∞)∩T. It
follows from the variation of constants,u(t) andv(t) also satisfy (.). Differentiating both sides of (.), we get
It follows from (.) and a similar method in the proof of part (i) that we have
and hencev(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 solutionynot to be inLw(ρ(), +∞). In fact, it can be concluded fromy(·) /∈Lw(ρ(), +∞)and (.) thatydoes not satisfy (.). (ii) By (i) of this remark and (i) of Theorem ., if equation (.) is in the case I with
respect to someη,K, then the same is true for allη,K, that is, case I is
independent ofη,K.
Remark .
(i) In the continuous case:μ(t)≡. Theorem .-., Corollary ., and Definition . are the same as those obtained by Brownet al.for formally non-self-adjoint second-order differential equations [], Section .
(ii) In the discrete case: All the points of[ρ(), +∞)∩Tare isolated. Let
[ρ(), +∞)∩T={t–,t,t, . . .}, wheret–<t<t<· · ·. In this case, equation (.) can be written as
–p˜(tn)y(tn)
+q˜(tn)y(tn+) =λw˜(nj)y(tn+), n∈ {–, , , . . .},
wherep˜(tn) =p(tn)/μ(tn),q˜(tn) =q(tn)μ(tn), andw˜(tn) =w(tn)μ(tn). By setting
x(n) =y(tn),pˆ(n) =p˜(tn),qˆ(n) =q˜(tn), andwˆ(n) =w˜(tn), the above problems can be
rewritten as
–pˆ(n)x(n)+qˆ(n)x(n+ ) =λwˆ(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 ofη,Kby (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 de-pendent onη,K, that is, there existη,Kandη,K withη,K∩η,K =∅such that equation (.) is in the case II with respect toη,K and case III with respect toη,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 thatη,η∈Bandη= η(modπ).If equation(.)is in the caseIIIwith respect toη,K andη,K,respectively,then equation(.)is in the caseIIIwith respect to all
η,K.
Proof Letp(t) =|p(t)|eiφ(t). Then we have
eiηjp(t)=p(t)cosη
j+φ(t)
Usingsin(η–η) =sinηcosη–cosηsinηandcos(ηj+φ(t)) =cosηjcosφ(t) –sinηj×
sinφ(t),j= , , and noting thatsin(η–η)= byη=η(modπ), we have
cosφ(t) = sinη
sin(η–η)
cosη+φ(t)
– sinη
sin(η–η)
cosη+φ(t)
,
sinφ(t) = cosη
sin(η–η)
cosη+φ(t)
– cosη
sin(η–η)
cosη+φ(t)
.
(.)
It follows from (.) and (.) that
p(t)= sinη
sin(η–η)
eiηp(t)– sinη
sin(η–η)
eiηp(t),
p(t)= cosη
sin(η–η)
eiηp(t)– cosη
sin(η–η)
eiηp(t).
(.)
Hence, it follows from (.) and (.) that there exists a positive constantMsuch that
p(t)≤Meiηp(t)+eiηp(t). (.)
With a similar argument, we can prove that there exists a positive constantM˜ such that forλ∈C,
q(t) –λw(t)≤ ˜Meiηq(t) –λw(t)+eiηq(t) –λw(t). (.)
Now, let (η,K)∈S(α) andλ∈η,K. Letu(t) be a solution of (.). Suppose that equa-tion (.) is in the case III with respect toη,Kandη,K, respectively. Then, it follows from (ii) of Theorem . thatu(t) satisfies (.) withη,Kreplaced byη,Kandη,K,
respectively. Then, we can get from (.), (.), andu(·)∈Lw(ρ(), +∞),
+∞
ρ()
eiηjp(t)u(t)t< +∞,
+∞
ρ()
eiηjq(t) –λw(t)uσ(t)t< +∞, j= , .
(.)
It follows from (.)-(.) that
+∞
ρ()
p(t)u(t)t< +∞ and
+∞
ρ()
q(t) –λw(t)uσ(t)t< +∞. (.)
Clearly, (.) gives
+∞
ρ()
eiηp(t)u(t)t< +∞ and
+∞
ρ()
eiηq(t) –λw(t)uσ(t)t< +∞.
(.)
Note thatλ∈η,K. We then see from (.) and (.) that (.) holds for each solution
u(t) of (.). Hence, equation (.) is in the case III with respect toη,K. This completes
The following result is a direct consequence of Theorem ..
Corollary . If equation(.)is in the caseIIwith respect to anη,K,then there exists at
most oneη∈B(modπ)such that equation(.)is in the caseIIIwith respect toη,K.
4 Several criteria of the limit cases Denote yσ
(t) :=y(σ(t)), yσ
(t) := (yσ(t)), and yσ(t) :=yσ(σ(t)). In this section, we assume that yσ
(t) =yσ
(t) for allt∈[ρ(), +∞)∩Tin order to establish several cri-teria for equation (.) to be in the case I on the special time scales. It follows from
yσ(t) = ( +μ(t))yσ(t) and the assumptionyσ(t) =yσ(t) thatμ(t)≡, that is, the graininessμ(t) is constant, which impliesT=RorT=hZ. Sinceμ(t) is a constant, the integrals+∞f(t)tand+∞fσ(t)tare convergent or divergent simultaneously.
Theorem . Assume thatBcontains at least two different elementsηandη(modπ).
If
+∞
ρ()
(w(t)wσ(t))
|pσ(t)| t= +∞,
then equation(.)is in the caseI.
Proof Sinceη,η∈Bare different (modπ), the imaginary axes of the two half-planes η,K andη,K intersect. Therefore, we haveη,K∩η,K =∅. Chooseλ∈η,K∩
η,K. Letu(t) be the solution of equation (.) satisfying the initial value conditions
uρ()= , uρ()= . (.)
Then we get from (.)
p(t)u(t)uσ(t)=q(t) –λw(t)uσ
(t)+pσ(t)uσ(t),
which together with (.), (.), and (.) yields, forj= , ,
eiηjp(t)u(t)uσ(t)
= t
ρ()
eiηjq(s) –λw(s)uσ(s)s+
t
ρ()
eiηjpσ(s)uσ(s)s
≥δj t
ρ()
w(s)uσ(s)s, (.)
whereδjis the distance fromλto∂ηj,Kj. It is clear that, fort∈[ρ(), +∞)∩T,
p(t)u(t)uσ(t)=eiηjp(t)u(t)uσ(t)≥ eiηjp(t)u(t)uσ(t), j= , . (.)
Chooset∈(ρ(), +∞)∩Tand leth:=min{δ
t
ρ()w(s)|u
σ(s)|s,δ
t
ρ()w(s)|u
σ(s)|s}.
Then we get from (.) and (.)
≥ |p(t)u(t)uσ(t)||pσ(t)u
Integrating (.) fromttotand using the Schwarz inequality, we get
Clearly,Bcontains at least two different elementsηandη(modπ). Further, it is clear
thatρ+()∞ bounded from below.Then equation(.)is in the caseI.
Proof From the assumptions, we have
Q=co
Clearly,Bcontains at least two different elementsηandη(modπ) since all the points
x≥inf{qw((tt))}form a half line in the complex planexoy. In addition,ρ+()∞
+∞ ρ()
(w(t)wσ(t))t= +∞. So, equation (.) is in the case I by Theorem .. This
com-pletes the proof.
In the following, denotep(t) :=[p(t)],q(t) :=[q(t)], p(t) :=[p(t)], andq(t) := [q(t)]. It is noted that Theorem . cannot be used for equation (.) for which there is only one element inB. 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
-dif-ferentiable function M(t)on[t, +∞)∩Tfor some t∈[ρ(), +∞)∩Tand 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)≥–kMσ(t)w(t),
(iv) |p(t)|M(t)M–(t)(Mσ(t))–≤kw(t),
(v)
+∞
t
pσ
(s)(w(s)wσ(s))
|pσ(s)|(Mσ(s))
s= +∞, (.)
then equation(.)is in the caseI.
Proof Let (η,K)∈S(α) andη,K be the corresponding half-plane. Chooseλ∈η,K. Let
u(t) be the solution of (.) satisfying the initial value conditions (.). Then (.) and (.) hold withηjandδjreplaced byηandδ, whereδis the distance fromλto∂η,K. Then it follows that there exist˜t∈(ρ(), +∞)∩Tand a positive constanth˜such that
p(t)u(t)uσ(t)≥ ˜h for allt∈[˜t
, +∞)∩T. (.)
On the other hand, from the fact thatu(t) is the solution of (.), we get
–(p(t)u
(t))uσ(t)
M(t) +
q(t)|uσ(t)| M(t) =λ
w(t)|uσ(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)|u
σ(t)|–λw(t)|uσ(t)|)M(t) +pσ(t)|uσ(t)|M(t) –M(t)p(t)u(t)uσ(t)
M(t)Mσ(t)
=p
σ(t)|uσ(t)| Mσ(t) –
M(t)p(t)u(t)uσ(t)
M(t)Mσ(t)
+q(t)|u σ(t)| Mσ(t) –λ
w(t)|uσ(t)|
Integrating both sides of (.) fromt:=max{t,˜t}totand then taking the real part, we
w(ρ(), +∞). Then, using the assumptions (ii), (iv), and the Schwarz inequality, we have
t
Hence, we get from the assumptions (i), (iii), and (.)
Furthermore, it can be seen from (.),p(t) > , the Schwarz inequality, and a similar
(t, +∞)∩Tsuch that
Integrating (.) from˜ttotand using the Schwarz inequality, we get
Remark .. This completes the proof.
Remark . Theorem . extended the related result Theorem . of [] for
second-order differential equation with complex coefficients to the time scales. In addition, let
p, qbe 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 [].
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)≥–kMσ(t)w(t), |p(t)|M(t)M–(t)(Mσ(t))–≤
kw(t),and+∞ t
pσ(s)(w(s)wσ(s))
|pσ(s)|(Mσ(s)) s= +∞,then equation(.)is in the caseI.
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, andw(t)≡. Then
Q:=co–t+i+rt,t∈ρ(), +∞∩T, <r< +∞=C.
By choosingM(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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