Sufficient and Necessary Conditions for the Classification of Sturm-Liouville Differential Equations with Complex Coefficients

Abstract

This paper gives sufficient and necessary conditions for the classification of Sturm-Liouville differential equations with complex coefficients given by Brown et al. These conditions involve weighted Sobolev subspaces and the asymptotic behavior of elements in the maximal domain. The results of the present paper generalize the corresponding results for formally symmetric Sturm-Liouville differential equations to non-self-adjoint cases.

1. Introduction

Consider the Sturm-Liouville differential expression 𝜏𝑦∶=𝑤⁻¹−𝑝𝑦^^+𝑞𝑦=𝜆𝑦on[𝑎,𝑏),(1.1) where 𝑝,𝑞 are both complex valued, 𝑤(𝑥) is a positive weight function, −∞<𝑎<𝑏≤+∞, and 𝜆 is the so-called spectral parameter. We call 𝜏 a formally symmetric differential expression if 𝑝,𝑞 are both real valued; otherwise 𝜏 is called formally nonsymmetric. In all cases, we call 𝜏 a formally differential expression or operator.

Let 𝐿^2𝑤 denote the Hilbert space 𝐿^2𝑤∶=𝑦ismeasurable∶[𝑎,𝑏)⟶ℂ∶^𝑏𝑎𝑤(𝑥)||𝑦(𝑥)||²d𝑥<∞(1.2) with inner product ⟨𝑦,𝑧⟩∶=∫^𝑏𝑎𝑧(𝑥)𝑤(𝑥)𝑦(𝑥)d𝑥 and the norm ‖𝑦‖²=⟨𝑦,𝑦⟩ for 𝑦,𝑧∈𝐿^2𝑤. We call a solution 𝑦 of (1.1) an 𝐿^2𝑤-solution or square integrable solution if 𝑦∈𝐿^2𝑤. Set 𝒟(𝜏)=𝑦∈𝐿^2𝑤∶𝑦,𝑝𝑦^∈AC_loc,𝜏𝑦∈𝐿^2𝑤,(1.3) where AC_loc=AC_loc([𝑎,𝑏),ℂ) is the set of complex valued functions that are absolutely continuous on each compact subinterval of [𝑎,𝑏). We call 𝒟(𝜏) the natural (or maximal) domain associated with the formally differential operator 𝜏.

The aim of the present paper is to study the asymptotic behavior of elements of 𝒟(𝜏). This is closely related to the classification of (1.1) according to the number of square integrable solutions of (1.1) in suitable weighted integrable spaces. The study of this problem has a long history started with the pioneering work of Weyl in 1910 [1]. When 𝑝(𝑥) and 𝑞(𝑥) are all real valued, Weyl classified (1.1) into the limit point and limit circle cases in the geometric point of view by introducing the 𝑚(𝜆)-functions, where we say that 𝜏 or (1.1) is in the limit point case at 𝑏 if there exists exactly one 𝐿^2𝑤-solution (up to constant multiple) for 𝜆∈ℂ with Im𝜆≠0 and is in the limit circle case if all solutions belong to 𝐿^2𝑤 for 𝜆∈ℂ with Im𝜆≠0. This work has been greatly developed and generalized to formally symmetric higher-order differential equations and Hamiltonian differential systems. For this line, the reader is referred to [2–10] and references therein.

The same problem was also studied by Sims in 1957 for the case where 𝑞(𝑥) is complex valued [11]. He considered the case where 𝑝(𝑥)=𝑤(𝑥)≡1 and Im𝑞(𝑥) is semibounded and classified (1.1) into three cases. Recently, this work has been extensively generalized by Brown et al. [12] under mild assumptions on weighted function 𝑤(𝑥) and the complex valued coefficients 𝑝(𝑥), 𝑞(𝑥). They proved that there exist three distinct possible cases for (1.1).

For formally symmetric 𝜏, it is well known (see [13, 14]) that (1.1) is in the limit point case at 𝑏 if and only if 𝑝(𝑥)𝑦₂(𝑥)𝑦^1(𝑥)−𝑦₁(𝑥)𝑦^2(𝑥)⟶0as𝑥⟶𝑏(1.4) for 𝑦₁,𝑦₂∈𝒟(𝜏). This kind of characterization (1.4) plays an important role in spectral theory of differential operators since (1.4) gives a natural boundary condition of functions in 𝒟(𝜏) at the end point 𝑏. In this case every self-adjoint extension associated with the differential expression needs not a boundary condition at 𝑏. The analogues of the result (1.4) are also valid for both formally symmetric higher-order differential equations and Hamiltonian differential systems (see, e.g. [4, 5, 7, 8, 15, 16]). By using the asymptotic behavior of elements in 𝒟(𝜏), the further classification of the limit point case into the strong limit point case and the weak limit point case for high-order scalar differential equations was given by Everitt et al. in [17–19] and further studied in [14, 20]. It was generalized to Hamiltonian differential systems by Qi and Chen [21] and well studied in [22]. For real valued functions 𝑝(𝑥) and 𝑞(𝑥), we say that (1.1) is in the strong limit point case at the end point 𝑏 if, for 𝑦₁,𝑦₂∈𝒟(𝜏), 𝑝(𝑥)𝑦₁(𝑥)𝑦^2(𝑥)⟶0as𝑥⟶𝑏.(1.5)

In the present paper, we attempt to set up the analogues of the results (1.4) and (1.5) for (1.1) with complex valued coefficients 𝑝 and 𝑞. In the classification of Brown et al. in [12], Cases II and III depend on the admissible rotation angles (see Theorem 2.1). The exact dependence is set up in Theorem 2.5. We find that the asymptotic behavior of elements in 𝒟(𝜏) also depends on the admissible rotation angles. So we first study the properties of the admissible angle set 𝐸 (defined in (2.10)) and prove that 𝐸 either contains a single point or is an interval. See Lemma 3.1. Then we introduce a pencil of Hamiltonian differential expressions with a new spectral parameter corresponding to (1.1) and set up the relationship between classifications of Hamiltonian differential expressions and (1.1). See Lemma 4.3. Applying the results mentioned in (1.4) and (1.5), we obtain sufficient and necessary conditions for Cases I and II involving weighted Sobolev spaces and the asymptotic behavior of elements in 𝒟(𝜏). See Theorems 4.1 and 4.11. The main results of the present paper cover the result (1.4) (see Remark 4.2) and indicate that (1.4) means (1.5) when 𝐸 has more than one point; see Corollary 4.9.

Following this section, Section 2 gives some preliminary knowledge for (1.1) with complex valued coefficients, and Section 3 presents properties of the admissible rotation angle set 𝐸. The main results are given in Section 4.

2. Preliminary Knowledge

Throughout this paper, we always assume that(i)𝑝(𝑥)≠0, 𝑤(𝑥)>0 a.e. on [𝑎,𝑏) and 1/𝑝,𝑞,𝑤 are all locally integrable on [𝑎,𝑏),(ii)𝑝 and 𝑞 are complex valued, and Ω=co𝑞(𝑥)𝑤(𝑥)+𝑟𝑝(𝑥)∶𝑟>0,𝑥∈[𝑎,𝑏)≠ℂ,(2.1) where co denotes the closed convex hull (i.e., the smallest closed convex set containing the exhibited set). Then, for each point on the boundary 𝜕Ω, there exists a line through this point such that every point of Ω either lies in the same side of this line or is on it. That is, there exists a supporting line through this point. Let 𝐾 be a point on 𝜕Ω. Denote by 𝐿 an arbitrary supporting line touching Ω at 𝐾, which may be the tangent to Ω at 𝐾 if it exists. We then perform a transformation of the complex plane 𝑧↦𝑧−𝐾 and a rotation through an appropriate angle 𝜃 so that the image of 𝐿 coincides with the imaginary axis now and the set Ω is contained in the new right nonnegative half-plane.

For this purpose we introduce the set 𝑆 defined by 𝑆=(𝜃,𝐾)∶𝐾∉Ω^∘,Re𝑒^𝑖𝜃(𝜇−𝐾)≥0∀𝜇∈Ω,(2.2) where Ω^∘ is the interior of Ω, and define the corresponding half-plane Λ_𝜃,𝐾=𝜇∈ℂ∶Re𝑒^𝑖𝜃(𝜇−𝐾)<0.(2.3) Then, Λ_𝜃,𝐾⊂ℂ⧵Ω. From the definition of 𝑆, for all 𝑥∈[𝑎,𝑏) and 0<𝑟<∞, Re𝑒^𝑖𝜃𝑤𝑞((𝑥𝑥))+𝑟𝑝(𝑥)−𝐾≥0.(2.4)

The definition of 𝑆 is different from the corresponding one given by Brown et al. [12], but they are equivalent in describing square integrable solutions.

Besides, for (𝜃,𝐾)∈𝑆Re𝑒^𝑖𝜃(𝜇−𝐾)≥0⟺cos(𝜃+𝛾)≥0where𝜇−𝐾=||𝜇−𝐾||𝑒^𝑖𝛾.(2.5)

Using a nesting circle method based on that of both Weyl [1] and Sims, Brown et al. [12] divided (1.1) into three cases with respect to the corresponding half-planes Λ_𝜃,𝐾 as follows. The uniqueness referred to in the theorem and the following sections is only up to constant multiple.

Theorem 2.1 (cf. [12, Theorem  2.1]). Given a (𝜃,𝐾)∈𝑆, the following three distinct cases are possible. Case I. For all 𝜆∈Λ_𝜃,𝐾, equation (1.1) has unique solution 𝑦 satisfying ^𝑏𝑎Re𝑒^𝑖𝜃𝑝||𝑦^||²+Re𝑒^𝑖𝜃(𝑞−𝐾𝑤)||𝑦||²+^𝑏𝑎𝑤||𝑦||²<∞(2.6) and this is the only solution satisfying 𝑦∈𝐿^2𝑤.Case II. For all 𝜆∈Λ_𝜃,𝐾, all solutions of (1.1) belong to 𝐿^2𝑤, and there exists unique solution of (1.1) satisfying (2.6).Case III. For all 𝜆∈Λ_𝜃,𝐾, all solutions of (1.1) satisfy (2.6).

Since every Λ_𝜃,𝐾 is a half-plane, it holds that Λ_𝜃1,𝐾1Λ^𝜃2,𝐾2≠∅(2.7) for (𝜃_𝑗,𝐾_𝑗)∈𝑆, 𝑗=1,2, with 𝜃₁≠𝜃₂ (mod𝜋). Note that (2.4) implies that, for 0<𝑟<∞ and 𝑥∈[𝑎,𝑏), Re𝑒^𝑖𝜃𝑟𝑞𝑤(𝑥(𝑥))+𝑝(𝑥)−𝐾𝑟≥0.(2.8) Letting 𝑟→0 and 𝑟→∞ in (2.4) and (2.8), respectively, we have the following.

Lemma 2.2. For every (𝜃,𝐾)∈𝑆 and 𝜆∈Λ_𝜃,𝐾, there exists 𝛿_𝜆(𝜃)>0 such that Re𝑒^𝑖𝜃(𝑞−𝐾𝑤)≥0,Re𝑒^𝑖𝜃(𝑞−𝜆𝑤)≥𝛿_𝜆(𝜃)𝑤,Re𝑒^𝑖𝜃𝑝≥0(2.9) on [𝑎,𝑏).

Using variation of parameters method, we can verify that, if all solutions of (1.1) belong to 𝐿^2𝑤 for some 𝜆₀∈ℂ, then it is true for all 𝜆∈ℂ. This also means the following.

Lemma 2.3. If there exists a (𝜃₀,𝐾₀)∈𝑆 such that (1.1) is in Case I with respect to (with respect to for short) Λ_𝜃0,𝐾0, then (1.1) is in Case I with respect to Λ_𝜃,𝐾 for every (𝜃,𝐾)∈𝑆.

This indicates that Case I is independent of the choice of (𝜃,𝐾)∈𝑆. But Cases II and III depend on the choice of (𝜃,𝐾)∈𝑆 in general, that is, there may exist (𝜃₁,𝐾₁),(𝜃₂,𝐾₂)∈𝑆 such that (1.1) is in Case II with respect to Λ_𝜃1,𝐾1 and is in Case III with respect to Λ_𝜃2,𝐾2. In order to make clear the dependence, we introduce the admissible angle set 𝐸 defined by 𝐸={𝜃∶∃𝐾∉Ω^∘,(𝜃,𝐾)∈𝑆}.(2.10)

Remark 2.4. For given 𝜃∈𝐸, there exist many 𝐾 such that (𝜃,𝐾)∈𝑆. In fact, if 𝜃₀∈𝐸 with (𝜃₀,𝐾₀)∈𝑆 for some 𝐾₀∉Ω^∘, then for all 𝐾∈𝐿₀, (𝜃₀,𝐾)∈𝑆𝐿₀=𝜆∈ℂ∶Re𝑒^𝑖𝜃0𝜆−𝐾₀=0.(2.11)

The exact dependence of Cases II and III on (𝜃,𝐾) can be given with the similar proof in [23, Theorem  2.1].

Theorem 2.5 (cf. [23, Theorem  2.1]). If there exists a (𝜃₀,𝐾₀)∈𝑆 such that (1.1) is in Case II with respect to Λ_𝜃0,𝐾0, then (1.1) is in Case II with respect to Λ_𝜃,𝐾 for all (𝜃,𝐾)∈𝑆 except for at most one 𝜃₁∈𝐸(mod𝜋) such that (1.1) is in Case III with respect to Λ_𝜃1,𝐾1.

Remark 2.6. Theorem 2.5 means that, if there exist 𝜃_𝑗∈𝐸, 𝑗=1,2, such that 𝜃₁≠𝜃₂ (mod𝜋) and (1.1) is in Case III with respect to Λ_{𝜃𝑗,𝐾𝑗} for 𝑗=1,2, then (1.1) is in Case III with respect to Λ_𝜃,𝐾 for all (𝜃,𝐾)∈𝑆.

3. Properties of the Angel Set 𝐸

This section gives some properties of the set 𝐸, which will be used in the proof of our main results in Section 4. In what follows, we say that 𝐸 has more than one point if there exist 𝜃₁,𝜃₂∈𝐸 with 𝜃₁≠𝜃₂ (mod𝜋).

Lemma 3.1. Let 𝐸 be defined as in (2.10). (i)The set 𝐸 is connected in the sense of mod2𝜋. (ii)If 𝐸 has more than one point, then, for every 𝜆∈ℂ⧵Ω, there exist 𝜃₁,𝜃₂∈𝐸 with 𝜃₁<𝜃₂ such that 𝜆∈Λ_𝜃,𝐾 for 𝜃∈(𝜃₁,𝜃₂)⊂𝐸.

Proof. (i) Suppose that 𝐸 has more than one point. Let 𝜃₁,𝜃₂∈𝐸 with 𝜃₁≠𝜃₂ (mod𝜋); then 0<𝜃₂−𝜃₁<𝜋 (mod2𝜋) or 𝜋<𝜃₂−𝜃₁<2𝜋 (mod2𝜋).
If 0<𝜃₂−𝜃₁<𝜋 (mod2𝜋) and (𝜃_𝑗,𝐾_𝑗)∈𝑆, 𝑗=1,2, then we claim that [𝜃₁,𝜃₂]⊂𝐸 (mod2𝜋). Let 𝐿_𝑗 be the line similarly defined as 𝐿₀ with 𝐾₀ and 𝜃₀ replaced by 𝐾_𝑗 and 𝜃_𝑗, 𝑗=1,2. That is, 𝐿_𝑗=𝜆∈ℂ∶Re𝑒^𝑖𝜃𝑗𝜆−𝐾_𝑗=0,𝐾_𝑗∉Ω^∘,𝑗=1,2.(3.1) Let 𝐾 be the intersection point of 𝐿₁ and 𝐿₂. Set 𝜇−𝐾=||𝜇−𝐾||𝑒^{𝑖𝛾(𝜇,𝐾)},𝜇∈Ω.(3.2) It follows from (2.5) that cos𝛾(𝜇,𝐾)+𝜃_𝑗≥0,𝜇∈Ω,𝑗=1,2.(3.3)
By 0<𝜃₂−𝜃₁<𝜋 (mod2𝜋) and (3.3), we can get cos(𝛾(𝜇,𝐾)+𝜃)≥0 for 𝜃∈[𝜃₁,𝜃₂] (mod2𝜋) on Ω, which means (𝜃,𝐾)∈𝑆 and 𝜃∈𝐸.
According to the similar method, we can verify that, if 𝜋<𝜃₂−𝜃₁<2𝜋 (mod2𝜋) and (𝜃_𝑗,𝐾_𝑗)∈𝑆, 𝑗=1,2, then [0,𝜃₁]∪[𝜃₂,2𝜋]⊂𝐸 (mod2𝜋), that is, [𝜃₂,𝜃₁]⊂𝐸 (mod2𝜋).
(ii) For 𝜆₀∈ℂ⧵Ω, choose (𝜃₁,𝐾₁)∈𝑆 and 𝛿₀>0 such that 𝜆₀∈Λ_𝜃1,𝐾1 and Re𝑒^𝑖𝜃1𝐾₁−𝜆₀=𝛿₀>0.(3.4) Since 𝐸 has more than one point, we can choose 𝜃₂∈𝐸 with 𝜃₂≠𝜃₁ (mod𝜋). Without loss of generality, we suppose that 0<𝜃₂−𝜃₁<𝜋 (mod2𝜋). Let 𝐾 be defined as in the proof of (i).
If 𝜆₀∈Λ_𝜃1,𝐾∩Λ_𝜃2,𝐾, then it follows from (2.3) that cos𝛾+𝜃_𝑗<0,𝑗=1,2,where𝜆₀−𝐾=||𝜆₀−𝐾||𝑒^𝑖𝛾.(3.5) By 0<𝜃₂−𝜃₁<𝜋 (mod2𝜋) and (3.5), we can get cos(𝛾+𝜃)<0 for 𝜃∈[𝜃₁,𝜃₂] (mod2𝜋), which means 𝜆₀∈Λ_𝜃,𝐾 for 𝜃∈[𝜃₁,𝜃₂].
Suppose that 𝜆₀∉Λ_𝜃2,𝐾. Let 𝜆₁∈𝐿₁ be the unique point such that 𝛿₀=dist(𝜆₀,𝐿₁)=dist(𝜆₁,𝜆₀). Let 𝛼=arctan|𝜆₀−𝜆₁|/2|𝐾−𝜆₁|; then 𝛼+𝜃₁∈(𝜃₁,𝜃₂)⊂𝐸 by 𝜆₀∉Λ_𝜃2,𝐾, and 𝜆₀∈Λ_{𝛼+𝜃1,𝐾} by the definition of 𝛼 (see Figure 1).
So, we can get that 𝜆₀∈Λ_𝜃,𝐾 for 𝜃∈[𝜃₁,𝜃₁+𝛼] by 𝜆₀∈Λ_𝜃1,𝐾∩Λ_{𝜃1+𝛼,𝐾}, and the lemma is proved.

Figure 1 

Figure of Lemma 3.1(ii).

4. Asymptotic Behavior

In this section, we will give asymptotic behavior of elements in the natural domain of the formally differential operator 𝜏 defined on the interval [0,∞) with 0 being a regular end point and +∞ being implicitly a singular end point. All results in this section can be stated for any singular end point, left or right on an arbitrary interval (𝑎,𝑏), where −∞≤𝑎<𝑏≤+∞. Recall that (1.1) on (𝑎,𝑏) is said to be regular at 𝑎 if 1/𝑝, 𝑞 and 𝑤 are integrable on (𝑎,𝑐) for some (and hence any) 𝑐∈(𝑎,𝑏) and singular at 𝑎 otherwise; the regularity and singularity at 𝑏 are defined similarly (cf. [24]). Note that the regularity (resp., singularity) of an end point is solely determined by the integrability (resp., nonintegrability) of the coefficients in (1.1) at the end point, not the finiteness (resp., infiniteness) of the end point, as already remarked by Atkinson at the end of [13, Section  9.1]. See also [10, Theorem 2.3.1]. Recall the definition of 𝒟(𝜏) in (1.3). We also define 𝒟𝜏=𝑦∈𝐿^2𝑤∶𝑦,𝑝𝑦^∈AC_loc,𝜏𝑦∈𝐿^2𝑤,(4.1) where 𝜏𝑦∶=𝑤⁻¹−𝑝𝑦′′+𝑞𝑦on[0,∞).(4.2) The first result of this section is as follows.

Theorem 4.1. (i) 𝜏 is in Case I if and only if for 𝑦₁,𝑦₂∈𝒟(𝜏) and 𝜃∈𝐸𝑝(𝑥)𝑦₂(𝑥)𝑦^1(𝑥)+𝑒^2𝑖𝜃𝑝(𝑥)𝑦₁(𝑥)𝑦^2(𝑥)⟶0as𝑥⟶∞.(4.3)
(ii) 𝜏 is in Case I if and only if for 𝑦₁∈𝒟(𝜏),𝑦₂∈𝒟(𝜏)𝑝(𝑥)𝑦₂(𝑥)𝑦^1(𝑥)−𝑦₁(𝑥)𝑦^2(𝑥)⟶0as𝑥⟶∞.(4.4)

Remark 4.2. Clearly 𝒟(𝜏)=𝒟(𝜏), by the definition of 𝒟(𝜏). It is easy to see that (4.4) is equivalent to 𝑝(𝑥)𝑦₂(𝑥)𝑦^1(𝑥)−𝑦₁(𝑥)𝑦^2(𝑥)⟶0as𝑥⟶∞(4.5) for 𝑦₁,𝑦₂∈𝒟(𝜏).

We will use spectral theory of Hamiltonian differential systems to prove Theorem 4.1, so that we first prepare some known results for the Hamiltonian differential system 𝑣𝑢′^==𝐶𝐴𝑢𝑢−+𝐴𝐵_∗𝑣+𝜉𝑊₂𝑣,𝑣−𝜉𝑊₁𝑢,on[0,∞),(4.6) where 𝑢,𝑣 are ℂ^𝑛 valued functions, 𝑢^𝑇 is the transpose of 𝑢, 𝐴,𝐵,𝐶,𝑊₁, and 𝑊₂ are locally integrable, complex valued 𝑛×𝑛 matrices on [0,∞), 𝐵,𝐶,𝑊₁, and 𝑊₂ are Hermit matrices and 𝑊₁(𝑡)>0,𝑊₂(𝑡)≥0 on [0,∞), and 𝜉 is the spectral parameter. Assume that the definiteness condition (see, e.g., [13, Chapter  9, page 253]) holds: ^∞0𝑦^∗𝑊𝑦>0foreachnontrivialsolution𝑦of(4.6),(4.7) where 𝑊=diag(𝑊₁,𝑊₂). Let 𝐿^2𝑊∶=𝐿^2𝑊[0,∞) denote the space of Lebesgue measurable 2𝑛-dimensional functions 𝑓 satisfying ∫^∞0𝑓^∗(𝑠)𝑊(𝑠)𝑓(𝑠)𝑑𝑠<∞. We say that (4.6) is in the limit point case at infinity if there exists exactly 𝑛's solutions of (4.6) belonging to 𝐿^2𝑊 for 𝜉∈ℂ with Im𝜉≠0.

Let 𝒟 be the maximal domain associated with (4.6), that is, (𝑢^𝑇,𝑣^𝑇)^𝑇∈𝒟 if and only if (𝑢^𝑇,𝑣^𝑇)^𝑇∈AC_loc∩𝐿^2𝑊, and there exists an element (𝑓^𝑇,𝑔^𝑇)^𝑇∈𝐿^2𝑊 such that 𝑣𝑢′^==𝐶𝐴𝑢𝑢−+𝐴𝐵_∗𝑣+𝜉𝑊₂𝑣+𝑊₂𝑔,𝑣−𝜉𝑊₁𝑢−𝑊₁𝑓,on[0,∞).(4.8) It is well known (cf. [5, 7]) that (4.6) is in the limit point case at infinity if and only if 𝑌^∗1(𝑥)𝐽𝑌₂(𝑥)⟶0as𝑥⟶∞,𝐽=0−𝐼^𝑛𝐼_𝑛0(4.9) for 𝑌₁,𝑌₂∈𝒟, and for every 𝜉∈ℂ with Im𝜉≠0 there exists a Green function 𝐺(𝑡,𝑠,𝜉) such that, for 𝐹=(𝑓^𝑇,𝑔^𝑇)^𝑇∈𝐿^2𝑊, 𝑌=𝑣𝑢=𝑇_𝜉𝐹∈𝐿^2𝑊,satisfies(4.8),(4.10) where (𝑇_𝜉𝐹)(𝑥)=∫^∞0𝐺(𝑥,𝑠,𝜉)𝑊(𝑠)𝐹(𝑠)d𝑠.

Let (𝜃,𝐾)∈𝑆 and choose 𝜆₀∈Λ_𝜃,𝐾. Then from (2.9), one sees that Re𝑒^𝑖𝜃𝑞−𝜆₀𝜔≥𝛿₀𝜔>0,Re𝑒^𝑖𝜃𝑞−𝐾₀𝑤≥0,Re𝑒^𝑖𝜃𝑝≥0(4.11) for some 𝛿₀>0. Set 𝑟₁(𝑥)=||𝑞𝑟(𝑥)−𝜆₀𝑤(𝑥)||,𝑞(𝑥)−𝜆₀𝑤(𝑥)=𝑟₁(𝑥)𝑒^{𝑖𝛼(𝑥)},𝛼₁(𝑥)=𝜃+𝛼(𝑥),₂(𝑥)=||𝑝(𝑥)||,𝑝(𝑥)=𝑟₂(𝑥)𝑒^{𝑖𝛽(𝑥)},𝛽₁(𝑥)=𝜃+𝛽(𝑥).(4.12) Consider the Hamiltonian differential system (4.6) with 𝑛=1, 𝐴(𝑥)≡0 and 𝐶(𝑥)𝐵=(𝑟𝑥₁)(𝑥=)ssiinn𝛼𝛽₁₁(𝑥),𝑊₁(𝑥)∶=𝑤₁(𝑥)=𝑟₁(𝑥)cos𝛼₁(𝑥),(𝑥)𝑟₂(𝑥),𝑊²(𝑥)∶=𝑤₂(𝑥)=cos𝛽¹(𝑥)𝑟₂(𝑥),(4.13) that is, the 2-dimensional Hamiltonian differential system 𝐻(𝜃)∶𝑢′=𝐵𝑣+𝜉𝑤₂𝑣,𝑣^=𝐶𝑢−𝜉𝑤₁𝑢.(4.14) It follows from (4.11) that 𝑤₁=Re𝑒^𝑖𝜃𝑞−𝜆₀𝑤≥𝛿₀𝑤>0,𝑤₂=Re𝑒^𝑖𝜃𝑝(𝑡)𝑟²²≥0,(4.15) and it is easy to verify that the definiteness condition holds for the system (4.14). In fact, 𝑦 is a solution of (1.1) if and only if (𝑢,𝑣)^𝑇 is a solution of (4.14) with 𝑢=𝑦,𝑣=−𝑖𝑒^𝑖𝜃𝑝𝑦′.(4.16) This fact immediately yields the following result which is frequently used in the proof of Theorems 4.1 and 4.11.

Lemma 4.3. (i) 𝜏 is in Case I or Case II with respect to (𝜃,𝐾)∈𝑆 if and only if 𝐻(𝜃) is in the limit point case at ∞.
(ii) 𝜏 is in Case III with respect to (𝜃,𝐾)∈𝑆 if and only if 𝐻(𝜃) is in the limit circle case at ∞.

Lemma 4.4. If 𝐸 has more than one point, then 𝒟_𝜃(𝜏)≡𝒟_𝑠(𝜏) on 𝐸^𝑜, the interior of 𝐸, where 𝒟_𝜃(𝜏)=𝒟𝑦∈𝒟(𝜏)∶^∞0Re𝑒^𝑖𝜃𝑝||𝑦^||²+Re𝑒^𝑖𝜃𝑞||𝑦||²<∞_𝑠,(𝜏)=𝑦∈𝒟(𝜏)∶^∞0||𝑝||||𝑦^||²+||𝑞||||𝑦||²<∞.(4.17)

Proof. Let 𝜃₁∈𝐸^𝑜 be fixed. There exist 𝜃₂,𝜃₃∈𝐸^𝑜 such that 𝜃₃<𝜃₁<𝜃₂mod(2𝜋),0<𝜃₂−𝜃₃<𝜋2mod(2𝜋),³_𝑗=1Λ_{𝜃𝑗,𝐾𝑗}≠∅(4.18) by Lemma 3.1. Choose 𝜆₀∈⋂^3𝑗=1Λ_{𝜃𝑗,𝐾𝑗}. Letting 𝛽∶=𝛽(𝑥) be defined as in (4.12) and solving cos(𝜃₁+𝛽) from the equations cos𝜃_𝑗+𝛽=cos𝜃₁+𝛽cos𝜃_𝑗−𝜃₁−sin𝜃₁+𝛽sin𝜃_𝑗−𝜃₁,𝑗=2,3,(4.19) we have that cos(𝜃₁+𝛽)=𝐶₁cos(𝜃₂+𝛽)+𝐶₂cos(𝜃₃+𝛽) with 𝐶₁=sin𝜃¹−𝜃₃sin𝜃₂−𝜃₃>0,𝐶²=sin𝜃²−𝜃₁sin𝜃₂−𝜃₃>0(4.20) by (4.18). Since ∫^∞0Re{𝑒^𝑖𝜃1𝑝}|𝑦^|²<∞ for 𝑦∈𝒟_𝜃1(𝜏), we have that 𝐶₁^∞0Re𝑒^𝑖𝜃2𝑝||𝑦^||²+𝐶₂^∞0Re𝑒^𝑖𝜃3𝑝||𝑦^||²=^∞0Re𝑒^𝑖𝜃1𝑝||𝑦^||²<∞,(4.21) and hence ∫^∞0Re{𝑒^𝑖𝜃2𝑝}|𝑦^|²<∞ for 𝑦∈𝒟_𝜃1(𝜏). The same proof as the above with 𝛽 replaced by 𝛼 also proves ∫^∞0Re{𝑒^𝑖𝜃2(𝑞−𝜆₀𝑤)}|𝑦|²<∞ for 𝑦∈𝒟_𝜃1(𝜏), where 𝛼∶=𝛼(𝑥) is defined as in (4.12). Therefore, for 𝑦∈𝒟_𝜃1(𝜏), ^∞0Re𝑒^𝑖𝜃𝑗𝑝||𝑦^||²,^∞0Re𝑒^𝑖𝜃𝑗𝑞−𝜆₀𝑤||𝑦||²<∞,𝑗=1,2.(4.22) Set 𝑝_𝜃=𝑒^𝑖𝜃𝑝 and 𝑞_𝜃=𝑒^𝑖𝜃(𝑞−𝜆₀𝑤). It follows from sin²𝜃₂−𝜃₁=cos²𝜃₂+cos²𝜃₁−2cos𝜃₂cos𝜃₁cos𝜃₁−𝜃₂≤cos𝜃₂+cos𝜃₁²(4.23) and (4.15) that Re𝑝_𝜃1+𝑝_𝜃2≥𝜀₀||𝑝||,Re𝑞_𝜃1+𝑞_𝜃2≥𝜀₀||𝑞−𝜆𝑤||,𝜀₀=sin𝜃₂−𝜃₁.(4.24) Then (4.24) and (4.22) yield that, for 𝑦∈𝒟_𝜃1(𝜏), ^∞0||𝑝||||𝑦^||²,^∞0||𝑞−𝜆₀𝑤||||𝑦||²<∞.(4.25) Note that 𝑦∈𝐿^2𝑤. Then (4.25) gives 𝑦∈𝒟_𝑠(𝜏), or 𝒟_𝜃1(𝜏)⊂𝒟_𝑠(𝜏). Clearly, 𝒟_𝑠(𝜏)⊂𝒟_𝜃1(𝜏). Thus 𝒟_𝜃1(𝜏)=𝒟_𝑠(𝜏).

Lemma 4.4 indicates the following.

Corollary 4.5. If 𝜏 is in Case II with respect to some (𝜃₀,𝐾₀)∈𝑆 and 𝐸 has more than one point, then Case III only occurs at the end point of 𝐸.

Proof. If 𝜏 is in Case III with respect to some (𝜃₁,𝐾₁)∈𝑆 with 𝜃₁∈𝐸^𝑜, then 𝒟(𝜏)=𝒟_𝜃1(𝜏) is restricted in the solution space of (1.1) by the definition of Case III. Since 𝒟_𝜃1(𝜏)=𝒟_𝑠(𝜏) by Lemma 4.4, we have that 𝒟(𝜏)=𝒟_𝑠(𝜏) restricted in the solution space of (1.1). This means that all solutions of (1.1) with 𝜆∈Λ_𝜃1,𝐾1 satisfy ^∞0||𝑝||||𝑦^||²+||𝑞||||𝑦||²<∞.(4.26) Using variation of parameters method we can prove that it is true for all 𝜆∈ℂ, and hence 𝜏 is in Case III with respect to (𝜃₀,𝐾₀), a contradiction.

Lemma 4.6. If 𝜏 is in Case I and 𝑦∈𝒟(𝜏), then (𝑦,𝑣)^𝑇∈𝒟(𝜃) with 𝑣=−𝑖𝑒^𝑖𝜃𝑝𝑦^, where 𝒟(𝜃) is the maximal domain associated with (4.14).

Proof. Suppose that 𝜏 is in Case I with respect to (𝜃,𝐾)∈𝑆. We claim that 𝒟(𝜏)=𝒟_𝜃(𝜏). Set 𝜏−𝜆₀𝑦₀=𝑤⁻¹−𝑝𝑦^0^+𝑞−𝜆₀𝑤𝑦₀=𝑔⁰,(4.27) for 𝑦₀∈𝒟(𝜏) and 𝜆₀∈Λ_𝜃,𝐾.
Set 𝑢₀=𝑦₀, 𝑣₀=−𝑖𝑒^𝑖𝜃𝑝𝑦^0. Then (𝑢₀,𝑣₀) satisfies 𝑢′=𝐵𝑣+𝑖𝑤₂𝑣,𝑣^=𝐶𝑢−𝑖𝑤₁𝑢−𝑤₁𝑓₁,𝑓₁=𝑤𝑤₁−𝑖𝑒^𝑖𝜃𝑔₀.(4.28) Conversely, if (𝑢,𝑣) satisfies (4.28), then 𝑦=𝑢 solves (4.27). Note that 𝑔₀∈𝐿^2𝑤, or −𝑖𝑒^𝑖𝜃0𝑔₀∈𝐿^2𝑤, and 𝑤₁≥𝛿𝑤 implies 𝑓₁∈𝐿^2𝑤₁.
Considering (4.28), we get from (4.10) that (4.28) has a solution (𝑢₁,𝑣₁)^𝑇 such that 𝑢₁∈𝐿^2𝑤₁,𝑣₁∈𝐿^2𝑤₂ and 𝑣₁=−𝑖𝑒^𝑖𝜃0𝑝𝑢^1. Set 𝑦₁=𝑢₁. Then 𝑦₁ satisfies (4.27), and hence (𝜏−𝜆₀)(𝑦₀−𝑦₁)=0. Note that 𝑦₁=𝑢₁∈𝐿^2𝑤₁, and 𝑤₁≥𝛿𝑤 implies that 𝑦₁∈𝐿^2𝑤. Thus, 𝑦₁−𝑦₀ is an 𝐿^2𝑤-solution of 𝜏𝑦=𝜆₀𝑦. Since 𝜏 is in Case I with respect to (𝜃₀,𝐾₀), it follows from (2.6) that 𝑦₁−𝑦₀∈𝐿^2𝑤₁ and 𝑣₁−𝑣₀∈𝐿^2𝑤₂. This together with 𝑦₁∈𝐿^2𝑤₁ and 𝑣₁∈𝐿^2𝑤₂ gives 𝑦₀∈𝐿^2𝑤₁ and 𝑣₀∈𝐿^2𝑤₂. In fact, we have proved that, for 𝑦∈𝒟(𝜏), ^∞0||𝑞−𝜆₀𝑤||cos𝛼₁||𝑦||²<∞,^∞0||𝑝||cos𝛽₁||𝑦^||²<∞,(4.29) or ^∞0Re𝑒^𝑖𝜃𝑝||𝑦^||²+Re𝑒^𝑖𝜃𝑞−𝜆₀𝑤||𝑦||²<∞,(4.30) where 𝛼₁ and 𝛽₁ are defined in (4.12) or (4.13). Since 𝑦∈𝐿^2𝑤, (4.30) means that ^∞0Re𝑒^𝑖𝜃𝑝||𝑦^||²+Re𝑒^𝑖𝜃𝑞||𝑦||²<∞(4.31) or 𝑦∈𝒟_𝜃(𝜏), and hence 𝒟(𝜏)=𝒟_𝜃(𝜏). Recall that 𝑓₁∈𝐿^2𝑤₁. Then (4.30) and (4.28) imply that, if 𝑦∈𝒟(𝜏), then (𝑦,𝑣)^𝑇∈𝒟(𝜃).

Corollary 4.7. If 𝜏 is in Case I and 𝑦∈𝒟(𝜏), then (𝑦,𝑣)^𝑇∈𝒟(𝜃) with 𝑣=𝑖𝑒^−𝑖𝜃𝑝𝑦^.

Proof. For 𝑦∈𝒟(𝜏), 𝑦∈𝒟(𝜏) by 𝒟(𝜏)=𝒟(𝜏). So (𝑦,𝑣)^𝑇∈𝒟(𝜃) with 𝑣=−𝑖𝑒^𝑖𝜃𝑝𝑦^ by Lemma 4.6. Clearly 𝒟(𝜃)=𝒟(𝜃) since 𝐻(𝜃) is symmetrical. Then we have that (𝑦,𝑣)^𝑇∈𝒟(𝜃) with 𝑣=𝑖𝑒^−𝑖𝜃𝑝𝑦^.

Proof of Theorem 4.1. The proof of (i): suppose that 𝜏 is in Case I. Since (4.14) is in the limit point case at infinity by Lemma 4.3, we know that (4.9) holds for all 𝑌₁,𝑌₂∈𝒟(𝜃). For 𝑦₁,𝑦₂∈𝒟(𝜏), since (𝑦_𝑗,𝑣_𝑗)^𝑇∈𝒟(𝜃) with 𝑣_𝑗=−𝑖𝑒^𝑖𝜃𝑝𝑦^𝑗, 𝑗=1,2, by Lemma 4.6, it follows from (4.9) that 𝑦₁,𝑣₁01−01𝑦²𝑣₂=𝑖𝑒^−𝑖𝜃𝑝𝑦^1𝑦₂+𝑒^2𝑖𝜃𝑝𝑦₁𝑦^2⟶0(4.32) as 𝑥→∞.
Conversely, assume that (4.3) holds for all elements of 𝒟(𝜏). We claim that (1.1) must be in Case I. Suppose on the contrary that (1.1) is not in Case I. Then all solutions of (1.1) belong to 𝐿^2𝑤 for 𝜆∈ℂ. Choose 𝜆₀∈Λ_𝜃,𝐾, and let 𝑦₀ be a nontrivial solution of (1.1) satisfying 𝑦₀(0)=0. Then 𝑦₀∈𝒟(𝜏) by 𝑦₀∈𝐿^2𝑤. Furthermore, it follows from (𝜏−𝜆₀)𝑦₀=0 that −𝑝𝑦^0^𝑦₀+𝑞−𝜆₀𝑤||𝑦₀||²=0,−𝑝𝑦^0^𝑦₀+𝑞−𝜆₀𝑤||𝑦₀||²=0.(4.33) Integrating (4.33) on [0,𝑥] we have that −−((𝑝𝑦^0)𝑦₀||^𝑥0+^𝑥0𝑝||𝑦^0||²+𝑞−𝜆₀𝑤||𝑦₀||²=0,𝑝𝑦^0)𝑦₀||^𝑥0+^𝑥0𝑝||𝑦^0||²+𝑞−𝜆₀𝑤||𝑦₀||²=0.(4.34) Multiplying 𝑒^𝑖𝜃 and 𝑒^−𝑖𝜃 to the first and second equalities in (4.34), respectively, and adding them together, we have that −𝑒^𝑖𝜃𝑝𝑦^0𝑦₀+𝑒^−𝑖𝜃𝑝𝑦^0𝑦₀(𝑥)+^𝑥0𝑝^𝜃+𝑝_𝜃||𝑦^0||²+𝑞_𝜃+𝑞_𝜃||𝑦₀||²=0(4.35) since 𝑦₀(0)=0, where 𝑝_𝜃=𝑒^𝑖𝜃𝑝 and 𝑞_𝜃=𝑒^𝑖𝜃(𝑞−𝜆₀𝑤). Note that 𝑒^𝑖𝜃𝑝𝑦^0𝑦₀+𝑒^−𝑖𝜃𝑝𝑦^0𝑦₀(𝑥)=𝑒^−𝑖𝜃𝑝𝑦^0𝑦₀+𝑒^2𝑖𝜃𝑝𝑦^0𝑦₀(𝑥)⟶0(4.36) as 𝑥→∞ by assumption (4.3) and Re𝑝_𝜃=Re𝑒^𝑖𝜃𝑝≥0,Re𝑞_𝜃=Re𝑒^𝑖𝜃𝑞−𝜆₀𝑤≥𝛿₀𝑤(4.37) by (4.15). Then letting 𝑥→∞ in (4.35), we have a contradiction. This proves the first part of this theorem.
The proof of (ii): suppose that 𝜏 is in Case I. Set 𝑣₁=−𝑖𝑒^𝑖𝜃𝑝𝑦^1, 𝑣₂=𝑖𝑒^−𝑖𝜃𝑝𝑦^2 for 𝑦₁∈𝒟(𝜏), 𝑦₂∈𝒟(𝜏). Then, we can get (𝑦₁,𝑣₁)^𝑇∈𝒟(𝜃) by Lemma 4.6 and (𝑦₂,𝑣₂)^𝑇∈𝒟(𝜃) by Corollary 4.7. Hence 𝑦₁,𝑣₁01−01𝑦²𝑣₂=𝑖𝑒^−𝑖𝜃𝑝𝑦^1𝑦₂−𝑝𝑦₁𝑦₂⟶0(4.38) as 𝑥→∞ by (4.9), that is, 𝑝(𝑥)[𝑦₂(𝑥)𝑦^1(𝑥)−𝑦₁(𝑥)𝑦^2(𝑥)]→0 as 𝑥→∞.
Conversely, if 𝜏 is not in Case I, then all solutions of (1.1) belong to 𝐿^2𝑤 for 𝜆∈ℂ. Let 𝑦_𝑖,𝑖=1,2, be the solution of (𝜏−𝜆₀)𝑦=0 such that 𝑝𝑦^1𝑦(0)₁(0)=10,𝑝𝑦^2𝑦(0)₂(0)=01.(4.39) Since 𝑦_𝑖∈𝐿^2𝑤,𝑦_𝑖∈𝐷(𝜏),𝑖=1,2. Then the Wronskian |||||𝑝𝑦_1𝑦₁𝑝𝑦^2𝑦₂|||||=𝑝𝑦_1𝑦₂−𝑦₁𝑦^2≡1,(4.40) which contradicts condition (4.5). See Remark 4.2.

Remark 4.8. If 𝑞(𝑥) and 𝑝(𝑥) are real valued, then Ω⊂ℝ and (𝜃,𝐾)=(±𝜋/2,0)∈𝑆 with Re{𝑒^𝑖𝜃𝑝(𝑥)}=Re{𝑒^𝑖𝜃(𝑞(𝑥)−𝐾𝑤(𝑥))}≡0. This means that Case I, Cases II and III reduce to Weyl's limit point, limit-circle cases, respectively. For this case, we know that (1.1) is in the limit point case at ∞ if and only if 𝑝(𝑥)𝑦₂(𝑥)𝑦^1(𝑥)−𝑦₁(𝑥)𝑦^2(𝑥)⟶0as𝑥⟶∞(4.41) for 𝑦₁,𝑦₂∈𝒟(𝜏), that is, (1.4). Clearly, if 𝑝 is real valued and 𝜋/2∈𝐸, then (4.3) reduces to (1.4). Therefore, (4.3) is a generalization of (1.4).

Corollary 4.9. If 𝐸 has more than one point, then 𝜏 is in Case I if and only if, for 𝑦₁,𝑦₂∈𝒟(𝜏), 𝑝(𝑥)𝑦₁(𝑥)𝑦^2(𝑥)⟶0as𝑥⟶∞.(4.42) That is 𝜏 is in the strong limit point case at ∞.

Proof. Suppose that 𝐸 has more than one point and 𝜏 is in Case I. Choose 𝜃_𝑗∈𝐸, 𝑗=1,2, with 𝜃₁≠𝜃₂ (mod𝜋). Then (4.3) holds for 𝜃=𝜃_𝑗, 𝑗=1,2. This gives that for 𝑦₁,𝑦₂∈𝒟(𝜏)𝑒^2𝑖𝜃1−𝑒^2𝑖𝜃2𝑝𝑦₁𝑦^2⟶0as𝑥⟶∞,(4.43) and hence (4.42) holds since 𝜃₁≠𝜃₂ (mod𝜋).
Conversely, assume that (4.42) holds for all 𝑦_𝑖∈𝒟(𝜏), 𝑖=1,2. Since (4.42) implies (4.3), we conclude from (i) of Theorem 4.1 that 𝜏 is in Case I.

Corollary 4.10. If 𝜏 is symmetric and 𝑞(𝑥)≥𝑞₀𝑤(𝑥) on [0,∞), then 𝜏 is in the limit point case at ∞ if and only if it is in the strong limit point case at ∞.

Proof. Note that for, 𝜃∈[−𝜋/2,𝜋/2], (𝜃,𝑞₀)∈𝑆. Then [−𝜋/2,𝜋/2]∈𝐸. Therefore, (4.3) holds if and only if (4.42) holds by Corollary 4.9.

Theorem 4.11. 𝜏 is in Case II with respect to (𝜃₀,𝐾₀)∈𝑆 if and only if 𝒟(𝜏)≠𝒟_𝜃0(𝜏)≠∅ and (4.3) holds for 𝑦₁,𝑦₂∈𝒟_𝜃0(𝜏).

Proof. Suppose that 𝜏 is in Case II with respect to some (𝜃,𝐾)∈𝑆. By the definition of Case II we know that 𝒟_𝜃(𝜏) is nonempty and 𝒟(𝜏)≠𝒟_𝜃(𝜏). With a similar proof to that one in the first part of (i) in Theorem 4.1, we can get that (4.3) holds for 𝑦₁,𝑦₂∈𝒟_𝜃0(𝜏) by Lemma 4.3 and (4.32).
Conversely, suppose that 𝒟(𝜏)≠𝒟_𝜃(𝜏) for some (𝜃,𝐾)∈𝑆 and (4.3) holds for 𝑦∈𝒟_𝜃(𝜏). By the proof of Lemma 4.6, we know that 𝜏 is not in Case I with respect to (𝜃,𝐾). We only need to prove that 𝜏 is not in Case III with respect to this (𝜃,𝐾). If it is not true, then all solutions of (1.1) with 𝜆∈Λ_𝜃,𝐾 satisfy (2.6) and so belong to 𝒟_𝜃(𝜏). Let 𝑦₀ be a nontrivial solution of (1.1) with 𝑦(0)=0. Then 𝑦₀∈𝒟_𝜃(𝜏), and hence the same proof as in (4.33)–(4.35) yields a contradiction.

Corollary 4.12. If 𝐸 has more than one point, then 𝜏 is in case II with respect to some (𝜃,𝐾)∈𝑆 if and only if 𝒟(𝜏)≠𝒟_𝑠(𝜏) and (4.42) holds for 𝑦₁,𝑦₂∈𝒟_𝜃(𝜏) with 𝜃∈𝐸^𝑜.

Proof. If 𝐸 has more than one point and 𝜏 is in Case II with respect to some (𝜃₀,𝐾₀)∈𝑆, then there exists 𝜃₁∈𝐸^𝑜 such that 𝜏 is in Case II with respect to (𝜃₁,𝐾₁)∈𝑆 by Theorem 2.5. Since 𝒟_𝜃1(𝜏)⊊𝒟(𝜏) by Theorem 4.11 and 𝒟_𝑠(𝜏)=𝒟_𝜃1(𝜏) by Lemma 4.4, one sees that 𝒟(𝜏)≠𝒟_𝑠(𝜏).
Choose 𝜃₂∈𝐸^𝑜 with 𝜃₁≠𝜃₂ (mod𝜋) such that 𝜏 is in Case II with respect to (𝜃_𝑗,𝐾_𝑗)∈𝑆 for 𝑗=1,2 by Theorem 2.5. Then (4.3) holds for 𝜃=𝜃_𝑗, 𝑗=1,2 by Theorem 4.11. Since 𝒟_𝜃𝑗(𝜏)=𝒟_𝑠(𝜏), the same proof as in (4.43) gives that (4.42) holds for 𝑦₁,𝑦₂∈𝒟_𝑠(𝜏).
Conversely, suppose that 𝒟(𝜏)≠𝒟_𝑠(𝜏) and (4.42) holds for 𝑦∈𝒟_𝑠(𝜏). Since 𝒟_𝜃(𝜏)≡𝒟_𝑠(𝜏) on 𝐸^𝑜 by Lemma 4.4, we conclude that 𝒟_𝜃(𝜏)≠𝒟(𝜏) on 𝐸^𝑜 and (4.42) holds for 𝑦∈𝒟_𝜃(𝜏). So (4.3) holds for 𝑦∈𝒟_𝜃(𝜏) by (4.42). Then, we have that 𝜏 is in Case II with respect to (𝜃,𝐾)∈𝑆 with 𝜃∈𝐸^𝑜 by Theorem 4.11.

Acknowledgments

This work was supported by the NSF of Shandong Province (Grant Y2008A02) and the IIFSDU (Grant 2010ZRJQ002).