Limit Circle/Limit Point Criteria for Second-Order Superlinear Differential Equations with a Damping Term

Abstract

The purpose of the present paper is to establish some new criteria for the classifications of superlinear differential equations as being of the nonlinear limit circle type or of the nonlinear limit point type. The criteria presented here generalize some known results in literature.

1. Introduction

In 1910, Weyl [1] published his now classic paper on eigenvalue problems for second-order linear differential equations of the form 𝑎(𝑡)𝑦^′′+𝑟(𝑡)𝑦=𝜃𝑦,𝜃∈𝐶. (1.1) He classified this equation to be of the limit circle type if each solution 𝑦(𝑡) is square integrable (belongs to 𝐿²), that is,^∞₀𝑦²(𝑡)𝑑𝑡<∞, (1.2) and to be of the limit point type if at least one solution 𝑦(𝑡) does not belong to 𝐿², that is,^∞₀𝑦²(𝑡)𝑑𝑡=∞. (1.3) He showed that the linear equation (1.1) always has at least one square integrable solution if Im 𝜃≠0. Thus, for second-order linear equations with Im 𝜃≠0, the problem reduces to whether (1.1) has one (limit point type) or two (limit circle type) square integrable solutions. This is known as the Weyl Alternative. Weyl also proved that if (1.1) is of the limit circle type for some 𝜃₀∈𝐶, then it is of the limit circle type for all 𝜃∈𝐶. In particular, this is true for 𝜃=0; that is, if we can show that𝑎(𝑡)𝑦^′′+𝑟(𝑡)𝑦=0 (1.4) is of limit circle type, then (1.1) is of the limit circle type for all values of 𝜃. There is considerable interest in this problem over the years. The classification is important in spectral theory for linear equation (1.1) since it characterizes the number boundary conditions for differential operator generated by (1.1) being a self-adjoint operator. The analogous problem for nonlinear equations is relatively new and not as extensively studied as the linear cases. For a survey of known results on the linear and nonlinear problems as well as their relationships to other properties of solutions such as boundedness, oscillation, and convergence to zero, we refer the readers to [2–9] and the recent monograph [10]. In this paper, we will discuss the equation with a damping term:𝑎(𝑡)𝑦^′′+𝑝(𝑡)𝑦^′+𝑟(𝑡)𝑦^2𝑘−1=0, (1.5) where 𝑎,𝑟∶𝑅₊↦𝑅 and 𝑝∶𝑅₊↦𝑅₊ are continuous, 𝑎′,𝑟′∈𝐴𝐶_loc(𝑅₊), 𝑎^′′,𝑟^′′∈𝐿²_loc(𝑅₊), 𝑎(𝑡)>0, 𝑟(𝑡)>0, and 𝑘>0 is a positive integer. When 𝑝(𝑡)≡0, then (1.5) turns into𝑎(𝑡)𝑦^′′+𝑟(𝑡)𝑦^2𝑘−1=0, (1.6) which is widely researched by many authors (see [10] and references cited therein).

Definition 1.1 (see [2]). A nontrivial solution 𝑦(𝑡) of (1.5) is said to be of the nonlinear limit circle type if ^∞₀𝑦^2𝑘(𝑡)𝑑𝑡<∞, (1.7) and it is of the nonlinear limit point type otherwise, that is, ^∞₀𝑦^2𝑘(𝑡)𝑑𝑡=∞. (1.8) Equation (1.5) is said to be of the nonlinear limit circle type if all its solutions satisfy (1.7), and it is said to be of the nonlinear limit point type if there is at least one nontrivial solution satisfying (1.8).

In this paper, we will give necessary and sufficient conditions to guarantee the nonlinear limit circle type or nonlinear point type for (1.5).

2. Main Results

To simplify notations, let 𝛼=1/2(𝑘+1), and𝛽=(2𝑘+1)/2(𝑘+1). We make the transformation𝑠=^𝑡₀𝑟^𝛼(𝑢)𝑎^𝛽(𝑢)𝑑𝑢,𝑥(𝑠)=𝑦(𝑡). (2.1) Then (1.5) becomes𝑥̈+𝐴(𝑡)𝑥̇+𝐵(𝑡)𝑥^2𝑘−1=0, (2.2) where 𝐴(𝑡)=𝑝(𝑡)/𝑎^𝛼(𝑡)𝑟^𝛼(𝑡)+𝛼((𝑎(𝑡)𝑟(𝑡))^′/𝑎^𝛼(𝑡)𝑟^𝛼+1(𝑡)), and 𝐵(𝑡)=(𝑎(𝑡)𝑟(𝑡))^𝛽−𝛼.

Moreover, (2.2) can be rewritten as the system𝑥̇=𝑧−𝐴(𝑡)𝑥,𝑧̇=𝐴̇(𝑡)𝑥−𝐵(𝑡)𝑥^2𝑘−1. (2.3) We are now ready to prove the first result for system (2.3).

Theorem 2.1. Assume that ^∞₀𝐴^′(𝑢)𝐵^1/2(𝑢)𝑑𝑢<∞,_∞0𝐴^′(𝑢)𝐵^1/2(𝑢)𝑑𝑢<∞,_∞01𝐵(𝑢)𝑑𝑢<∞. (2.4)(2.5)(2.6) Then (1.5) is of the nonlinear limit circle type; that is, any solution 𝑦(𝑡) of (1.5) satisfies ^∞₀𝑦^2𝑘(𝑡)𝑑𝑡<∞. (2.7)

Proof. Define 𝑉()=𝑧𝑥,𝑧,𝑠²2(𝑡)𝑥+𝐵^2𝑘+2𝑘^𝑡₀𝑝(𝜉)𝑟^𝛽−𝛼(𝜉)𝑎^2𝛼(𝜉)𝑦^2𝑘(𝜉)𝑑𝜉. (2.8) Then we have 𝑉̇=𝑧𝑧̇+𝐵(𝑡)𝑥^2𝑘(𝑡𝑎(𝑡)𝑟(𝑡2𝑘+𝑝)[)]^𝛽−2𝛼𝑦^2𝑘(𝑡)=𝑧𝐴̇(𝑡)𝑥−𝐵(𝑡)𝑥^2𝑘−1+𝐵(𝑡)𝑥^2𝑘−1𝑥̇+𝐵̇(𝑡)𝑥^2𝑘(𝑡𝑎(𝑡)𝑟(𝑡2𝑘+𝑝)[)]^𝛽−2𝛼𝑦^2𝑘(𝑡)=𝐴̇(𝑡)𝑥𝑧−𝐵(𝑡)𝑥^2𝑘−1𝑧+𝐵(𝑡)𝑥^2𝑘−1[(𝑡)𝑥]+𝐵̇(𝑡)𝑥𝑧−𝐴_2𝑘(𝑡𝑎(𝑡)𝑟(𝑡2𝑘+𝑝)[)]^𝛽−2𝛼𝑦^2𝑘(𝑡)=𝐴̇(𝑡)𝑥𝑧−𝐴(𝑡)𝐵(𝑡)𝑥^2𝑘+𝐵̇(𝑡)𝑥^2𝑘(𝑡𝑎(𝑡)𝑟(𝑡2𝑘+𝑝)[)]^𝛽−2𝛼𝑦^2𝑘(𝑡)=𝐴̇(𝑡)𝐵̇(𝑡)𝑥𝑧+(𝑡)𝐵(𝑡𝑥2𝑘−𝐴)^2𝑘+𝑝(𝑡)[𝑎(𝑡)𝑟(𝑡)]^𝛽−2𝛼𝑦^2𝑘(𝑡)=𝐴̇(𝑡)𝑥𝑧−𝑝(𝑡)[𝑎(𝑡)𝑟(𝑡)]^𝛽−2𝛼𝑥^2𝑘(𝑠)+𝑝(𝑡)[𝑎(𝑡)𝑟(𝑡)]^𝛽−2𝛼𝑦^2𝑘(𝑡)=𝐴̇(𝑡)𝑥𝑧. (2.9) Since |𝑥𝑧|=|𝐵^1/2(𝑡)𝑥𝑧/𝐵^1/2(𝑡)|≤(𝐵(𝑡)𝑥²/2+𝑧²/2)/𝐵^1/2(𝑡)≤(𝐵(𝑡)[𝑥²/2𝑘+𝐾₁]+𝑧²/2)/𝐵^1/2(𝑡)≤𝑉(𝑠)/𝐵^1/2(𝑡)+𝐾₁𝐵^1/2(𝑡) for some constant 𝐾₁≥0, we have 𝑉̇(𝑠)≤𝐴̇(𝑡)𝑉(𝑠)𝐵^1/2(𝑡)+𝐾₁𝐴̇(𝑡)𝐵^1/2(𝑡). (2.10) Since 𝐴̇(𝑡)=𝐴^′(𝑡)𝑑𝑡𝑑𝑠=𝐴^′(𝑡)𝑎^𝛽(𝑡)𝑟^𝛼(𝑡), (2.11) letting 𝜏(𝑠) denote the inverse function of 𝑠(𝑡), we obtain 𝐴̇^𝑠₀(𝜏(𝑣))𝐵^1/2(𝜏(𝑣))𝑑𝑣=^𝑡₀𝐴^′(𝑢)𝐵^1/2(𝑢)𝑑𝑢. (2.12) Moreover, 𝐴̇^𝑠₀(𝜏(𝑣))𝐵^1/2(𝜏(𝑣))𝑑𝑣=^𝑡₀𝐴^′(𝑢)𝐵^1/2(𝑢)𝑑𝑢. (2.13) Integrating (2.10) from 0 to 𝑠, we obtain 𝑉(𝑠)≤𝑉(0)+^𝑠₀𝐴̇(𝜏(𝑣))𝐵^1/2(𝜏(𝑣𝑉(𝑣)))𝑑𝑣+𝐾₁^𝑠₀𝐴̇(𝜏(𝑣))𝐵^1/2(𝜏(𝑣))𝑑𝑣. (2.14) Condition (2.5) implies that the second integral on the right-hand side of (2.10) is convergent. By Gronwall’s inequality, we have 𝑉(𝑠)≤𝑀₁exp^𝑠₀𝐴̇(𝜏(𝑣))𝐵^1/2(𝜏(𝑣))𝑑𝑣, (2.15) for some constant 𝑀₁>0. By condition (2.4), the aforementioned integral is convergent, and we have that 𝑉(𝑠) is bounded, say, 𝑉(𝑠)≤𝑀₂ for some 𝑀₂>0. Therefore, 𝐵(𝑡)𝑦^2𝑘(𝑡)=𝐵(𝑡)𝑥^2𝑘(𝑠)≤2𝑘𝑀₂, (2.16) from which it follows that ^∞₀𝑦^2𝑘(𝑢)𝑑𝑢≤2𝑘𝑀₂^∞₀1𝐵(𝑢)𝑑𝑢<∞ (2.17) by condition (2.6), so all solutions of (1.5) are of the nonlinear limit circle type, and this completes the proof of Theorem 2.1.

If 𝑎(𝑡)≡1 in (1.5), then it turns into𝑦^′′+𝑝(𝑡)𝑦^′+𝑟(𝑡)𝑦^2𝑘−1=0, (2.18) and we have the following corollary.

Corollary 2.2. Assume that _∞0𝑝(𝑢)/𝑟^𝛼(𝑢)+𝛼𝑟^′(𝑢)/𝑟^1+𝛼(𝑢)^′𝑟^{(𝛽−𝛼)/2}(𝑢)𝑑𝑢<∞,_∞0𝑝(𝑢)𝑟^𝛼(𝑢)+𝛼𝑟^′(𝑢)𝑟^1+𝛼(𝑢)^′𝑟^{(𝛽−𝛼)/2}(𝑢)𝑑𝑢<∞,_∞01𝑟^𝛽−𝛼(𝑢)𝑑𝑢<∞. (2.19) Then (2.18) is of the nonlinear limit circle type.

The aforementioned theorem and corollary offer sufficient conditions to guarantee (1.5) and (2.18) to be of the nonlinear limit circle type, respectively. The next theorem gives necessary conditions to guarantee that (1.5) is of the nonlinear limit circle type.

Lemma 2.3 (see [10]). Assume that there exists 𝑁₁>0 such that [𝑎(𝑡)𝑟(𝑡)]^′𝑎^1/2(𝑡)𝑟^3/2(𝑡)≤𝑁₁,_∞0(𝑎(𝑡)𝑟(𝑡))^′2𝑎(𝑡)𝑟³(𝑡)𝑑𝑡<∞. (2.20)(2.21) If 𝑦(𝑡) is a nonlinear limit circle type solution of (1.5), then (𝑎_∞0(𝑡)𝑟(𝑡))^′2𝑦²(𝑡)𝑎(𝑡)𝑟³(𝑡)𝑑𝑡<∞. (2.22)

Proof. We have(𝑎_∞0(𝑡)𝑟(𝑡))^′2𝑦²(𝑡)𝑎(𝑡)𝑟³(𝑡)𝑑𝑡≤𝑁²₁^∞₀𝑦^2𝑘(𝑡)𝑑𝑡+^∞₀(𝑎(𝑡)𝑟(𝑡))^′2𝑎(𝑡)𝑟³(𝑡)𝑑𝑡<∞ (2.23) by (1.7) and (2.21).

Theorem 2.4. Suppose that there exist constants 𝑁₂>0 and 𝑀₃>0 such that 𝑎^1/2(𝑡)𝑟^′(𝑡)𝑟^3/2(𝑡)≤𝑁₂,_∞0𝑎(𝑢)𝑟^′(𝑢)²𝑟³(𝑢)𝑝𝑑𝑢<∞,₂(𝑡)𝑎(𝑡)𝑟(𝑡)≤𝑀₃,_∞0𝑝²(𝑡)𝑎(𝑡)𝑟(𝑡)𝑑𝑡<∞. (2.24)(2.25)(2.26)(2.27) If 𝑦(𝑡) is a nonlinear limit circle type solution of (1.5), then _∞0𝑎(𝑢)𝑦^′(𝑢)²𝑟(𝑢)<∞. (2.28)

Proof. If we multiply (1.5) by 𝑦(𝑡)/𝑟(𝑡), use the identity 𝑎(𝑡)𝑦^′′=𝑎(𝑡)𝑦^′𝑦^′−𝑎(𝑡)𝑦^′2, (2.29) and to integrate by parts, we obtain 𝑎(𝑡)𝑦^′(𝑡)𝑦(𝑡)𝑟(𝑡)−𝑎𝑡₁𝑦^′𝑡₁𝑦𝑡₁𝑟𝑡₁+^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)𝑦(𝑢)𝑟^′(𝑢)𝑟²(𝑢)+𝑑𝑢_𝑡𝑡1𝑦^2𝑘(𝑢)𝑑𝑢+^𝑡_𝑡1𝑝(𝑢)𝑦^′(𝑢)𝑦(𝑢)𝑟(𝑢)𝑑𝑢−^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)²𝑟(𝑢)𝑑𝑢=0 (2.30) for any 𝑡₁≥0. Now conditions (2.26), (2.27), and (1.7) imply ^∞_𝑡1𝑝²(𝑢)𝑦^2𝑘(𝑢)+1𝑎(𝑢)𝑟(𝑢)𝑑𝑢≤𝑀₃^∞_𝑡1𝑦^2𝑘(𝑢)𝑑𝑢+^∞_𝑡1𝑝²(𝑢)𝑎(𝑢)𝑟(𝑢)𝑑𝑢<∞. (2.31) Therefore, there exists a constant 𝑘₀>0,  subject to ^𝑡_𝑡1𝑝²(𝑢)𝑦^2𝑘(𝑢)+1𝑎(𝑢)𝑟(𝑢)𝑑𝑢≤𝑀₃^𝑡_𝑡1𝑦^2𝑘(𝑢)𝑑𝑢+^𝑡_𝑡1𝑝²(𝑢)𝑎(𝑢)𝑟(𝑢)𝑑𝑢<𝑘²₀. (2.32) Applying Schwartz inequality, we obtain ^𝑡_𝑡1𝑝(𝑢)𝑦^′(𝑢)𝑦(𝑢)𝑟(𝑢)𝑑𝑢≤^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)²𝑟(𝑢)𝑑𝑢^1/2𝑡_𝑡1𝑝²(𝑢)𝑦^2𝑘(𝑢)+1𝑎(𝑢)𝑟(𝑢)𝑑𝑢^1/2≤𝑘₀^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)²𝑟(𝑢)𝑑𝑢^1/2. (2.33) By the Schwartz inequality, ^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)𝑦(𝑢)𝑟^′(𝑢)𝑟²(𝑢)𝑑𝑢≤^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)²𝑟(𝑢)𝑑𝑢^1/2𝑡_𝑡1𝑎(𝑢)𝑦²(𝑢)𝑟^′(𝑢)²𝑟³(𝑢)𝑑𝑢^1/2. (2.34) Now (2.24) implies 𝑎(𝑡)𝑦²(𝑡)𝑟^′(𝑡)²𝑟³(𝑡)≤𝑎(𝑡)𝑟^′(𝑡)²𝑟³(𝑡)𝑦^2𝑘(𝑡)+1≤𝑁²₂𝑦^2𝑘(𝑡)+𝑎(𝑡)𝑟^′(𝑡)²𝑟³(𝑡). (2.35) So integrating the previous inequality and applying (2.25) and (1.7), we obtain _∞𝑡1𝑎(𝑢)𝑦²(𝑢)𝑟^′(𝑢)²𝑟³(𝑢)𝑑𝑢≤𝐾²₂<∞ (2.36) for some constant 𝐾₂>0.
If 𝑦(𝑡) is not eventually monotonic, let {𝑡_𝑗}→∞ be an increasing sequence of zeros of 𝑦′(𝑡). Then from (2.30), we have𝑘₀+𝐾₂𝐻^1/2𝑡_𝑗+𝐾₃≥𝐻𝑡_𝑗, (2.37) where 𝐻(𝑡)=^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)²𝑟(𝑢)𝑑𝑢, (2.38) and 𝐾₃>0 is a constant. It follows that 𝐻(𝑡_𝑗)≤𝐾₄<∞ for all 𝑗 and for some constant 𝐾₄>0, so (2.28) holds.
If 𝑦(𝑡) is eventually monotonic, then 𝑦(𝑡)𝑦′(𝑡)≤0 for all 𝑡≥𝑡₁ for sufficiently large 𝑡₁≥0 since otherwise (1.7) would be violated. Using this fact in (2.30), we can repeat the type of argument used previously to obtain that (2.28) holds.

The following theorem gives sufficient conditions to ensure that (1.5) is of the nonlinear limit point type.

Theorem 2.5. Suppose that conditions (2.4), (2.5), and (2.20)–(2.27) hold. If^∞₀1𝐵(𝑢)𝑑𝑢=∞, (2.39) then (1.5) is of the nonlinear limit point type.

Proof. As in the proof of Theorem 2.1, define 𝑉()=𝑧𝑥,𝑧,𝑠²2(𝑡)𝑥+𝐵^2𝑘2𝑘 (2.40) and differentiate it to obtain ̇𝐴̇(𝑡𝑉(𝑠)𝑉≥−)𝐵^1/2(𝑡)−𝐾₁𝐴̇(𝑡)𝐵^1/2(𝑡), (2.41) and so we then have ̇𝐴̇(𝑡𝑉(𝑠)𝑉+)𝐵^1/2(𝑡)≥−𝐾₁𝐴̇(𝑡)𝐵^1/2(𝑡). (2.42) If we define functions 𝐻 and ℎ∶𝑅₊↦𝑅 by 𝐻(𝑡)=|𝐴̇(𝑡)|/𝐵^1/2(𝑡), and ℎ(𝑡)=𝐾₁|𝐴̇(𝑡)|𝐵^1/2(𝑡), then we have 𝑑𝑑𝑠exp^𝑠₀𝐻(𝜏(𝜉))𝑑𝜉≥−ℎ(𝑡)exp^𝑠₀𝐻(𝜏(𝜉))𝑑𝜉. (2.43) Now condition (2.4) guarantees that exp^∞₀𝐻(𝜏(𝜉))𝑑𝜉≤𝐾₅<∞, (2.44) for some constant 𝐾₅>0, while condition (2.5) implies that 𝐾₅^∞₀ℎ(𝜏(𝜉))𝑑𝜉≤𝐾₆<∞, (2.45) for some 𝐾₆>0. Let 𝑦(𝑡) be any solution of (1.5) such that 𝑉(𝑥(0),𝑧(0),0)>𝐾₆+1. Integrating (2.43), we get 𝑉(𝑠)exp^𝑠₀𝐻(𝜏(𝜉))𝑑𝜉≥𝑉(0)−𝐾₆>1, (2.46) and so 𝑉(𝑠)≥1𝐾₅ (2.47) for 𝑠≥0. Dividing both sides of this last inequality by 𝐵(𝑡) and rewriting the left-hand side in terms of 𝑡, we have 𝑎(𝑡)𝑦^′(𝑡)²(𝑡)+𝑦2𝑟^2𝑘(𝑡)+𝑝(𝑡)2𝑘𝑟(𝑡)(𝑎(𝑡)𝑟(𝑡+𝛼))^′𝑟²(𝑡)𝑦(𝑡)𝑦^′(𝑡)+𝑝²(𝑡)(𝑡)𝑟(𝑡)𝑝(𝑡𝑎(𝑡)𝑟(𝑡2𝑎+𝛼)())^′𝑎(𝑡)𝑟²(𝑡)+𝛼²(𝑎(𝑡)𝑟(𝑡))^′22𝑎(𝑡)𝑟³(𝑡)𝑦²(𝑡)≥1𝐾₅[𝑎(𝑡)𝑟(𝑡)]^𝛽−𝛼. (2.48) If 𝑦(𝑡) is a limit circle type solution of (1.5), then Theorem 2.4 implies _∞0𝑎(𝑡)𝑦^′(𝑡)²𝑟(𝑡)𝑑𝑡<∞, (2.49) and Lemma 2.3 implies (𝑎_∞0(𝑡)𝑟(𝑡))^′2𝑦²(𝑡)𝑎(𝑡)𝑟³(𝑡)𝑑𝑡<∞. (2.50) By the Schwartz inequality, ^∞₀(𝑎(𝑡)𝑟(𝑡))^′𝑟²(𝑡)𝑦(𝑡)𝑦^′(𝑡)𝑑𝑡≤^∞₀(𝑎(𝑡)𝑟(𝑡))^′2𝑦²(𝑡)𝑎(𝑡)𝑟³(𝑡)𝑑𝑡^1/2×^∞₀𝑎(𝑡)𝑦^′(𝑡)²𝑟(𝑡)𝑑𝑡^1/2<∞. (2.51) From condition (2.27), using Schwartz inequality, we get ^∞₀𝑝(𝑡)(𝑎(𝑡)𝑟(𝑡))^′𝑎(𝑡)𝑟²(𝑡)𝑑𝑡≤^∞₀(𝑎(𝑡)𝑟(𝑡))^′2𝑦²(𝑡)𝑎(𝑡)𝑟³(𝑡)𝑑𝑡^1/2×^∞₀𝑝²(𝑡)𝑎(𝑡)𝑟(𝑡)𝑑𝑡^1/2<∞. (2.52) So that if 𝑦(𝑡) is a limit circle solution of (1.5), from conditions (2.26) and (2.27), we obtain ^∞₀𝑝²(𝑡)𝑦²(𝑡)2𝑎(𝑡)𝑟(𝑡)𝑑𝑡<^∞₀𝑝²(𝑡)𝑦^2𝑘(𝑡)+12𝑎(𝑡)𝑟(𝑡)𝑑𝑡<∞,_∞0𝛼𝑝(𝑡)(𝑎(𝑡)𝑟(𝑡))^′𝑦²(𝑡)𝑎(𝑡)𝑟²(𝑡)𝑑𝑡≤𝛼^∞₀𝑝²(𝑡)𝑦²(𝑡)𝑎(𝑡)𝑟(𝑡)𝑑𝑡^1/2∞₀(𝑎(𝑡)𝑟(𝑡))^′2𝑦²(𝑡)𝑎(𝑡)𝑟³(𝑡)𝑑𝑡^1/2<∞. (2.53) Furthermore, ^∞₀𝑝(𝑡)𝑦^′(𝑡)𝑦(𝑡)𝑟(𝑡)𝑑𝑡≤^∞₀𝑎(𝑡)𝑦^′(𝑡)²𝑟(𝑡)𝑑𝑡^1/2∞₀𝑝²(𝑡)𝑦²(𝑡)𝑎(𝑡)𝑟(𝑡)𝑑𝑡^1/2<∞. (2.54) Consequently, integrating both sides of (2.48), we see that the integrand of the left side of (2.48) is bounded, but the integrand of the right side of (2.48) tends to infinity according to condition (2.39). This leads to a contradiction, so 𝑦(𝑡) is a limit point type solution of (1.5), and (1.5) is of the nonlinear limit point type.

The last theorem and corollary give the sufficient and necessary conditions to guarantee (1.5) and (2.18) to be of the nonlinear limit circle type, respectively.

Theorem 2.6. Assume that conditions (2.4), (2.5), (2.20), (2.21), (2.24), (2.25), (2.26), and (2.27) hold. Then (1.5) is of the nonlinear limit circle type if and only if ^∞₀1(𝑎(𝑡)𝑟(𝑡))^{𝑘/(𝑘+1)}𝑑𝑡<∞. (2.55)

When one specializes this theorem to (2.18), that is, 𝑎(𝑡)≡1 in (1.5), one obtains the following corollary.

Corollary 2.7. Assume that _∞0𝑝(𝑡)/𝑟^𝛼(𝑡)+𝛼𝑟^′(𝑡)/𝑟^𝛼+1(𝑡)^′𝑟^{(𝛽−𝛼)/2}(𝑡)𝑑𝑡<∞,^∞₀𝑝(𝑡)𝑟^𝛼(𝑡)+𝛼𝑟^′(𝑡)𝑟^𝛼+1(𝑡)^′𝑟^{(𝛽−𝛼)/2}(𝑡)𝑟𝑑𝑡<∞,_′(𝑡)𝑟^3/2(𝑡)≤𝑁₁,^∞₀𝑟^′(𝑡)²𝑟³(𝑡)𝑝𝑑𝑡<∞,²(𝑡)𝑟(𝑡)≤𝑀₃,^∞₀𝑝²(𝑡)𝑟(𝑡)𝑑𝑡<∞. (2.56) Then (2.18) is of the nonlinear limit circle type if and only if ^∞₀1𝑟(𝑡)^{𝑘/(𝑘+1)}𝑑𝑡<∞. (2.57)

Acknowledgments

The authors would like to thank the referee for his helpful comments on this paper. This research was partially supported by the NSF of China (Grant 61104007) and NSF of Qufu Normal University (Grant xj201023).