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

Abstract

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

1. Introduction

In 1910, Weyl [1] published his now classical 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 (denoted by 𝑦(𝑡)∈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)

Weyl 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) linearly independent 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 the following equation𝑎(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 has been considerable interest in this problem over the years (see [1–10] and references cited therein). 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 reader to the recent monograph [10]. In this paper, we will discuss the equation with damping term 𝑎(𝑡)𝑦^′′+𝑏(𝑡)𝑦^′+𝑟(𝑡)𝑦^𝛾=0, (1.5)

where 𝑎,𝑟∶𝑅₊↦𝑅 and 𝑏∶𝑅₊↦𝑅₊ are continuous, 𝑎^′,𝑟^′𝜖𝐴𝐶_loc(𝑅₊), 𝑎^′′,𝑟^′′𝜖𝐿²_loc(𝑅₊), 𝑎(𝑡)>0, 𝑟(𝑡)>0, and 0<𝛾≤1, 𝛾=odd/odd, say 𝛾=(2𝑀−1)/(2𝑁−1), 𝑀 and 𝑁 are positive integers, we can write 𝛾=2𝑘−1, where 𝑘=(𝑀+𝑁−1)/(2𝑁−1). When 𝑏(𝑡)≡0, then (1.5) turns into the following equation𝑎(𝑡)𝑦^′′+𝑟(𝑡)𝑦^𝛾=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^𝛾+1(𝑡)𝑑𝑡<∞, (1.7) and it is of the nonlinear limit point type otherwise, that is, there exists a nontrivial solution 𝑦(𝑡) satisfying ^∞₀𝑦^𝛾+1(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 solution satisfying (1.8).
In this paper, we will give sufficient and necessary conditions to guarantee the nonlinear limit circle type or nonlinear limit point type for (1.5).

2. Main Results

To simplify notations, let 𝛼=1/2(𝑘+1) and 𝛽=(2𝑘+1)/2(𝑘+1). We define 𝑠=^𝑡₀𝑟^𝛼(𝑢)𝑎^𝛽(𝑢)𝑑𝑢,𝑥(𝑠)=𝑦(𝑡). (2.1)

Then (1.5) becomes 𝑥̈+𝐴(𝑡)𝑥̇+𝐵(𝑡)𝑥^𝛾=0, (2.2)

where 𝐴(𝑡)=𝑏(𝑡)𝑟^𝛼(𝑡)𝑎^𝛽(𝑡)+𝛼(𝑎(𝑡)𝑟(𝑡))^′𝑎^𝛼(𝑡)𝑟^𝛼+1(𝑡)^′,𝐵(𝑡)=(𝑎(𝑡)𝑟(𝑡))^𝛽−𝛼. (2.3)

Note that in the transformation (2.1), 𝑘 is no longer an integer here. In fact, 1/2<𝑘≤1.

We begin with a boundedness result.

Theorem 2.1. If the condition ^∞₀(𝑎(𝑡)𝑟(𝑡))^′₋𝑎(𝑡)𝑟(𝑡)𝑑𝑡<∞ (2.4) holds, then each solution of (1.5) is bounded.

Proof. We rewrite (1.5) as the system 𝑦^′=𝑤,𝑤^′=−𝑎^′(𝑡)𝑤−𝑟(𝑡)𝑦^𝛾−𝑏(𝑡)𝑤𝑎(𝑡), (2.5) and we define 𝑉(𝑡)=𝑎(𝑡)𝑤²(𝑡)+𝑦2𝑟^𝛾+1(𝑡)𝛾+1, (2.6) then we have 𝑉^′(𝑡)=𝑎(𝑡)𝑤𝑟(𝑡)𝑤^′+𝑎(𝑡)2𝑟(𝑡)^′𝑤²+𝑦^𝛾(𝑡)𝑤𝑎=−^′(𝑡)𝑟(𝑡)𝑤²−𝑏(𝑡)𝑟(𝑡)𝑤²+𝑎(𝑡)2𝑟(𝑡)^′𝑤²𝑎=−^′(𝑡)𝑟(𝑡)𝑤²−𝑏(𝑡)𝑟(𝑡)𝑤²+𝑎^′(𝑡)(𝑡)𝑤2𝑟²−𝑎(𝑡)𝑟^′(𝑡)2𝑟²(𝑡)𝑤²𝑎=−^′(𝑡)(𝑡)+𝑎(𝑡)𝑟2𝑟^′(𝑡)2𝑟²(𝑡)𝑤²−𝑏(𝑡)𝑟(𝑡)𝑤²(𝑎(𝑡)𝑟(𝑡=−))^′2𝑟²(𝑡)𝑤²−𝑏(𝑡)𝑟(𝑡)𝑤²(𝑎(𝑡)𝑟(𝑡≤−))^′2𝑟²(𝑡)𝑤²≤(𝑎(𝑡)𝑟(𝑡))^′₋2𝑟²(𝑡)𝑤²≤(𝑎(𝑡)𝑟(𝑡))^′₋2𝑟²(𝑡)𝑤²+(𝑎(𝑡)𝑟(𝑡))^′₋𝑎(𝑡)𝑟(𝑡)𝑦^𝛾+1(𝑡)=(𝑎(𝑡)𝑟(𝑡𝛾+1))^′₋𝑎(𝑡)𝑟(𝑡)𝑉(𝑡). (2.7) Gronwall’ inequality and condition (2.4) imply that 𝑉(𝑡) is bounded, so 𝑦(𝑡) is bounded.

To prove our main limit-circle result, we write (2.2) as the system 𝑥̇=𝑧−𝐴(𝑡)𝑥,𝑧̇=𝐴̇(𝑡)𝑥−𝐵(𝑡)𝑥^𝛾. (2.8)

Theorem 2.2. Assume that condition (2.4) holds and ^∞₀𝛼(𝑎(𝑡)𝑟(𝑡))^′𝑎^1/2(𝑡)𝑟^3/2(𝑡)^′+𝛼2−𝛼²(𝑎(𝑡)𝑟(𝑡))^′2𝑎^3/2(𝑡)𝑟^5/2(𝑡)+𝑏(𝑡)𝑟^𝛼(𝑡)/𝑎^𝛽(𝑡)^′(𝑎(𝑡)𝑟(𝑡))^{𝑘/(2(𝑘+1))}𝑑𝑡<∞, (2.9) and condition 𝐵^′(𝑡)≤𝛾+1𝐴(𝑡)𝐵(𝑡), (2.10) is satisfied. If ^∞₀1𝐵(𝑡)𝑑𝑡<∞, (2.11) then (1.5) is of the nonlinear limit circle type, that is, each solution 𝑦(𝑡) of (1.5) satisfies ^∞₀𝑦^𝛾+1(𝑡)𝑑𝑡<∞. (2.12)

Proof. Define 𝑉()=𝑧𝑥,𝑧,𝑠²2(𝑡)𝑥+𝐵^𝛾+1+𝛾+1^𝑡₀𝐴(𝜉)𝐵(𝜉)−𝐵̇(𝜉)𝑟𝛾+1^𝛼(𝜉)𝑎^𝛽(𝜉)𝑦^𝛾+1(𝜉)𝑑𝜉. (2.13) By direct calculation, we have 𝑉̇=𝑧𝑧̇+𝐵̇(𝑡)𝑥^𝛾+1(𝑡)𝑥𝛾+1+𝐵^𝛾̇𝐴(𝑡)𝐵(𝑡)−𝐵̇(𝑡)𝑥+𝑦𝛾+1^𝛾+1(𝑡)=𝑧𝐴̇(𝑡)𝑥−𝐵(𝑡)𝑥^𝛾+𝐵̇(𝑡)𝑥^𝛾+1(𝑡)𝑥𝛾+1+𝐵^𝛾((𝑡)𝑥)+𝐴(𝑡)𝐵(𝑡)−𝐵̇(𝑡)𝑧−𝐴𝑦𝛾+1^𝛾+1(𝑡)=𝐴̇(𝑡)𝑥𝑧. (2.14) Since condition (2.4) and Theorem 2.1 are satisfied, the solution of (1.5) 𝑦(𝑡)=𝑥(𝑠) is bounded, that is, there exists a constant 𝐾₁≥0 such that |𝑥(𝑡)|≤𝐾₁. So 𝐴̇(𝑡)=𝐴̇(𝑡𝑥|𝑥𝑧)|^(1−𝛾)/2[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}×(𝑎(𝑡)𝑟(𝑡))^{(𝛽−𝛼)/2}|𝑥|^(1+𝛾)/2|𝑧|≤𝐾₁^(1−𝛾)/2𝐴̇(𝑡)[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}×(𝑎(𝑡)𝑟(𝑡))^𝛽−𝛼𝑥^1+𝛾+𝑧²2≤𝐾₁^(1−𝛾)/2𝐴̇(𝑡)[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}×𝐵𝑥^1+𝛾+𝑧1+𝛾2²≤𝐾₁^(1−𝛾)/2𝐴̇(𝑡)[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}×𝑉(𝑡). (2.15) Now 𝐴̇(𝑡)=𝐴^′(𝑡)𝑑𝑡𝑑𝑠=𝐴^′(𝑡)𝑎^𝛽(𝑡)𝑟^𝛼(𝑡),𝐴′(𝑡)[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}=𝑏(𝑡)𝑟^𝛼(𝑡)/𝑎^𝛽(𝑡)+𝛼(𝑎(𝑡)𝑟(𝑡))^′/𝑎^𝛼(𝑡)𝑟^𝛼+1(𝑡)[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}=𝛼(𝑎(𝑡)𝑟(𝑡))^′′𝑎^1/2(𝑡)𝑟^3/2(𝑡)−𝛼²𝑎^′(𝑡)(𝑎(𝑡)𝑟(𝑡))^′/(𝑎(𝑡)𝑟(𝑡))^1+𝛼+(1+𝛼)𝛼𝑟^′(𝑡)(𝑎(𝑡)𝑟(𝑡))^′/𝑎^𝛼(𝑡)𝑟^𝛼+2(𝑡)[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}+𝑏(𝑡)𝑟^𝛼(𝑡)/𝑎^𝛽(𝑡)^′[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}=𝛼(𝑎(𝑡)𝑟(𝑡))^′′𝑎^1/2(𝑡)𝑟^3/2(𝑡)−𝛼²𝑎^′(𝑡)(𝑎(𝑡)𝑟(𝑡))^′(𝑎(𝑡)𝑟(𝑡))^3/2−()𝛼𝑟1+𝛼^′(𝑡)(𝑎(𝑡)𝑟(𝑡))^′𝑎^1/2(𝑡)𝑟^5/2(𝑡)+𝑏(𝑡)𝑟^𝛼(𝑡)/𝑎^𝛽(𝑡)^′[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}=𝛼(𝑎(𝑡)𝑟(𝑡))^′𝑎^1/2(𝑡)𝑟^3/2(𝑡)^′+𝛼2(𝑎(𝑡)𝑟(𝑡))^′𝑎^′(𝑡)𝑟(𝑡)+𝑎(𝑡)𝑟^′(𝑡)𝑎^3/2(𝑡)𝑟^5/2(𝑡)−𝛼²(𝑎(𝑡)𝑟(𝑡))^′2𝑎^3/2(𝑡)𝑟^5/2(𝑡)+𝑏(𝑡)𝑟^𝛼(𝑡)/𝑎^𝛽(𝑡)^′[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}=𝛼(𝑎(𝑡)𝑟(𝑡))^′𝑎^1/2(𝑡)𝑟^3/2(𝑡)^′+𝛼2(𝑎(𝑡)𝑟(𝑡))^′2𝑎^3/2(𝑡)𝑟^5/2(𝑡)−𝛼²(𝑎(𝑡)𝑟(𝑡))^′2𝑎^3/2(𝑡)𝑟^5/2(𝑡)+𝑏(𝑡)𝑟^𝛼(𝑡)/𝑎^𝛽(𝑡)^′[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}=𝛼(𝑎(𝑡)𝑟(𝑡))^′𝑎^1/2(𝑡)𝑟^3/2(𝑡)^′+𝛼2−𝛼²(𝑎(𝑡)𝑟(𝑡))^′2𝑎^3/2(𝑡)𝑟^5/2(𝑡)+𝑏(𝑡)𝑟^𝛼(𝑡)/𝑎^𝛽(𝑡)^′[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}. (2.16) Let 𝜏(𝑠) denote the inverse function of 𝑠(𝑡), we obtain that 𝐴̇_𝑠0(𝜏(𝑣))𝐵^1/2(𝜏(𝑣))𝑑𝑣=^𝑠₀𝐴^′(𝜏(𝑣))𝑎^𝛽(𝜏(𝑣))/𝑟^𝛼(𝜏(𝑣))𝐵^1/2(𝜏(𝑣=))𝑑𝑣_𝑡0𝛼(𝑎(𝑢)𝑟(𝑢))^′𝑎^1/2(𝑢)𝑟^3/2(𝑢)^′+𝛼2−𝛼²(𝑎(𝑢)𝑟(𝑢))^′2𝑎^3/2(𝑢)𝑟^5/2(𝑢)+𝑏(𝑢)𝑟^𝛼(𝑢)/𝑎^𝛽(𝑢)^′(𝑎(𝑢)𝑟(𝑢))^{𝑘/(2(𝑘+1))}𝑑𝑢 (2.17) is convergent by condition (2.9). Hence, integrating 𝑉̇(𝑠), applying Gronwall’s inequality, and using condition (2.9), we obtain that 𝑉(𝑠) is bounded, so 𝐵(𝑡)𝑦^𝛾+1=(𝑎(𝑡)𝑟(𝑡𝛾+1))^𝛽−𝛼𝑦^𝛾+1𝛾+1≤𝐾₂ (2.18) for some constant 𝐾₂>0. Condition (2.11) then implies that 𝑦(𝑡) is of the nonlinear limit circle type.
When 𝑎(𝑡)≡1, the (1.5) becomes 𝑦^′′+𝑏(𝑡)𝑦^′+𝑟(𝑡)𝑦^𝛾=0. (2.19) In this case, 𝐴(𝑡)=𝑏(𝑡)𝑟^𝛼(𝑡)+𝛼𝑟^′(𝑡)/𝑟^𝛼+1(𝑡), 𝐵(𝑡)=𝑟^𝛽−𝛼(𝑡). We get the following corollary.

Corollary 2.3. Assume condition (2.4) and 𝛼𝑟(^∞₀𝑡)^′𝑟^3/2(𝑡)^′+𝛼2−𝛼²𝑟(𝑡)^′2𝑟^5/2(𝑡)+𝑏(𝑡)𝑟^𝛼(𝑡)^′𝑟(𝑡)^{𝑘/(2(𝑘+1))}𝑑𝑡<∞, (2.20) and condition 𝐵̇(𝑡)≤𝛾+1𝐴(𝑡)𝐵(𝑡) (2.21) is satisfied. If ^∞₀1𝐵(𝑡)𝑑𝑡<∞, (2.22) then (2.19) is of the nonlinear limit circle type, that is, each solution 𝑦(𝑡) of (2.19) satisfies ^∞₀𝑦^𝛾+1(𝑡)𝑑𝑡<∞. (2.23)

Example 2.4. Consider the following second-order nonlinear differential equation 𝑦^′′+1+𝑡^21/4𝑦^′+1+𝑡^25/2𝑦^1/3=0, (2.24) here 𝑏(𝑡)=(1+𝑡²)^1/4, 𝑟(𝑡)=(1+𝑡²)^5/2, 𝛾=1/3. We can easily verify that all the conditions in Corollary 2.3 are fulfilled, so each solution of (2.24) is of the nonlinear limit circle type. We note further that the type of (2.24) cannot be determined since 𝑏(𝑡)≠0.

Next, we give a necessary condition for the sublinear (1.5) to be of the nonlinear limit circle type.

Theorem 2.5. Suppose condition (2.4), ^∞₀𝑏²(𝑡)𝑎(𝑡)𝑟(𝑡)𝑑𝑡<∞,_∞0𝑎(𝑡)𝑟^′(𝑡)²𝑟³(𝑡)𝑑𝑡<∞ (2.25) hold. If 𝑦(𝑡) is a nonlinear limit circle type solution of (1.5), then ^∞₀𝑎(𝑡)𝑦^′(𝑡)²𝑟(𝑡)𝑑𝑡<∞. (2.26)

Proof. Let 𝑦(𝑡) be a nonlinear limit circle type solution of (1.5), then 𝑦(𝑡) is bounded by Theorem 2.1. Multiplying (1.5) by 𝑦(𝑡)/𝑟(𝑡), noting that 𝑎(𝑡)𝑦^′′=𝑎(𝑡)𝑦^′𝑦^′−𝑎(𝑡)𝑦^′2, (2.27) and integrating 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_𝑡𝑡1𝑎(𝑢)𝑦^′(𝑢)^′𝑦(𝑢)𝑟(𝑢)𝑑𝑢−^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)²𝑟(𝑢)𝑑𝑢=0. (2.28) Using (1.5), we have 𝑎(𝑡)𝑦^′(𝑡)𝑦(𝑡)𝑟(𝑡)−𝑎𝑡₁𝑦^′𝑡₁𝑦𝑡₁𝑟𝑡₁+^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)𝑦(𝑢)𝑟^′(𝑢)𝑟²(𝑢)𝑑𝑢^𝑡_𝑡1𝑏(𝑢)𝑦^′(𝑢)𝑦(𝑢)𝑟(𝑢)+𝑑𝑢_𝑡𝑡1𝑦^𝛾+1(𝑢)−^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)²𝑟(𝑢)𝑑𝑢=0. (2.29) Denote 𝐻(𝑡)=^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)²𝑟(𝑢)𝑑𝑢, (2.30) by Schwartz inequality, the boundedness of 𝑦(𝑡), and condition (2.14), we obtain ^𝑡_𝑡1𝑎(𝑢)𝑦^′(𝑢)𝑦(𝑢)𝑟^′(𝑢)𝑟²(𝑢)𝑑𝑢≤𝐻^1/2(𝑡)^𝑡_𝑡1𝑎(𝑢)𝑟^′(𝑢)²𝑦²(𝑢)𝑟³(𝑢)𝑑𝑢^1/2<𝑀₁𝐻^1/2(𝑡) (2.31) for some constant 𝑀₁>0. By Schwartz inequality, the boundedness of 𝑦(𝑡), and condition (2.25), ^𝑡_𝑡1𝑏(𝑢)𝑦^′(𝑢)𝑦(𝑢)𝑟(𝑢)𝑑𝑢≤^𝑡_𝑡1𝑏²(𝑢)𝑦²(𝑢)𝑎(𝑢)𝑟(𝑢)𝑑𝑢^1/2𝐻^1/2(𝑡)≤𝑀₂𝐻^1/2(𝑡), (2.32) for some constant 𝑀₂>0.
If 𝑦(𝑡) is not eventually monotonic, let {𝑡_𝑗}→∞ be an increasing sequence of zeros of 𝑦^′(𝑡). Then, by (2.28), there exists some constant 𝑀₃>0, such that 𝑀₁+𝑀₂𝐻^1/2𝑡_𝑗+𝑀₃≥𝐻𝑡_𝑗. (2.33) This implies 𝐻(𝑡_𝑗)≤𝑀₄<∞ for all 𝑗 and some 𝑀₄>0, so (2.26) holds.
If 𝑦(𝑡) is eventually monotonic, then 𝑦(𝑡)𝑦^′(𝑡)≤0 for all 𝑡≥𝑡₁ (𝑡₁≥0 large enough). Using (2.28), we can repeat the type of argument used above to obtain that (2.26) holds. This completes the proof of Theorem 2.5.

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

Theorem 2.6. Suppose condition (2.4), (2.9), (2.10), (2.25) hold. If (𝑎^∞₀(𝑡)𝑟(𝑡))^′2𝑎(𝑡)𝑟³(𝑡)𝑑𝑡<∞, (2.34)^∞₀𝑏²(𝑡)𝑟^3𝛼−𝛽(𝑡)𝑎^3𝛽−𝛼(𝑡)𝑑𝑡<∞,_∞0𝑡0𝐴(𝜉)𝐵(𝜉)−𝐴̇(𝜉)/𝛾+1𝑟^𝛼(𝜉)/𝑎^𝛽(𝜉)𝐵(𝑡)𝑑𝜉𝑑𝑡<∞, (2.35)^∞₀1𝐵(𝑢)𝑑𝑢=∞, (2.36) then (1.5) is of the nonlinear limit point type.

Proof. As in the proof of Theorem 2.2, we define 𝑉()=𝑧𝑥,𝑧,𝑠²2(𝑡)𝑥+𝐵^𝛾+1+𝛾+1^𝑡₀𝐴(𝜉)𝐵(𝜉)−𝐴̇(𝜉)𝑟𝛾+1^𝛼(𝜉)𝑎^𝛽(𝜉)𝑦^𝛾+1(𝜉)𝑑𝜉, (2.37) we differentiate it to obtain 𝑉̇=𝑧𝑧̇+𝐵̇(𝑡)𝑥^𝛾+1(𝑡)𝑥𝛾+1+𝐵^𝛾̇𝐴(𝑡)𝐵(𝑡)−𝐵̇(𝑡)𝑥+𝑦𝛾+1^𝛾+1(𝑡)=𝑧𝐴̇(𝑡)𝑥−𝐵(𝑡)𝑥^𝛾+𝐵̇(𝑡)𝑥^𝛾+1(𝑡)𝑥𝛾+1+𝐵^𝛾((𝑡)𝑥)+𝐴(𝑡)𝐵(𝑡)−𝐵̇(𝑡)𝑧−𝐴𝑦𝛾+1^𝛾+1(𝑡)=𝐴̇(𝑡)𝑥𝑧. (2.38) Let 𝑦(𝑡)=𝑥(𝑠) be any nontrivial solution of (1.5) with 𝑦(𝑡₁)=𝑥(𝑠(𝑡₁))=𝑥(𝑠₁)≠0. Theorem 2.1 implies that 𝑦(𝑡)=𝑥(𝑠) is bounded, so |𝑥(𝑠)|^(1−𝛾)/2≤𝐾₁ for some constant 𝐾₁>0. Hence 𝑉̇(𝑠)≥−𝐴̇(𝑡)|𝑥(𝑠)|^(1−𝛾)/2|𝑥(𝑠)|^(1+𝛾)/2|𝑧(𝑠)|≥−𝐾₁𝐴̇(𝑡)[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}×𝑧²2+𝐵(𝑡)𝑥^𝛾+1(𝑡)2≥−𝐾₁𝐴̇(𝑡)[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}×𝑧²2+𝐵(𝑡)𝑥^𝛾+1(𝑡)𝛾+1≥−𝐾₁𝐴̇(𝑡)[𝑎(𝑡)𝑟(𝑡)]^{(𝛽−𝛼)/2}×𝑉(𝑡). (2.39) Let 𝐻(𝑡)=𝐾₁|𝐴̇(𝑡)|/𝐵^1/2(𝑡), then we can write 𝑉̇+𝐻(𝑡)𝑉≥0. So 𝑑𝑉(𝑠)𝑑𝑠exp^𝑠_𝑠1𝐻(𝜏(𝜉))𝑑𝜉≥0. (2.40) Integrating the above inequality, we obtain 𝑉(𝑠)exp^𝑠_𝑠1𝐻𝜏(𝜉)≥𝑉𝑠₁. (2.41) Condition (2.9) implies ^𝑠_𝑠1𝐻(𝜏(𝜉))𝑑𝜉<∞, (2.42) and since 𝑉(𝑠₁)>0, we have 𝑉(𝑠)≥𝐾₂>0for𝑠≥𝑠₁. (2.43) Rewriting 𝑉(𝑥,𝑧,𝑠) in terms of 𝑦 and 𝑡 and dividing (2.42) by 𝐵(𝑡), we obtain 𝑦^′(𝑡)²𝑎(𝑡)(𝑡)+𝑦2𝑟^𝛾+1(𝑡)+𝑏(𝑡)𝛾+1[𝑎(𝑡)𝑟(𝑡)]^𝛽−𝛼[𝑎(𝑡)𝑟(𝑡+𝛼)]^′𝑟²(𝑡)𝑦(𝑡)𝑦^′(𝑡)+𝑏²(𝑡)𝑟^3𝛼−𝛽(𝑡)2𝑎^3𝛽−𝛼(𝑡)+𝛼²(𝑎(𝑡)𝑟(𝑡))^′22𝑎(𝑡)𝑟³(𝑡)𝑏(𝑡𝑎(𝑡)𝑟(𝑡+𝛼)[)]^′[𝑎(𝑡)𝑟(𝑡)]^2𝛽𝑦²(𝑡)+^𝑡₀𝐴(𝜉)𝐵(𝜉)−𝐵̇(𝜉)/𝛾+1𝑟^𝛼(𝜉)/𝑎^𝛽(𝜉)𝑦^𝛾+1(𝜉)𝐵(𝑡)𝐾𝑑𝜉≥²𝐵(𝑡). (2.44) If 𝑦(𝑡) is a limit circle type solution, then 𝑦(𝑡) is bounded, ∫^∞_𝑡1𝑦^𝛾+1(𝑡)𝑑𝑡<∞, and by condition (2.4) and (2.25), we get ^∞_𝑡1𝑦^′(𝑡)²𝑎(𝑡)2𝑟(𝑡)𝑑𝑡<∞, (2.45) according to Theorem 2.5. By condition (2.34), we get ^∞_𝑡1𝛼²(𝑎(𝑡)𝑟(𝑡))^′22𝑎(𝑡)𝑟³(𝑡)𝑦²(𝑡)𝑑𝑡<∞. (2.46) By the Schwartz inequality, ^∞_𝑡1𝛼[𝑎(𝑡)𝑟(𝑡)]^′𝑟²(𝑡)𝑦(𝑡)𝑦^′(𝑡)𝑑𝑡≤^∞_𝑡1𝛼²(𝑎(𝑡)𝑟(𝑡))^′2𝑦²(𝑡)𝑎(𝑡)𝑟³(𝑡)𝑑𝑡^1/2×^∞_𝑡1𝑦^′(𝑡)²𝑎(𝑡)𝑟(𝑡)𝑑𝑡^1/2<∞. (2.47) Noticing condition (2.35) and 𝑦(𝑡) being bounded, we get ^∞_𝑡1𝑏²(𝑡)𝑟^3𝛼−𝛽(𝑡)𝑎^3𝛽−𝛼(𝑡)𝑦²(𝑡)𝑑𝑡<∞, (2.48) so, by the Schwartz inequality, we have ^∞_𝑡1𝑏(𝑡)[𝑎(𝑡)𝑟(𝑡)]^′[𝑎(𝑡)𝑟(𝑡)]^2𝛽𝑦²(𝑡)𝑑𝑡≤^∞_𝑡1𝑏²(𝑡)𝑟^3𝛼−𝛽(𝑡)𝑎^3𝛽−𝛼(𝑡)𝑦²(𝑡)𝑑𝑡^1/2×^∞_𝑡1(𝑎(𝑡)𝑟(𝑡))^′2𝑎(𝑡)𝑟³(𝑡)𝑦²(𝑡)𝑑𝑡^1/2<∞,_∞𝑡1𝑏(𝑡)[𝑎(𝑡)𝑟(𝑡)]^𝛽−𝛼𝑦(𝑡)𝑦^′(𝑡)𝑑𝑡≤^∞_𝑡1𝑏²(𝑡)𝑟^3𝛼−𝛽(𝑡)𝑦²(𝑡)𝑎^3𝛽−𝛼(𝑡)𝑑𝑡^1/2×^∞_𝑡1𝑦^′(𝑡)²𝑎(𝑡)𝑟(𝑡)𝑑𝑡^1/2<∞, (2.49) Consequently, integrating both sides of (2.44), we see that the integrand of the left side of (2.44) is bounded, but the integrand of the right side of (2.44) tends to infinity according to condition (2.36). This leads to a contradiction, so 𝑦(𝑡) is a limit point case solution of (1.5), and (1.5) is of the nonlinear limit point type.

Similarly, when 𝑎(𝑡)≡1, (1.5) becomes (2.19) and we get the following corollary.

Corollary 2.7. Suppose condition (2.4), (2.10), (2.20) hold. If ^∞₀𝑏²(𝑡)𝑟(𝑡)𝑑𝑡<∞,^∞₀(𝑟(𝑡))^′2𝑟³(𝑡)𝑑𝑡<∞,^∞₀𝑏²(𝑡)𝑟^3𝛼−𝛽(𝑡)𝑑𝑡<∞,_∞0𝑡0𝐴(𝜉)𝐵(𝜉)−𝐴̇(𝜉)/𝛾+1𝑟^𝛼(𝜉)𝐵(𝑡)𝑑𝜉𝑑𝑡<∞,^∞₀1𝑟^𝛽−𝛼(𝑢)𝑑𝑢=∞, (2.50) then (2.19) is of the nonlinear limit point type.

Acknowledgments

The authors thank the referee for his helpful suggestions on this paper which improved some of our results. This research was partially supported by the NSF of China (Grants 10801089 and 11171178).