Two-Point Oscillation for a Class of Second-Order Damped Linear Differential Equations

Abstract

Using the comparison theorem, the two-point oscillation for linear differential equation with damping term 𝑦^+(𝑓(𝑥)/(𝑥−𝑥²)^𝛼)𝑦^ + (𝑔(𝑥)/(𝑥−𝑥²)^𝛽)𝑦=0 is considered, where 𝛼,𝛽>0;𝑓(𝑥),𝑔(𝑥)>0, and 𝑓(𝑥),𝑔(𝑥)∈𝐶(𝐼),𝐼=(0,1). Results are obtained that 0<𝛼<3/2,𝛽>3 or 𝛼>3/2,𝛽>2𝛼 imply the two-point oscillation of the equation.

1. Introduction

Under the solution 𝑦(𝑥) of a differential equation appearing in the paper, we mean a function 𝑦=𝑦(𝑥) such that 𝑦∈𝐶²(𝐼). Here, we allow that 𝑦∉𝐶(𝐼).

Recently, two-point oscillation of the differential equations caused the concern of many scholars ([1, 2]). In paper [1], Pašić and Wong construct the equation: 𝑦^+12𝑆𝑞^(𝑥)+𝑞^²(𝑥)𝑦=0,𝑥∈𝐼,(1.1) where 𝐼=(0,1), 𝑞∈𝐶³(𝐼), |𝑞(0+)|=|𝑞(1−)|=+∞, |𝑞^(0+)|=|𝑞^(1−)|=+∞, 𝑞^(𝑥)<0, 𝑥∈𝐼, 𝑆(𝑞^)∈𝐶(𝐼), 𝑆(𝑞^)(𝑥)=(𝑞^(𝑥)/𝑞^(𝑥))−(3/2)[𝑞^(𝑥)/𝑞^(𝑥)]², they study the following equation: 𝑦^+𝑐(𝑥)𝑥−𝑥²^𝜎𝑦=0,𝑥∈𝐼,(1.2) by comparison theorem (where 𝑐(𝑥)>0, 𝑐(𝑥)∈𝐶(𝐼)), and they obtain that when 𝜎>2, (1.2) is two-point oscillatory.

In this paper, we construct the following equation with damping: 𝑝^(𝑥)𝑦^^−2𝑝^(𝑥)𝑦^+𝑝^(𝑥)³𝑦=0,𝑥∈𝐼,(1.3) where 𝑝(𝑥)∈𝐶²(𝐼) and ||𝑝(𝑥)(0+)||=||𝑝(𝑥)(1−)||=∞.(1.4) we study the two-point oscillation of the following damped equation by comparison theorem 𝑦^+𝑓(𝑥)𝑥−𝑥²^𝛼𝑦^+𝑔(𝑥)𝑥−𝑥²^𝛽𝑦=0,(1.5) where 𝑥∈𝐼,𝛼,𝛽>0; 𝑓(𝑥),𝑔(𝑥)>0, 𝑓(𝑥),𝑔(𝑥)∈𝐶(𝐼), the result we obtained is new, and it continues the results obtained in [1].

2. Two-Point Oscillation of (1.3)

Definition 2.1. A function 𝑦=𝑦(𝑥), 𝑦(𝑥)∈𝐶(𝐼) is said to be two-point oscillation on the interval 𝐼, if there exist a decreasing sequence 𝑎_(𝑘)∈𝐼 and an increasing sequence 𝑏_(𝑘)∈𝐼 of consecutive zeros of 𝑦(𝑥) such that 𝑎_(𝑘)↘0 and 𝑏_(𝑘)↗1.

Definition 2.2. A linear differential equation is said to be two-point oscillation on 𝐼 if all its nontrivial solutions 𝑦=𝑦(𝑥), 𝑦(𝑥)∈𝐶²(𝐼) are two-point oscillatory on 𝐼.

By Sturm separation theorem, all nontrivial solutions of a linear differential equation are two-point oscillatory if there is a nontrivial solution is two-point oscillatory on 𝐼.

We know that 𝑦₁(𝑥)=cos𝑝(𝑥),𝑦₂(𝑥)=sin𝑝(𝑥) are two linearly independent solutions of (1.3), so the general solution of (1.3) can be expressed as 𝑦(𝑥)=𝑐₁cos𝑝(𝑥)+𝑐₂sin𝑝(𝑥).(2.1) Because of |𝑝(𝑥)(0+)|=|𝑝(𝑥)(1−)|=∞, the function 𝑦(𝑥) is two-point oscillatory on 𝐼, then (1.3) is two-point oscillatory on 𝐼.

Example 2.3. Let 𝑝(𝑥)=−lnln(1/𝑥), 𝑥∈𝐼, then 𝑝^(𝑥)=1/(𝑥ln(1/𝑥)), 𝑝^(𝑥)=(1−ln(1/𝑥))/(𝑥ln(1/𝑥))², 𝑝(𝑥) satisfies the condition (1.4), so the following equation: 𝑥ln1(1/𝑥)𝑦^^−21−ln(1/𝑥)(𝑥ln(1/𝑥))²𝑦^+1𝑥ln(1/𝑥)³𝑦=0(2.2) is two-point oscillatory on 𝐼.

Example 2.4. Let 𝑝(𝑥)=−(1−2𝑥)/(𝑥−𝑥²)^𝜀, 𝑥∈𝐼, where 𝜀>0. Then, 𝑝^(𝑥)=2𝑥−𝑥²+𝜀(1−2𝑥)²𝑥−𝑥²^(𝜀+1)𝑝_,(𝑥)=−(𝜀+1)(1−2𝑥)𝑥−𝑥²^𝜀2𝑥−𝑥²+𝜀(1−2𝑥)²−(2−4𝜀)(1−2𝑥)𝑥−𝑥²^𝜀+1𝑥−𝑥²^2𝜀+2,(2.3) when 𝑥→0, 𝑝(𝑥)→−∞; when 𝑥→1, 𝑝(𝑥)→−∞, which satisfies the condition (1.4); 𝑝^(𝑥)∈𝐶(𝐼), 𝑝^(𝑥)∈𝐶(𝐼). Substituting 𝑝^(𝑥) and 𝑝^(𝑥) into (1.3), we obtain that the following equation: 2𝑥−𝑥²+𝜀(1−2𝑥)²(𝑥−𝑥²)^(𝜀+1)𝑦^^−2(2−4𝜀)(1−2𝑥)𝑥−𝑥²^𝜀+1−(𝜀+1)(1−2𝑥)𝑥−𝑥²^𝜀2𝑥−𝑥²+𝜀(1−2𝑥)²𝑥−𝑥²^2𝜀+2𝑦^+2𝑥−𝑥²+𝜀(1−2𝑥)²𝑥−𝑥²^(𝜀+1)³𝑦=0(2.4) is two-point oscillatory on 𝐼.

3. A New Comparison Theorem

Theorem 3.1. Suppose that the second order differential equations 𝑝𝑝₁(𝑥)𝑦^(𝑥)^+𝑟₁(𝑥)𝑦^(𝑥)+𝑞₁(𝑥)𝑦(𝑥)=0,(3.1)₂(𝑥)𝑧^(𝑥)^+𝑟₂(𝑥)𝑧^(𝑥)+𝑞₂(𝑥)𝑧(𝑥)=0,(3.2) satisfy the existence and uniqueness theorem on 𝐼, and one of the following conditions holds: (1)when 0<𝑝₂<𝑝₁ and 𝑟₁≠𝑟₂, 𝑞₂≥𝑞₁+𝑟²²4𝑝₂+𝑟¹−𝑟₂²4𝑝₁−𝑝₂,(3.3)(2)when 0<𝑝₁=𝑝₂=𝑝 and 𝑟₁≠𝑟₂, there exists an 𝑛∈ℝ⁺, which satisfies 𝑞₂≥(1+𝑛)𝑞₁+(1+𝑛)𝑟¹−𝑟₂²+𝑟₂₂4𝑝,(3.4)then (3.2) has at least one zero point between two consecutive zero point 𝛼,𝛽 (𝛼<𝛽) of any nontrivial solution 𝑦(𝑥) of (3.1).

Proof. (1) We suppose that 𝑧(𝑥) has no zero point on [𝛼,𝛽] when 0<𝑝₂<𝑝₁. Without loss of generality, let 𝑦(𝑥)≥0, 𝑧(𝑥)>0, 𝑥∈[𝛼,𝛽], then we have 𝑑𝑑𝑥𝑧𝑦𝑧𝑝¹=𝑦𝑦^−𝑦𝑝₂𝑧^𝑧𝑧𝑝¹𝑦^^+𝑝₁𝑧^𝑦^−𝑦𝑝₂𝑧^^−𝑝₂𝑦^𝑧^+𝑦^𝑧−𝑧^𝑦𝑧²=𝑦𝑝₁𝑦^𝑧−𝑝₂𝑦𝑧^𝑧𝑧−𝑞¹𝑦−𝑟₁𝑦^−𝑦−𝑞₂𝑧−𝑟₂𝑧^+𝑝₁−𝑝₂𝑦^𝑧^+𝑦^𝑧−𝑧^𝑦𝑧²=𝑞₂𝑝₁𝑦^𝑧−𝑝₂𝑦𝑧^−𝑞₁𝑦²+𝑟₂𝑦²𝑧^𝑧−𝑟¹𝑦𝑦^+𝑝₁−𝑝₂𝑦𝑦^𝑧^𝑧+𝑝¹𝑦^²−𝑝₁𝑦𝑦^𝑧^−𝑝₂𝑧𝑦𝑦^𝑧𝑧^𝑧²+𝑝₂𝑦𝑧^²𝑧²=𝑞₂−𝑞₁𝑦²+𝑟₂𝑦²𝑧^𝑧−𝑟¹𝑦𝑦^+𝑝₁−𝑝₂𝑦^²+𝑝₂𝑦^−𝑦𝑧^𝑧²=𝑞₂−𝑞₁𝑦²+√𝑝₁−𝑝₂𝑦^²−𝑟₁−𝑟₂𝑦𝑦^−𝑟₂𝑦𝑦^−𝑦𝑧^𝑧+√𝑝₂𝑦^−𝑦𝑧^𝑧²+𝑟¹−𝑟₂2√𝑝₁−𝑝₂𝑦²+𝑟²𝑦2√𝑝₂²−𝑟¹−𝑟₂²4𝑝₁−𝑝₂𝑦²−𝑟²²4𝑝₂𝑦²=√𝑝₁−𝑝₂𝑦^−𝑟¹−𝑟₂2√𝑝₁−𝑝₂𝑦²+𝑟²𝑦2√𝑝₂−√𝑝₂𝑦^−𝑦𝑧^𝑧²+𝑞₂−𝑞₁−𝑟²²4𝑝₂−𝑟¹−𝑟₂²4𝑝₁−𝑝₂𝑦².(3.5) Integrating the above equation from 𝛼 to 𝛽, we obtain 0=^𝛽𝛼√𝑝₁−𝑝₂𝑦^−𝑟¹−𝑟₂2√𝑝₁−𝑝₂+𝑦²𝑑𝑥_𝛽𝛼𝑟²𝑦2√𝑝₂−√𝑝₂𝑦^−𝑦𝑧^𝑧²+𝑑𝑥_𝛽𝛼𝑞²−𝑞₁−𝑟²²4𝑝₂−𝑟¹−𝑟₂²4𝑝₁−𝑝₂𝑦²𝑑𝑥,(3.6) that is, 0≥^𝛽𝛼𝑞²−𝑞₁−𝑟²²4𝑝₂−𝑟¹−𝑟₂²4𝑝₁−𝑝₂𝑦²𝑑𝑥.(3.7) From previous equality and assumption (3.3), we obtain the next equalities: √𝑝₁−𝑝₂𝑦′=𝑟¹−𝑟₂2√𝑝₁−𝑝₂𝑟𝑦,(3.8)₂𝑦2√𝑝₂=√𝑝₂𝑦^−𝑦𝑧^𝑧.(3.9) By (3.8), we obtain 𝑦^/𝑦=(𝑟₁−𝑟₂)/2(𝑝₁−𝑝₂), 𝑦=𝑒^{∫((𝑟1−𝑟2)/2(𝑝1−𝑝2))𝑑𝑥}. By (3.9), we obtain (𝑦^/𝑦)−(𝑧^/𝑧)=(𝑟₂/2𝑝₂). In summary, we obtain 𝑧=𝑦𝑒^{−∫(𝑟2/2𝑝2)𝑑𝑥}, that is, 𝑧(𝛼)=𝑧(𝛽)=0, which contradicts with the assumption.
(2) We suppose that 𝑧(𝑥) has no zero point on [𝛼,𝛽] when 0<𝑝₁=𝑝₂=𝑝. Without loss of generality, let 𝑦(𝑥)≥0, 𝑧(𝑥)>0, 𝑥∈[𝛼,𝛽], for all 𝑛∈ℝ⁺, then 𝑑𝑑𝑥𝑧𝑦𝑧𝑝𝑦^=𝑦−𝑦𝑝𝑧^+𝑛𝑦𝑝𝑦^𝑧𝑧𝑝𝑦^^−𝑦𝑝𝑧^^+𝑝𝑦^𝑧−𝑧^𝑦𝑧²=𝑦𝑦^𝑧−𝑦𝑧^+𝑝𝑦^²+𝑛𝑦−𝑞₁𝑦−𝑟₁𝑦^𝑧𝑧−𝑞¹𝑦−𝑟₁𝑦^−𝑦−𝑞₂𝑧−𝑟₂𝑧^+𝑝𝑦^𝑧−𝑧^𝑦²𝑧²=𝑞₂+𝑝𝑦^²+𝑛𝑦−𝑞₁𝑦−𝑟₁𝑦^−(1+𝑛)𝑞₁𝑦²+𝑟₂𝑦²𝑧^𝑧−(1+𝑛)𝑟¹𝑦𝑦^+𝑝𝑦^𝑧−𝑧^𝑦²𝑧²+𝑝𝑦^²=𝑞₂−(1+𝑛)𝑞₁𝑦²−(1+𝑛)𝑟₁−𝑟₂𝑦𝑦^−𝑟₂𝑦𝑦^−𝑦𝑧^𝑧+𝑝𝑦^𝑧−𝑧^𝑦²𝑧²+𝑝𝑦^²=𝑞₂−(1+𝑛)𝑞₁𝑦²−(1+𝑛)𝑟₁−𝑟₂𝑦𝑦^−𝑟₂𝑦𝑦^−𝑦𝑧^+𝑝𝑦_𝑧−𝑦𝑧^𝑧²+𝑝𝑦^²+𝑟²²4𝑝𝑦²+2𝑟¹−𝑟₂²4𝑝𝑦²−(1+𝑛)𝑟¹−𝑟₂²4𝑝𝑦²−𝑟²²4𝑝𝑦²=𝑟²𝑦2√𝑝−√𝑝𝑦^−𝑦𝑧^𝑧²+(1+𝑛)𝑟¹−𝑟₂𝑦2√𝑝−𝑦^√𝑝²+𝑞₂−(1+𝑛)𝑞₁−2𝑟¹−𝑟₂²+𝑟₂₂4𝑝𝑦².(3.10) Integrating the above equation from 𝛼 to 𝛽, we obtain 0≥^𝛽𝛼𝑞²−(1+𝑛)𝑞₁−(1+𝑛)𝑟¹−𝑟₂²+𝑟₂₂4𝑝𝑦²𝑑𝑥.(3.11) We can find the contradiction similarly; here, we delete the details. This completes the proof.

When 𝑧(𝑥)=𝑤(𝑧(𝑥)) is a nonlinear term, where 𝑤(𝑧)∶𝑅→𝑅 is a continuous function and 𝑧𝑤(𝑧)>0 for 𝑧≠0, Zhuang and Wu established some comparison theorems if 𝑤′(𝑧)≥𝐾>0 holds in [3]. The condition of Corollary 2.2 in [3] is identical with (3.3) when 𝑤^(𝑧) is smooth and 𝐾=1, but there's no condition about the situation of 𝑝₁=𝑝₂=𝑝. We put “𝑛𝑦𝑝^𝑦” added to Picone identity, which solve the problem of the vacuousness of (3.3) when 𝑝₁=𝑝₂=𝑝. Then, we obtain (3.4) and establish the integrated comparison theorem of second order damped linear differential equations.

We can easily obtain the following corollaries by Theorem 3.1.

Corollary 3.2. Suppose (3.1), (3.2) satisfy the existence and uniqueness theorem on 𝐼. If (3.1) is two-point oscillatory on 𝐼, r₁≠r₂ and satisfies one of the following conditions: (1)when 0<𝑝₂<𝑝₁, the following condition is satisfied on 𝐼, 𝑞₂≥𝑞₁+𝑟²²4𝑝₂+𝑟²−𝑟₁²4𝑝₁−𝑝₂,(3.12)(2)when 0<𝑝₁=𝑝₂=𝑝, the following condition is satisfied on 𝐼, 𝑞₂≥(1+𝑛)𝑞₁+(1+𝑛)𝑟¹−𝑟₂²+𝑟₂₂4𝑝,(3.13)then (3.2) is two-point oscillatory on 𝐼.

Corollary 3.3. Consider the second order equation (1.3) and the following equation: 𝐴(𝑥)𝑦^^+𝐵(𝑥)𝑦^+𝐶(𝑥)𝑦=0,(3.14) where 𝑥∈𝐼, 𝑝(𝑥) satisfies condition (1.4), 𝐵(𝑥)≠−2𝑝^(𝑥). Suppose they satisfy the existence and uniqueness theorem on 𝐼. When 𝑥→0+ and 𝑥→1−, if 0<𝐴(𝑥)<𝑝^(𝑥) is satisfied on 𝐼, and 𝐶(𝑥)≥𝑝^(𝑥)³+𝐵(𝑥)²4𝐴(𝑥)+𝐵(𝑥)+2𝑝^(𝑥)²4(𝑝^(𝑥)−𝐴(𝑥)),(3.15) then (3.14) is two-point oscillatory on 𝐼.

Remark 3.4. The two-point oscillation of (1.2) is studied by comparison theorem and two-point oscillatory equation in [1]. When 𝑝₁(𝑥)=𝑝₂(𝑥)=1 and 𝑟₁(𝑥)=𝑟₂(𝑥)=0, Theorem 3.1 reduces to Theorem 2.1 in [1].

As an application of Corollary 3.3, we discuss the two-point oscillation of (1.5). Since Example 2.4 is the known two-point oscillatory equation, that is 𝑝(𝑥)=−(1−2𝑥)/(𝑥−𝑥²)^𝜀, 𝑥∈𝐼, where 𝜀>0, 𝑝^(𝑥)∼(𝑥−𝑥²)^−(𝜀+1) as 𝑥→0+ or 𝑥→1−; 𝑝^(𝑥)∼(𝑥−𝑥²)^−(𝜀+2) as 𝑥→0+ or 𝑥→1−. For (1.5), 𝐴(𝑥)=1, 𝐵(𝑥)=𝑓(𝑥)/(𝑥−𝑥²)^𝛼, 𝐶(𝑥)=𝑔(𝑥)/(𝑥−𝑥²)^𝛽. Because of 𝑓(𝑥)>0, 𝑓(𝑥)∈𝐶(𝐼), there exists 𝑀>0 such that 𝑓(𝑥)<𝑀 for all 𝑥∈𝐼. Therefore, 𝐵(𝑥)=𝑓(𝑥)𝑥−𝑥²^𝛼𝐶(𝑥)=𝑔(𝑥)∼𝑥−𝑥²^−𝛼,as𝑥⟶0+or𝑥⟶1−,𝑥−𝑥²^𝛽∼𝑥−𝑥²^−𝛽,as𝑥⟶0+or𝑥⟶1−,(3.16) Thus, when 𝑥→0+ and 𝑥→1−, 𝑝^(𝑥)→+∞, 𝐴(𝑥)≡1, −2𝑝^(𝑥)≠𝐵(𝑥), 𝐵(𝑥)+2𝑝^(𝑥)²4(𝑝^(𝑥)−𝐴(𝑥))∼𝑥−𝑥²^−𝛼+𝑥−𝑥²^−𝜀−2²𝑥−𝑥²^−(𝜀+1)𝐵(𝑥)₂∼𝑥−𝑥²^−(3+𝜀)+𝑥−as𝑥𝑥²→0+o^{−(2𝛼−𝜀−1)}r+𝑥𝑥→−1𝑥−²,^−(𝛼+1),4𝐴(𝑥)∼𝑥−𝑥²^−2𝛼𝑝,as𝑥⟶0+or𝑥⟶1−,_³=2𝑥−𝑥²+𝜀(1−2𝑥)²𝑥−𝑥²^(𝜀+1)³∼𝑥−𝑥²^−3(𝜀+1),as𝑥⟶0+or𝑥⟶1−.(3.17) By (3.17), the following condition need to be satisfied if (3.15) holds, 𝑥−𝑥²^−𝛽≥𝑥−𝑥²^−3(𝜀+1)+𝑥−𝑥²^−2𝛼+𝑥−𝑥²^−(3+𝜀)+𝑥−𝑥²as𝑥⟶^{−(2𝛼−𝜀−1)}+0+𝑥or−𝑥𝑥²⟶1−^−(𝛼+1),,(3.18) that is, 𝛽>max{3(𝜀+1),2𝛼,𝛼+1}.(3.19) In summary, let 𝜀→0, then, when 0<𝛼<3/2, 3+3𝜀>max{2𝛼,𝛼+1}, condition (3.19) holds with 𝛽>3, (1.5) is two-point oscillatory on 𝐼 in this case,when 𝛼>3/2, 2𝛼>max{3+3𝜀,𝛼+1}, condition (3.19) holds with 𝛽>2𝛼, (1.5) is two-point oscillatory on 𝐼 in this case.

Acknowledgments

This work was supported by the National Nature Science Foundation of China (10801089, 11171178) and the National Nature Science Foundation of Shandong Province (ZR2009AQ010).