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

Authors: Kong Xiang-Cong; Zheng Zhao-Wen
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Using the comparison theorem, the two-point oscillation for linear
differential equation with damping termy′′+(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.


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## Body

## 1. Introduction

Under the solutiony(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: (1.1)y′′+[12S(q′)(x)+(q′)2(x)]y=0,x∈I,
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: (1.2)y′′+c(x)(x-x2)σy=0,x∈I,
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:(1.3)(p′(x)y′)′-2p′′(x)y′+(p′(x))3y=0,x∈I,
where p(x)∈C2(I) and (1.4)|p(x)(0+)|=|p(x)(1-)|=∞.
we study the two-point oscillation of the following damped equation by comparison theorem (1.5)y′′+f(x)(x-x2)αy′+g(x)(x-x2)βy=0,
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 functiony=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 onI 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 onI.We know thaty1(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 (2.1)y(x)=c1cos⁡⁡p(x)+c2sin⁡p(x).
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.
Letp(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:
(2.2)(1xln⁡⁡(1/x)y′)′-21-ln⁡⁡(1/x)(xln⁡⁡(1/x))2y′+(1xln⁡⁡(1/x))3y=0
is two-point oscillatory on I.Example 2.4.
Letp(x)=-(1-2x)/(x-x2)ε,x∈I,where ε>0.Then,
(2.3)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,
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:
(2.4)(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
is two-point oscillatory on I.


## 3. A New Comparison Theorem

Theorem 3.1.
Suppose that the second order differential equations(3.1)(p1(x)y′(x))′+r1(x)y′(x)+q1(x)y(x)=0,(3.2)(p2(x)z′(x))′+r2(x)z′(x)+q2(x)z(x)=0,
satisfy the existence and uniqueness theorem on I, and one of the following conditions holds:(1)
when0<p2<p1 and r1≠r2,
(3.3)q2≥q1+r224p2+(r1-r2)24(p1-p2),(2)
when0<p1=p2=p and r1≠r2, there exists an n∈ℝ+, which satisfies
(3.4)q2≥(1+n)q1+((1+n)r1-r2)2+r224p,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 thatz(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
(3.5)ddx[yz(zp1y′-yp2z′)]=yz[z(p1y′)′+p1z′y′-y(p2z′)′-p2y′z′]+y′z-z′yz2(p1y′z-p2yz′)=yz[z(-q1y-r1y′)-y(-q2z-r2z′)+(p1-p2)y′z′]+y′z-z′yz2(p1y′z-p2yz′)=(q2-q1)y2+r2y2z′z-r1yy′+(p1-p2)yy′z′z+p1(y′)2-p1yy′z′z-p2yy′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-r22p1-p2y)2+(r2y2p2)2-(r1-r2)24(p1-p2)y2-r224p2y2=[p1-p2y′-r1-r22p1-p2y]2+[r2y2p2-p2(y′-yz′z)]2+[q2-q1-r224p2-(r1-r2)24(p1-p2)]y2.
Integrating the above equation from α to β, we obtain
(3.6)0=∫αβ[p1-p2y′-r1-r22p1-p2y]2dx+∫αβ[r2y2p2-p2(y′-yz′z)]2dx+∫αβ[q2-q1-r224p2-(r1-r2)24(p1-p2)]y2dx,
that is,
(3.7)0≥∫αβ[q2-q1-r224p2-(r1-r2)24(p1-p2)]y2dx.
From previous equality and assumption (3.3), we obtain the next equalities:
(3.8)p1-p2y′=r1-r22p1-p2y,(3.9)r2y2p2=p2(y′-yz′z).
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 thatz(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
(3.10)ddx[yz(zpy′-ypz′)+nypy′]=yz(z(py′)′-y(pz′)′)+py′z-z′yz2(y′z-yz′)+p(y′)2+ny(-q1y-r1y′)=yz[z(-q1y-r1y′)-y(-q2z-r2z′)]+p(y′z-z′y)2z2+p(y′)2+ny(-q1y-r1y′)=[(q2-(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′z)+p(y′-yz′z)2+p(y′)2+r224py2+(2r1-r2)24py2-((1+n)r1-r2)24py2-r224py2=[r2y2p-p(y′-yz′z)]2+[((1+n)r1-r2)y2p-y′p]2+[q2-(1+n)q1-(2r1-r2)2+r224p]y2.
Integrating the above equation from α to β, we obtain
(3.11)0≥∫αβ[q2-(1+n)q1-((1+n)r1-r2)2+r224p]y2dx.
We can find the contradiction similarly; here, we delete the details. This completes the proof.Whenz(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 Theorem3.1.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)
when0<p2<p1, the following condition is satisfied on I,
(3.12)q2≥q1+r224p2+(r2-r1)24(p1-p2),(2)
when0<p1=p2=p, the following condition is satisfied on I,
(3.13)q2≥(1+n)q1+((1+n)r1-r2)2+r224p,then (3.2) is two-point oscillatory on I.Corollary 3.3.
Consider the second order equation (1.3) and the following equation:
(3.14)(A(x)y′)′+B(x)y′+C(x)y=0,
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
(3.15)C(x)≥(p′(x))3+B(x)24A(x)+(B(x)+2p′′(x))24(p′(x)-A(x)),
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 Corollary3.3, 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, (3.16)B(x)=f(x)(x-x2)α~(x-x2)-α,asx⟶0+orx⟶1-,C(x)=g(x)(x-x2)β~(x-x2)-β,asx⟶0+orx⟶1-,
Thus, when x→0+ and x→1-, p′(x)→+∞, A(x)≡1, -2p′′(x)≠B(x), (3.17)(B(x)+2p′′(x))24(p′(x)-A(x))~[(x-x2)-α+(x-x2)-ε-2]2(x-x2)-(ε+1)~(x-x2)-(3+ε)+(x-x2)-(2α-ε-1)+(x-x2)-(α+1),asx→0+orx→1-,B(x)24A(x)~(x-x2)-2α,asx⟶0+orx⟶1-,(p′)3=[2(x-x2)+ε(1-2x)2(x-x2)(ε+1)]3~(x-x2)-3(ε+1),asx⟶0+orx⟶1-.
By (3.17), the following condition need to be satisfied if (3.15) holds, (3.18)(x-x2)-β≥(x-x2)-3(ε+1)+(x-x2)-2α+(x-x2)-(3+ε)+(x-x2)-(2α-ε-1)+(x-x2)-(α+1),asx⟶0+orx⟶1-,
that is, (3.19)β>max⁡⁡{3(ε+1),2α,α+1}.
In summary, let ε→0, then, when0<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,whenα>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.



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