Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 420589, 8 pages
doi:10.1155/2011/420589
Research Article
Two-Point Oscillation for a Class of Second-Order
Damped Linear Differential Equations
Kong Xiang-Cong and Zheng Zhao-Wen
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Correspondence should be addressed to Zheng Zhao-Wen, zhwzheng@126.com
Received 20 May 2011; Revised 18 July 2011; Accepted 20 July 2011
Academic Editor: Svatoslav Staněk
Copyright q 2011 K. Xiang-Cong and Z. Zhao-Wen. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Using the comparison theorem, the two-point oscillation for linear differential equation with
α
β
damping term y fx/x − x2 y gx/x − x2 y 0 is considered, where α, β >
0; fx, gx > 0, and fx, gx ∈ CI, 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 yx of a differential equation appearing in the paper, we mean a function
/ CI.
y yx such that y ∈ C2 I. Here, we allow that y ∈
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:
2
1 y S q x q x y 0,
2
x ∈ I,
1.1
where I 0, 1, q ∈ C3 I, |q0| |q1−| ∞, |q 0| |q 1−| ∞, q x < 0,
x ∈ I, Sq ∈ CI, Sq x q x/q x − 3/2q x/q x2 , they study the following
equation:
y cx
σ y 0,
x − x2 x ∈ I,
1.2
2
Abstract and Applied Analysis
by comparison theorem where cx > 0, cx ∈ CI, and they obtain that when σ > 2, 1.2
is two-point oscillatory.
In this paper, we construct the following equation with damping:
3
p xy − 2p xy p x y 0,
x ∈ I,
1.3
where px ∈ C2 I and
px0 px1− ∞.
1.4
we study the two-point oscillation of the following damped equation by comparison theorem
y fx
αy
x − x2 gx
x − x2 β
y 0,
1.5
where x ∈ I, α, β > 0; fx, gx > 0, fx, gx ∈ CI, 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 yx, yx ∈ CI is said to be two-point oscillation on the
interval I, if there exist a decreasing sequence ak ∈ I and an increasing sequence bk ∈ I of
consecutive zeros of yx such that ak 0 and bk 1.
Definition 2.2. A linear differential equation is said to be two-point oscillation on I if all its
nontrivial solutions y yx, yx ∈ 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 cos px, y2 x sin px are two linearly independent solutions
of 1.3, so the general solution of 1.3 can be expressed as
yx c1 cos px c2 sin px.
2.1
Because of |px0| |px1−| ∞, the function yx is two-point oscillatory on I, then
1.3 is two-point oscillatory on I.
Example 2.3. Let px − ln ln1/x, x ∈ I, then p x 1/x ln1/x, p x 1 −
ln1/x/x ln1/x2 , px satisfies the condition 1.4, so the following equation:
1
y
x ln1/x
is two-point oscillatory on I.
−2
1 − ln1/x
y 2
x ln1/x
1
x ln1/x
3
y0
2.2
Abstract and Applied Analysis
3
ε
Example 2.4. Let px −1 − 2x/x − x2 , x ∈ I, where ε > 0. Then,
p x p x −
2 x − x2 ε1 − 2x2
,
ε1
x − x2 ε ε1
ε 11 − 2x x − x2 2 x − x2 ε1 − 2x2 − 2 − 4ε1 − 2x x − x2
x − x2 2ε2
,
2.3
when x → 0, px → −∞; when x → 1, px → −∞, which satisfies the condition 1.4;
p x ∈ CI, p x ∈ CI. Substituting p x and p x into 1.3, we obtain that the following
equation:
2 x − x2 ε1 − 2x2
x − x2 ε1
−2
y
ε1
ε − ε 11 − 2x x − x2 2 x − x2 ε1 − 2x2
2 − 4ε1 − 2x x − x2
x − x2 3
2 x − x2 ε1 − 2x2
ε1
x − x2 2ε2
y
y0
2.4
is two-point oscillatory on I.
3. A New Comparison Theorem
Theorem 3.1. Suppose that the second order differential equations
p1 xy x r1 xy x q1 xyx 0,
p2 xz x r2 xz x q2 xzx 0,
3.1
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 r22
r1 − r2 2
,
4p2 4 p1 − p2
3.3
2 when 0 < p1 p2 p and r1 /
r2 , there exists an n ∈ R , which satisfies
q2 ≥ 1 nq1 1 nr1 − r2 2 r2 2
,
4p
3.4
4
Abstract and Applied Analysis
then 3.2 has at least one zero point between two consecutive zero point α, β α < β of any nontrivial
solution yx of 3.1.
Proof. 1 We suppose that zx has no zero point on α, β when 0 < p2 < p1 . Without loss of
generality, let yx ≥ 0, zx > 0, x ∈ α, β, then we have
d y
zp1 y − yp2 z
dx z
y z − z y y z p1 y p1 z y − y p2 z − p2 y z p1 y z − p2 yz
2
z
z
y z − z y y p1 y z − p2 yz
z −q1 y − r1 y − y −q2 z − r2 z p1 − p2 y z 2
z
z
yy z
2
y2 z
yy z
q2 − q1 y2 r2
− r1 yy p1 − p2
p1 y − p1
z
z
z
2
yz
yy zz
p
− p2
2
z2
z2
2
y2 z
yz 2
− r1 yy p1 − p2 y p2 y −
q2 − q1 y2 r2
z
z
2
yz
yz 2
q2 − q1 y2 p1 − p2 y − r1 − r2 yy − r2 y y −
p2 y −
z
z
2
2 r2
r2 y
r1 − r2
r1 − r2 2 2
y − y − 2 y2
√
√
2 p1 − p2
2 p2
4p2
4 p1 − p2
2 2
r2 y
yz
r1 − r2
p1 − p2 y − √
y √ − p2 y −
2 p1 − p2
2 p2
z
r2
r1 − r2 2
q2 − q1 − 2 − y2 .
4p2 4 p1 − p2
3.5
Integrating the above equation from α to β, we obtain
0
β α
2
r1 − r2
p1 − p2 y − √
y dx
2 p1 − p2
β α
2
r2 y
yz
dx
√ − p2 y −
2 p2
z
β
r22
r1 − r2 2
q2 − q1 −
− y2 dx,
4p2 4 p1 − p2
α
3.6
Abstract and Applied Analysis
5
that is,
β
r22
r1 − r2 2
q2 − q1 −
0≥
− y2 dx.
4p2 4 p1 − p2
α
3.7
From previous equality and assumption 3.3, we obtain the next equalities:
r1 − r2
p1 − p2 y √
y,
2 p1 − p2
r2 y
yz
.
√ p2 y −
2 p2
z
3.8
3.9
2 /2p1 −p2 dx
. By 3.9, we obtain
By 3.8, we obtain y /y r1 − r2 /2p1 − p2 , y e r1 −r
− r2 /2p2 dx
, that is, zα zβ 0,
y /y − z /z r2 /2p2 . In summary, we obtain z ye
which contradicts with the assumption.
2 We suppose that zx has no zero point on α, β when 0 < p1 p2 p. Without
loss of generality, let yx ≥ 0, zx > 0, x ∈ α, β, for all n ∈ R , then
d y
zpy − ypz nypy
dx z
2
y y z − z y z py − y pz
p
y z − yz p y ny −q1 y − r1 y
z
z2
2
y z − z y
2
y p y ny −q1 y − r1 y
z −q1 y − r1 y − y −q2 z − r2 z p
2
z
z
2
y z − z y
2
y2 z
q2 − 1 nq1 y2 r2
− 1 nr1 yy p
p y
z
z2
2
y z − z y
2
yz
p
q2 − 1 nq1 y − 1 nr1 − r2 yy − r2 y y −
p y
2
z
z
yz
q2 − 1 nq1 y2 − 1 nr1 − r2 yy − r2 y y −
z
2 r 2
yz 2
2r1 − r2 2 2 1 nr1 − r2 2 2 r22 2
y −
y −
y
p y 2 y2 p y −
z
4p
4p
4p
4p
2
2 r2 y yz
1 nr1 − r2 y
−y p
√ − p y −
√
2 p
z
2 p
2r1 − r2 2 r2 2 2
y .
q2 − 1 nq1 −
4p
3.10
2
6
Abstract and Applied Analysis
Integrating the above equation from α to β, we obtain
β
1 nr1 − r2 2 r2 2 2
q2 − 1 nq1 −
y dx.
0≥
4p
α
3.11
We can find the contradiction similarly; here, we delete the details. This completes the proof.
When zx wzx is a nonlinear term, where wz : R → R is a continuous
function and zwz > 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
r2 and satisfies one of the following conditions:
two-point oscillatory on I, r1 /
1 when 0 < p2 < p1 , the following condition is satisfied on I,
q2 ≥ q1 r22
r2 − r1 2
,
4p2 4 p1 − p2
3.12
2 when 0 < p1 p2 p, the following condition is satisfied on I,
q2 ≥ 1 nq1 1 nr1 − r2 2 r2 2
,
4p
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:
Axy Bxy Cxy 0,
3.14
where x ∈ I, px satisfies condition 1.4, Bx /
− 2p x. Suppose they satisfy the existence and
uniqueness theorem on I. When x → 0 and x → 1−, if 0 < Ax < p x is satisfied on I, and
Cx ≥ p x
then 3.14 is two-point oscillatory on I.
3
2
Bx 2p x
Bx2
,
4Ax
4 p x − Ax
3.15
Abstract and Applied Analysis
7
Remark 3.4. The two-point oscillation of 1.2 is studied by comparison theorem and twopoint 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
ε
Example 2.4 is the known two-point oscillatory equation, that is px −1 − 2x/x − x2 ,
−ε1
as x → 0 or x → 1−; p x ∼ x − x2 −ε2 as
x ∈ I, where ε > 0, p x ∼ x − x2 x → 0 or x → 1−. For 1.5, Ax 1, Bx fx/x − x2 α , Cx gx/x − x2 β .
Because of fx > 0, fx ∈ CI, there exists M > 0 such that fx < M for all x ∈ I.
Therefore,
Bx Cx −α
fx
2
∼
x
−
x
,
α
x − x2 gx
x − x2 β
∼ x−x
2
−β
,
as x −→ 0 or x −→ 1−,
3.16
as x −→ 0 or x −→ 1−,
Bx,
Thus, when x → 0 and x → 1−, p x → ∞, Ax ≡ 1, −2p x /
2
2
2 −α
2 −ε−2
x
−
x
x
−
x
Bx 2p x
∼
−ε1
4 p x − Ax
x − x2 −3ε −2α−ε−1 −α1
∼ x − x2
x − x2
x − x2
,
as x → 0 or x → 1−,
3
p −2α
Bx2 ∼ x − x2
,
4Ax
3
2 x − x2 ε1 − 2x2
x − x2 ε1
as x −→ 0 or x −→ 1−,
−3ε1
∼ x − x2
,
as x −→ 0 or x −→ 1 − .
3.17
By 3.17, the following condition need to be satisfied if 3.15 holds,
−β −3ε1 −2α −3ε −2α−ε−1 −α1
x − x2
≥ x − x2
x − x2
x − x2
x − x2
x − x2
,
as x −→ 0 or x −→ 1−,
3.18
that is,
β > max{3ε 1, 2α, α 1}.
3.19
8
Abstract and Applied Analysis
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.
References
1 M. Pašić and J. S. W. Wong, “Two-point oscillations in second-order linear differential equations,”
Differential Equations & Applications, vol. 1, no. 1, pp. 85–122, 2009.
2 M. K. Kwong, M. Pašić, and J. S. W. Wong, “Rectifiable oscillations in second-order linear differential
equations,” Journal of Differential Equations, vol. 245, no. 8, pp. 2333–2351, 2008.
3 R.-K. Zhuang and H.-W. Wu, “Sturm comparison theorem of solution for second order nonlinear
differential equations,” Applied Mathematics and Computation, vol. 162, no. 3, pp. 1227–1235, 2005.
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