Hindawi Publishing Corporation
International Journal of Differential Equations
Volume 2011, Article ID 712703, 12 pages
doi:10.1155/2011/712703
Research Article
On the Upper Bounds of Eigenvalues for
a Class of Systems of Ordinary Differential
Equations with Higher Order
Gao Jia, Li-Na Huang, and Wei Liu
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Correspondence should be addressed to Gao Jia, gaojia79@yahoo.com.cn
Received 4 May 2011; Revised 16 July 2011; Accepted 19 July 2011
Academic Editor: Bashir Ahmad
Copyright q 2011 Gao Jia et al. 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.
The estimate of the upper bounds of eigenvalues for a class of systems of ordinary differential
equations with higher order is considered by using the calculus theory. Several results about the
upper bound inequalities of the n 1th eigenvalue are obtained by the first n eigenvalues. The
estimate coefficients do not have any relation to the geometric measure of the domain. This kind
of problem is interesting and significant both in theory of systems of differential equations and in
applications to mechanics and physics.
1. Introduction
In many physical settings, such as the vibrations of the general homogeneous or nonhomogeneous string, rod and plate can yield the Sturm-Liouville eigenvalue problems or other
eigenvalue problems. However, it is not easy to get the accurate values by the analytic
method. Sometimes, it is necessary to consider the estimations of the eigenvalues. And since
1960s, the problems of the eigenvalue estimates had become one of the hotspots of the
differential equations.
There are lots of achievements about the upper bounds of arbitrary eigenvalues for the
differential equations and uniformly elliptic operators with higher orders 1–9. However,
there are few achievements associated with the estimates of the eigenvalues for systems of
differential equations with higher order. In the following, we will obtain some inequalities
concerning the eigenvalue λn1 in terms of λ1 , λ2 , . . . , λn in the systems of ordinary differential
equations with higher order. In fact, the eigenvalue problems have great strong practical
backgrounds and important theoretical values 10, 11.
2
International Journal of Differential Equations
Let a, b ⊂ R1 be a bounded domain and t ≥ 2 be an integer. The following eigenvalue
problems are studied:
−1t Dt a11 Dt y1 a12 Dt y2 · · · a1n Dt yn λsxy1 ,
−1t Dt a21 Dt y1 a22 Dt y2 · · · a2n Dt yn λsxy2 ,
..
.
1.1
−1t Dt an1 Dt y1 an2 Dt y2 · · · ann Dt yn λsxyn ,
Dk yi a Dk yi b 0
i 1, 2, . . . , n, k 0, 1, 2, . . . , t − 1,
where D d/dx, Dk dk /dxk , aij x i, j 1, 2, . . . , n and sx satisfies the following
conditions:
1◦ aij x ∈ Ct a, b, aij x aji x, i, j 1, 2, . . . , n;
2◦ for the arbitrary ξ ξ1 , ξ2 , . . . , ξn ∈ Rn , we have
μ1 |ξ|2 ≤
n
aij xξi ξj ≤ μ2 |ξ|2 ,
∀x ∈ a, b,
1.2
i,j1
where μ2 ≥ μ1 > 0, μ1 , μ2 are both constants;
3◦ sx ∈ Ca, b, and there are constants ν1 ≤ ν2 , such that 0 < ν1 ≤ sx ≤ ν2 .
According to the theories of the differential equations 11, 12, the eigenvalues of 1.1
are all positive real numbers, and they are discrete.
We change 1.1 to the form of matrix. Let
⎛
y1
⎞
⎜ ⎟
⎜ y2 ⎟
⎜ ⎟
⎟
yT ⎜
⎜ .. ⎟,
⎜ . ⎟
⎝ ⎠
yn
⎛
Dt y1
⎞
⎜ t ⎟
⎜ D y2 ⎟
⎟
⎜
⎟
Dt yT ⎜
⎜ .. ⎟,
⎜ . ⎟
⎠
⎝
Dt yn
⎛
a11 x a12 x · · · a1n x
⎞
⎜
⎟
⎜ a21 x a22 x · · · a2n x ⎟
⎜
⎟
⎟.
Ax ⎜
⎜
⎟
.
..
⎜
⎟
⎝
⎠
an1 x an2 x · · · ann x
1.3
By virtue of aij x aji x, therefore AT x Ax, 1.1 can be changed into the
following form:
−1t Dt AxDt yT λsxyT ,
yk a yk b 0,
Obviously, 1.4-1.5 is equivalent to 1.1.
k 0, 1, 2, . . . , t − 1.
1.4
1.5
International Journal of Differential Equations
3
Suppose that 0 < λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · · are eigenvalues of 1.4-1.5,
y1 , y2 , . . . , yn , . . . are the corresponding eigenfunctions and satisfy the following weighted
orthogonal conditions:
b
a
sxyi yTj dx
δij ⎧
⎨1,
i j,
⎩0,
i/
j,
i, j 1, 2, . . . .
1.6
Multiplying yi in sides of 1.4, by using 1.5 and integration by parts, we have
λi b
a
Dt yi AxDt yTi dx,
i 1, 2, . . . .
1.7
λi
,
μ1
1.8
From 2◦ , we have
b
t 2
D yi dx a
b
a
Dt yi Dt yTi dx ≤
i 1, 2, . . . .
For fixed n, let
Φi xyi −
n
bij yj ,
i 1, 2, . . . , n,
1.9
j1
b
where bij a xsxyi yTj dx. Obviously, bij bji , and Φi are weighted orthogonal to
y1 , y2 , . . . , yn . Furthermore, Φi a Φi b 0, i, j 1, 2, . . . , n.
We can use the well-known Rayleigh theorem 11, 12 to obtain
λn1 ≤
−1t
t
t T
AxD
dx
Φ
D
Φ
i
i
a
.
b
2
sx|Φ
dx
|
i
a
b
1.10
It is easy to see that
−1t Dt AxDt ΦTi −1t tDt AxDt−1 yTi −1t tDt−1 AxDt yTi
n
−1t xDt AxDt yTi − −1t bij Dt AxDt yTj
j1
−1t tDt AxDt−1 yTi −1t tDt−1 AxDt yTi
λi xsxyTi − sx
n
j1
λj bij yTj .
1.11
4
International Journal of Differential Equations
We have
b
a
b
b
Φi −1t Dt AxDt ΦTi dx λi xsxΦi yTi dx −1t t Φi Dt AxDt−1 yTi dx
a
−1t t
−
b
a
b
a
Φi Dt−1 AxDt yTi dx
sxΦi
a
n
1.12
λj bij yTj dx.
j1
In addition, using the fact that Φi are weighted orthogonal to y1 , y2 , . . . , yn and
b
sx|Φi |2 dx b
a
a
xsxΦi yTi dx,
1.13
we know that the last term of 1.12 is equal to zero. Thus, we have
b
a
b
Φi −1t Dt AxDt ΦTi dx λi sx|Φi |2 dx
a
t
−1 t
b
a
t
−1 t
b
a
Φi Dt AxDt−1 yTi dx
1.14
Φi Dt−1 AxDt yTi dx.
Set
Ii −1t t
b
a
t
Ji −1 t
b
Φi Dt AxDt−1 yTi dx,
I
n
Ii ,
i1
Φi D
t−1
a
AxDt yTi
dx,
n
J
Ji .
1.15
i1
From 1.14, we have
n b
i1
a
b
n
Φi −1t Dt AxDt ΦTi dx λi sx|Φi |2 dx I J.
i1
1.16
a
By using 1.10 and 1.16, one can give
λn1
n b
i1
a
sx|Φi |2 dx ≤
b
n
λi sx|Φi |2 dx I J.
i1
a
1.17
International Journal of Differential Equations
5
Substituting λn for λi in 1.17, we get
n b
sx|Φi |2 dx ≤ I J.
λn1 − λn i1
1.18
a
In order to get the estimations of the eigenvalues, we only need to show the estimates
b
about I, J, and ni1 a sx|Φi |2 dx.
2. Lemmas
Lemma 2.1. Suppose that the eigenfunctions yi of 1.4-1.5 correspond to the eigenvalues λi . Then
one has
1
2
b
a
b
a
−1/p1
|Dp yi |2 dx ≤ ν1
−1−1/t
|Dyi |2 dx ≤ ν1
b
a
p/p1
|Dp1 yi |2 dx
, p 1, 2, . . . , t − 1;
λi /μ1 1/t .
Proof. 1 By induction. If p 1, using integration by parts and the Schwarz inequality, we
have
b
b
b
2
T
2 T
|Dyi | dx ≤ |Dyi | dx Dyi Dyi dx yi D yi dx
a
a
a
a
b
2
≤
b
|yi |2 dx
1/2 b
a
a
2 T 2
D yi dx
1/2
≤ ν1−1/2
b
a
2 T 2
D yi dx
2.1
1/2
.
Therefore, when p 1, 1 is true.
If for p k, 1 is true, that is,
b
k/k1
b k 2
k1 2
−1/k1
.
D yi dx ≤ ν1
D yi dx
a
2.2
a
For p k 1, using integration by parts, the Schwarz inequality and the result when p k,
one can give
b b
b
k1 2 k1 2
k
k2 T
D yi dx ≤ D yi dx D yi · D yi dx
a
a
a
≤
b
k 2
D yi dx
a
≤
−1/2k1
ν1
1/2 b
a
b
a
k2 T 2
D yi dx
k2 2
D yi dx
1/2 b
a
1/2
k1 2
D yi dx
2.3
k/2k1
.
6
International Journal of Differential Equations
By further calculating, one can give
k1/k11
b
b k1 2
k11 2
−1/k11
yi dx
.
D yi dx ≤ ν1
D
a
2.4
a
Therefore, when p k 1, 1 is true.
2 Using 1 and the inductive method, we have
b
2
|D yi | dx ≤
p
a
≤
−1/p1
ν1
−2/p2
ν1
≤ ··· ≤
b
p1 2
D yi dx
p/p1
a
b
p2 2
D yi dx
p/p2
2.5
a
−1−p/t
ν1
b
t 2
D yi dx
p/t
.
a
From 1.8 and 2.5, we get
b
a
2
|D yi | dx ≤
p
−1−p/t
ν1
b
t 2
D yi dx
p/t
−1−p/t
≤ ν1
a
λi
μ1
p/t
,
p 1, 2, . . . , t.
2.6
Taking p 1, we have
b
a
−1−1/t
|Dyi |2 dx ≤ ν1
λi
μ1
1/t
.
2.7
So Lemma 2.1 is true.
Lemma 2.2. Let λ1 , λ2 , . . . , λn be the eigenvalues of 1.4-1.5. Then one has
−1−1/t −1/t
ν1 μ2
I J ≤ t2t − 1μ1
n
1−1/t
λi
.
i1
2.8
International Journal of Differential Equations
7
Proof. Since
Ii −1t t
b
Φi Dt AxDt−1 yTi dx
a
−1t t
⎛
b
⎝xyi −
a
t
−1 t
n
b
bij
a
j1
b
t2
a
bij yj ⎠Dt AxDt−1 yTi dx
xyi Dt AxDt−1 yTi dx
a
− −1 t
⎞
j1
b
t
n
yj D
t
AxDt−1 yTi
Dt−1 yi AxDt−1 yTi dx t
b
a
2.9
dx
xDt yi AxDt−1 yTi dx
b
n
− t bij
Dt yj AxDt−1 yTi dx,
a
j1
Ji −1t t
b
Φi Dt−1 AxDt yTi dx
a
b
−tt − 1
a
Dt−2 yi AxDt yTi dx − t
b
n
t bij
Dt−1 yj AxDt yTi dx,
b
a
xDt−1 yi AxDt yTi dx
2.10
a
j1
we have
I J n
Ii Ji i1
n b
tDt−1 yi AxDt−1 yTi − t − 1Dt−2 yi AxDt yTi dx
t
a
i1
−t
n
bij
i,j1
2.11
b
a
Dt yj AxDt−1 yTi − Dt−1 yj AxDt yTi dx.
By aij x aji x, the last term of 2.11 is zero. Then we can get
I J n b
tDt−1 yi AxDt−1 yTi − t − 1Dt−2 yi AxDt yTi dx.
t
i1
a
2.12
8
International Journal of Differential Equations
Using 2◦ , Lemma 2.1, 1 and 2.6, we have
b
D
t−1
a
yi AxDt−1 yTi dx
b 1−1/t
t−1 2
−1/t λi
≤ μ2 D yi dx ≤ μ2 ν1
.
μ1
a
2.13
Using 2◦ , the Schwarz inequality, Lemma 2.1 1, and 2.6, one can give
b
1/2 b
1/2
b
t−2 2
t T 2
t−2
t T
− D yi AxD yi dx ≤ μ2
D yi dx
D yi dx
a
a
a
≤ μ2 ν1−1/t
λi
μ1
1−1/t
2.14
.
Therefore, we obtain
−1−1/t −1/t
ν1 μ2
I J ≤ t2t − 1μ1
n
1−1/t
λi
.
2.15
i1
Lemma 2.3. If Φi and λi i 1, 2, . . . , n as above, then one has
n b
2
sx|Φi | dx ≥
2−1/t 2 n
n μ1/t
1 ν1
4ν22
a
i1
−1
λ1/t
i
.
2.16
yj DyTi dx.
2.17
i1
Proof. By the definition of Φi , one has
n b
i1
a
Φi DyTi dx n b
i1
a
xyi DyTi dx −
n
b
bij
i,j1
a
b
b
Using bij bji and a yj DyTi dx − a yi DyTj dx, it is easy to see that the last term of 2.17 is
zero. Then we have
n b
i1
a
Φi DyTi dx
n b
i1
a
xyi DyTi dx.
2.18
Using integration by parts, one can give
b
a
xyi DyTi dx
b
a
−
b
2
|yi | dx −
a
xyi DyTi dx
1
−
2
b
a
b
a
xyi DyTi dx,
|yi |2 dx.
2.19
2.20
International Journal of Differential Equations
By 1/ν2 ≤
b
a
9
|yi |2 dx ≤ 1/ν1 , we have
1 b
b
1
T
.
|y |2 dx ≥
xyi Dyi dx 2 a i
a
2ν2
2.21
From 2.18 and 2.21, we can get
n b
n
T
.
Φi Dyi dx ≥
2ν2
a
i1
2.22
Using the Schwarz inequality, Lemma 2.1 2, and 3◦ , we have
b
n
|Dyi |2
dx
sx|Φi | dx
sx
i1 a
i1 a
b
n
n
−2−1/t −1/t
2
sx|Φi | dx ν1
μ1
λ1/t
≤
i .
n2
≤
4ν22
b
n
i1
2
a
2.23
i1
By further calculating, we can easily get Lemma 2.3.
3. Main Results
Theorem 3.1. If λi i 1, 2, . . . , n 1 are the eigenvalues of 1.4-1.5, then
n
n
4t2t − 1μ2 ν22 1−1/t
λ
λ1/t
i ;
i
μ1 ν12 n2
i1
i1
4t2t − 1μ2 ν22
λn .
≤ 1
μ1 ν12
1
λn1 ≤ λn 3.1
2
λn1
3.2
Proof. From 1.18, we can get
λn1 − λn ≤ I J
n b
i1
−1
2
sx|Φi | dx
.
3.3
a
Using Lemmas 2.2 and 2.3, we can easily get 3.1. In 3.1, Replacing λi with λn , by further
calculating, we can get 3.2.
Theorem 3.2. For n ≥ 1, one has
n
λ1/t
μ1 ν12 n2
i
≥
λ − λi 4t2t − 1μ2 ν22
i1 n1
n
i1
−1
1−1/t
λi
.
3.4
10
International Journal of Differential Equations
Proof. Choosing the parameter σ > λn , using 1.17, one can give
λn1
n b
i1
2
sx|Φi | dx ≤ σ
a
n b
n b
2
sx|Φi | dx a
i1
λi − σsx|Φi |2 dx I J.
3.5
a
i1
By 2.22 and the Young inequality, we obtain
n
δ
n
≤
σ − λi 2ν2 2 i1
b
sx|Φi |2 dx a
n
1 σ − λi −1
2δ i1
b
a
|Dyi |2
dx,
sx
3.6
where δ > 0 is a constant to be determined. Set
V n b
b
n
T
σ − λi sx|Φi |2 dx.
2
sx|Φi | dx,
a
i1
3.7
a
i1
Using Lemma 2.1, 3.5, and 3.6, we can get the following results, respectively,
λn1 − σV T ≤ I J,
3.8
n
n
1
−2−1/t
≤ δT μ−1/t
ν1
σ − λi −1 λ1/t
i .
1
ν2
δ
i1
3.9
In order to get the minimum of the right of 3.9, we can take
δT
−1/2
−2−1/t
μ−1/t
ν1
1
n
1/2
σ −
λi −1 λ1/t
i
3.10
.
i1
By 3.9, and 3.10, we can easily get
T≥
2−1/t 2 n
n λ1/t
i
σ − λi
i1
μ1/t
1 ν1
4ν22
−1
3.11
.
Using Lemma 2.2, 3.8, and 3.11, we have
λn1 − σV 2−1/t 2 n
n μ1/t
1 ν1
4ν22
λ1/t
i
σ
−
λi
i1
−1
−1−1/t −1/t
ν1 μ2
≤ t2t − 1μ1
n
1−1/t
λi
,
3.12
i1
that is,
λn1 − σV ≤ t2t −
−1−1/t −1/t
1μ1
ν1 μ2
n
i1
1−1/t
λi
−
2−1/t 2 n
n μ1/t
1 ν1
4ν22
λ1/t
i
σ
−
λi
i1
−1
.
3.13
International Journal of Differential Equations
11
Let the right term of 3.13 be fσ. It is easy to see that
lim fσ −∞,
σ → ∞
−1−1/t −1/t
ν1 μ2
lim fσ t2t − 1μ1
σ → λn
n
1−1/t
λi
> 0.
3.14
i1
Hence, there is σ0 ∈ λn , ∞, such that
n
λ1/t
μ1 ν12 n2
i
σ − λi 4t2t − 1μ2 ν22
i1 0
n
−1
1−1/t
λi
.
3.15
i1
On the other hand, letting
gσ n λ1/t
i
,
σ
−
λi
i1
3.16
we have
g σ −
n
λ1/t
i
i1
σ − λi 2
≤ 0.
3.17
It implies that gσ is the monotone decreasing and continuous function, and its value range
is 0, ∞. Therefore, there exits exactly one σ0 to satisfy 3.15. From 3.13, we know that
σ0 > λn1 . Replacing σ0 with λn1 in 3.15, we can get the result.
References
1 L. E. Payne, G. Polya, and H. F. Weinberger, “Sur le quotient de deux frquences propres conscutives,”
Comptes Rendus de l’Académie des Sciences de Paris, vol. 241, pp. 917–919, 1955.
2 L. E. Payne, G. Polya, and H. F. Weinberger, “On the ratio of consecutive eigenvalues,” Journal of
Mathematics and Physics, vol. 35, pp. 289–298, 1956.
3 G. N. Hile and M. H. Protter, “Inequalities for eigenvalues of the Laplacian,” Indiana University
Mathematics Journal, vol. 29, no. 4, pp. 523–538, 1980.
4 G. N. Hile and R. Z. Yeb, “Inequalities for eigenvalues of the biharmonic operator,” Pacific Journal of
Mathematics, vol. 112, no. 1, pp. 115–133, 1984.
5 S. M. Hook, “Domain independent upper bounds for eigenvalues of elliptic operators,” Transactions
of the American Mathematical Society, vol. 318, no. 2, pp. 615–642, 1990.
6 Z. C. Chen and C. L. Qian, “On the difference of consecutive eigenvalues of uniformly elliptic
operators of higher orders,” Chinese Annals of Mathematics B, vol. 14, no. 4, pp. 435–442, 1993.
7 Z. C. Chen and C. L. Qian, “On the upper bound of eigenvalues for elliptic equations with higher
orders,” Journal of Mathematical Analysis and Applications, vol. 186, no. 3, pp. 821–834, 1994.
8 G. Jia, X. P. Yang, and C. L. Qian, “On the upper bound of second eigenvalues for uniformly elliptic
operators of any orders,” Acta Mathematicae Applicatae Sinica, vol. 19, no. 1, pp. 107–116, 2003.
9 G. Jia and X. P. Yang, “The upper bounds of arbitrary eigenvalues for uniformly elliptic operators
with higher orders,” Acta Mathematicae Applicatae Sinica, vol. 22, no. 4, pp. 589–598, 2006.
10 M. H. Protter, “Can one hear the shape of a drum?” SIAM Review, vol. 29, no. 2, pp. 185–197, 1987.
12
International Journal of Differential Equations
11 R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience Publishers, New York, NY,
USA, 1989.
12 H. Weyl, “Ramifications, old and new, of the eigenvalue problem,” Bulletin of the American
Mathematical Society, vol. 56, pp. 115–139, 1950.
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