On the Upper Bounds of Eigenvalues for a Class of Systems of Ordinary Differential Equations with Higher Order

The estimate of the upper bounds of eigenvalues for a class of systems of ordinary di ﬀ erential equations with higher order is considered by using the calculus theory. Several results about the upper bound inequalities of the (cid:3) n (cid:4) 1 (cid:5) th eigenvalue are obtained by the ﬁrst n eigenvalues. The estimate coe ﬃ cients 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 di ﬀ erential equations and in applications to mechanics and physics.


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 λ n 1 in terms of λ 1 , λ 2 , . . ., λ n 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 .

International Journal of Differential Equations
Let a, b ⊂ R 1 be a bounded domain and t ≥ 2 be an integer.The following eigenvalue problems are studied: 1, 2, . . ., n and s x satisfies the following conditions: where μ 2 ≥ μ 1 > 0, μ 1 , μ 2 are both constants; , and there are constants 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 By virtue of a ij x a ji x , therefore A T x A x , 1.1 can be changed into the following form: Obviously, 1.4 -1.5 is equivalent to 1.1 .
International Journal of Differential Equations For fixed n, let where b ij b a xs x y i y T j dx.Obviously, b ij b ji , and Φ i are weighted orthogonal to y 1 , y 2 , . . ., y n .Furthermore, Φ i a Φ i b 0, i, j 1, 2, . . ., n.We can use the well-known Rayleigh theorem 11, 12 to obtain It is easy to see that

International Journal of Differential Equations
We have b a

1.12
In addition, using the fact that Φ i are weighted orthogonal to y 1 , y 2 , . . ., y n and we know that the last term of 1.12 is equal to zero.Thus, we have

1.15
From 1.14 , we have By using 1.10 and 1.16 , one can give 1.17 Substituting λ n for λ i in 1.17 , we get In order to get the estimations of the eigenvalues, we only need to show the estimates about I, J, and

Lemmas
Lemma 2.1.Suppose that the eigenfunctions y i of 1.4 -1.5 correspond to the eigenvalues λ i .Then one has Proof. 1 By induction.If p 1, using integration by parts and the Schwarz inequality, we have b a

2.2
For p k 1, using integration by parts, the Schwarz inequality and the result when p k, one can give

6 International Journal of Differential Equations
By further calculating, one can give .
2 Using 1 and the inductive method, we have b a

2.8
International Journal of Differential Equations 7 Proof.Since 2.9 2.10 we have

2.11
By a ij x a ji x , the last term of 2.11 is zero.Then we can get

2.14
Therefore, we obtain

2.16
Proof.By the definition of Φ i , one has

2.23
By further calculating, we can easily get Lemma 2.3.

International Journal of Differential Equations
Proof.Choosing the parameter σ > λ n , using 1.17 , one can give By 2.22 and the Young inequality, we obtain where δ > 0 is a constant to be determined.Set Using Lemma 2.1, 3.5 , and 3.6 , we can get the following results, respectively, 3.9 In order to get the minimum of the right of 3.9 , we can take

3.10
By 3.9 , and 3.10 , we can easily get

3.11
Using Lemma 2.2, 3.8 , and 3.11 , we have 12 that is, It implies that g σ is the monotone decreasing and continuous function, and its value range is 0, ∞ .Therefore, there exits exactly one σ 0 to satisfy 3.15 .From 3.13 , we know that σ 0 > λ n 1 .Replacing σ 0 with λ n 1 in 3.15 , we can get the result.
Dy T j dx, it is easy to see that the last term of 2.17 is zero.Then we have 5 , y 1 , y 2 , . . ., y n , . . .are the corresponding eigenfunctions and satisfy the following weighted orthogonal conditions: we have