International Journal of Differential EquationsVolume 2011 (2011), Article ID 712703, 12 pagesdoi: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

Received 4 May 2011; Revised 16 July 2011; Accepted 19 July 2011

Academic Editor: Bashir Ahmad

Copyright © 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.

Abstract.

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 ()th eigenvalue are obtained by the first 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 in terms of 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].

Let be a bounded domain and be an integer. The following eigenvalue problems are studied: where , and satisfies the following conditions: (1°); (2°) for the arbitrary , we have where , are both constants;(3°), and there are constants , such that .

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 , therefore , (1.1) can be changed into the following form: Obviously, (1.4)-(1.5) is equivalent to (1.1).

Suppose that are eigenvalues of (1.4)-(1.5), are the corresponding eigenfunctions and satisfy the following weighted orthogonal conditions:

Multiplying in sides of (1.4), by using (1.5) and integration by parts, we have From (2°), we have

For fixed , let where . Obviously, , and are weighted orthogonal to . Furthermore, , .

We can use the well-known Rayleigh theorem [11, 12] to obtain It is easy to see that

We have

In addition, using the fact that are weighted orthogonal to and we know that the last term of (1.12) is equal to zero. Thus, we have Set From (1.14), we have By using (1.10) and (1.16), one can give Substituting for in (1.17), we get

In order to get the estimations of the eigenvalues, we only need to show the estimates about , and

2. Lemmas

Lemma 2.1. Suppose that the eigenfunctions of (1.4)-(1.5) correspond to the eigenvalues . Then one has (1); (2).

Proof. (1) By induction. If , using integration by parts and the Schwarz inequality, we have Therefore, when , (1) is true.If for , (1) is true, that is, For , using integration by parts, the Schwarz inequality and the result when , one can give By further calculating, one can give Therefore, when , (1) is true.(2) Using (1) and the inductive method, we have From (1.8) and (2.5), we get Taking , we have So Lemma 2.1 is true.

Lemma 2.2. Let be the eigenvalues of (1.4)-(1.5). Then one has

Proof. Since we have By , the last term of (2.11) is zero. Then we can get Using (2°), Lemma 2.1, (1) and (2.6), we have Using (2°), the Schwarz inequality, Lemma 2.1  (1), and (2.6), one can give Therefore, we obtain

Lemma 2.3. If and as above, then one has

Proof. By the definition of , one has Using and , it is easy to see that the last term of (2.17) is zero. Then we have Using integration by parts, one can give By , we have From (2.18) and (2.21), we can get Using the Schwarz inequality, Lemma 2.1  (2), and (3°), we have By further calculating, we can easily get Lemma 2.3.

3. Main Results

Theorem 3.1. If are the eigenvalues of (1.4)-(1.5), then

Proof. From (1.18), we can get Using Lemmas 2.2 and 2.3, we can easily get (3.1). In (3.1), Replacing with , by further calculating, we can get (3.2).

Theorem 3.2. For , one has

Proof. Choosing the parameter , using (1.17), one can give By (2.22) and the Young inequality, we obtain where is a constant to be determined. Set Using Lemma 2.1, (3.5), and (3.6), we can get the following results, respectively, In order to get the minimum of the right of (3.9), we can take By (3.9), and (3.10), we can easily get Using Lemma 2.2, (3.8), and (3.11), we have that is, Let the right term of (3.13) be . It is easy to see that Hence, there is , such that On the other hand, letting we have It implies that is the monotone decreasing and continuous function, and its value range is . Therefore, there exits exactly one to satisfy (3.15). From (3.13), we know that . Replacing with in (3.15), we can get the result.

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