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International Journal of Differential Equations
Volumeย 2011, Article IDย 712703, 12 pages
http://dx.doi.org/10.1155/2011/712703
Research Article

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

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: Bashirย Ahmad

Copyright ยฉ 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 (๐‘›+1)th eigenvalue are obtained by the first ๐‘› 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 [1โ€“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 ๐œ†๐‘›+1 in terms of ๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘› 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 [๐‘Ž,๐‘]โŠ‚๐‘…1 be a bounded domain and ๐‘กโ‰ฅ2 be an integer. The following eigenvalue problems are studied: (โˆ’1)๐‘ก๐ท๐‘ก๎€ท๐‘Ž11๐ท๐‘ก๐‘ฆ1+๐‘Ž12๐ท๐‘ก๐‘ฆ2+โ‹ฏ+๐‘Ž1๐‘›๐ท๐‘ก๐‘ฆ๐‘›๎€ธ=๐œ†๐‘ (๐‘ฅ)๐‘ฆ1,(โˆ’1)๐‘ก๐ท๐‘ก๎€ท๐‘Ž21๐ท๐‘ก๐‘ฆ1+๐‘Ž22๐ท๐‘ก๐‘ฆ2+โ‹ฏ+๐‘Ž2๐‘›๐ท๐‘ก๐‘ฆ๐‘›๎€ธ=๐œ†๐‘ (๐‘ฅ)๐‘ฆ2,โ‹ฎ(โˆ’1)๐‘ก๐ท๐‘ก๎€ท๐‘Ž๐‘›1๐ท๐‘ก๐‘ฆ1+๐‘Ž๐‘›2๐ท๐‘ก๐‘ฆ2+โ‹ฏ+๐‘Ž๐‘›๐‘›๐ท๐‘ก๐‘ฆ๐‘›๎€ธ=๐œ†๐‘ (๐‘ฅ)๐‘ฆ๐‘›,๐ท๐‘˜๐‘ฆ๐‘–(๐‘Ž)=๐ท๐‘˜๐‘ฆ๐‘–(๐‘)=0(๐‘–=1,2,โ€ฆ,๐‘›,๐‘˜=0,1,2,โ€ฆ,๐‘กโˆ’1),(1.1) where ๐ท=๐‘‘/๐‘‘๐‘ฅ,๐ท๐‘˜=๐‘‘๐‘˜/๐‘‘๐‘ฅ๐‘˜, ๐‘Ž๐‘–๐‘—(๐‘ฅ)(๐‘–,๐‘—=1,2,โ€ฆ,๐‘›) and ๐‘ (๐‘ฅ) satisfies the following conditions: (1ยฐ)๐‘Ž๐‘–๐‘—(๐‘ฅ)โˆˆ๐ถ๐‘ก[๐‘Ž,๐‘],๐‘Ž๐‘–๐‘—(๐‘ฅ)=๐‘Ž๐‘—๐‘–(๐‘ฅ),๐‘–,๐‘—=1,2,โ€ฆ,๐‘›; (2ยฐ) for the arbitrary ๐œ‰=(๐œ‰1,๐œ‰2,โ€ฆ,๐œ‰๐‘›)โˆˆ๐‘…๐‘›, we have ๐œ‡1||๐œ‰||2โ‰ค๐‘›๎“๐‘–,๐‘—=1๐‘Ž๐‘–๐‘—(๐‘ฅ)๐œ‰๐‘–๐œ‰๐‘—โ‰ค๐œ‡2||๐œ‰||2[],โˆ€๐‘ฅโˆˆ๐‘Ž,๐‘,(1.2) where ๐œ‡2โ‰ฅ๐œ‡1>0, ๐œ‡1,๐œ‡2 are both constants;(3ยฐ)๐‘ (๐‘ฅ)โˆˆ๐ถ[๐‘Ž,๐‘], and there are constants ๐œˆ1โ‰ค๐œˆ2, such that 0<๐œˆ1โ‰ค๐‘ (๐‘ฅ)โ‰ค๐œˆ2.

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 ๐ฒ๐‘‡=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ฆ1๐‘ฆ2โ‹ฎ๐‘ฆ๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๐ท๐‘ก๐ฒ๐‘‡=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐ท๐‘ก๐‘ฆ1๐ท๐‘ก๐‘ฆ2โ‹ฎ๐ท๐‘ก๐‘ฆ๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘Ž,๐€(๐‘ฅ)=11(๐‘ฅ)๐‘Ž12(๐‘ฅ)โ‹ฏ๐‘Ž1๐‘›๐‘Ž(๐‘ฅ)21(๐‘ฅ)๐‘Ž22(๐‘ฅ)โ‹ฏ๐‘Ž2๐‘›(โ‹ฎ๐‘Ž๐‘ฅ)๐‘›1(๐‘ฅ)๐‘Ž๐‘›2(๐‘ฅ)โ‹ฏ๐‘Ž๐‘›๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ (๐‘ฅ).(1.3)

By virtue of ๐‘Ž๐‘–๐‘—(๐‘ฅ)=๐‘Ž๐‘—๐‘–(๐‘ฅ), therefore ๐€๐‘‡(๐‘ฅ)=๐€(๐‘ฅ), (1.1) can be changed into the following form: (โˆ’1)๐‘ก๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๎€ธ=๐œ†๐‘ (๐‘ฅ)๐ฒ๐‘‡๐ฒ,(1.4)(๐‘˜)(๐‘Ž)=๐ฒ(๐‘˜)(๐‘)=0,๐‘˜=0,1,2,โ€ฆ,๐‘กโˆ’1.(1.5) Obviously, (1.4)-(1.5) is equivalent to (1.1).

Suppose that 0<๐œ†1โ‰ค๐œ†2โ‰คโ‹ฏโ‰ค๐œ†๐‘›โ‰คโ‹ฏ are eigenvalues of (1.4)-(1.5), ๐ฒ1,๐ฒ2,โ€ฆ,๐ฒ๐‘›,โ€ฆ are the corresponding eigenfunctions and satisfy the following weighted orthogonal conditions:๎€œ๐‘๐‘Ž๐‘ (๐‘ฅ)๐ฒ๐‘–๐ฒ๐‘‡๐‘—๐‘‘๐‘ฅ=๐›ฟ๐‘–๐‘—=๎‚ป1,๐‘–=๐‘—,0,๐‘–โ‰ ๐‘—,๐‘–,๐‘—=1,2,โ€ฆ.(1.6)

Multiplying ๐ฒ๐‘– in sides of (1.4), by using (1.5) and integration by parts, we have ๐œ†๐‘–=๎€œ๐‘๐‘Ž๐ท๐‘ก๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ,๐‘–=1,2,โ€ฆ.(1.7) From (2ยฐ), we have ๎€œ๐‘๐‘Ž||๐ท๐‘ก๐ฒ๐‘–||2๎€œ๐‘‘๐‘ฅ=๐‘๐‘Ž๐ท๐‘ก๐ฒ๐‘–๐ท๐‘ก๐ฒ๐‘‡๐‘–๐œ†๐‘‘๐‘ฅโ‰ค๐‘–๐œ‡1,๐‘–=1,2,โ€ฆ.(1.8)

For fixed ๐‘›, let ๐šฝ๐‘–=๐‘ฅ๐ฒ๐‘–โˆ’๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๐ฒ๐‘—,๐‘–=1,2,โ€ฆ,๐‘›,(1.9) where ๐‘๐‘–๐‘—=โˆซ๐‘๐‘Ž๐‘ฅ๐‘ (๐‘ฅ)๐ฒ๐‘–๐ฒ๐‘‡๐‘—๐‘‘๐‘ฅ. Obviously, ๐‘๐‘–๐‘—=๐‘๐‘—๐‘–, and ฮฆ๐‘– are weighted orthogonal to ๐ฒ1,๐ฒ2,โ€ฆ,๐ฒ๐‘›. Furthermore, ฮฆ๐‘–(๐‘Ž)=ฮฆ๐‘–(๐‘)=0, ๐‘–,๐‘—=1,2,โ€ฆ,๐‘›.

We can use the well-known Rayleigh theorem [11, 12] to obtain ๐œ†๐‘›+1โ‰ค(โˆ’1)๐‘กโˆซ๐‘๐‘Ž๐šฝ๐‘–๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐šฝ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅโˆซ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅ.(1.10) It is easy to see that (โˆ’1)๐‘ก๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐šฝ๐‘‡๐‘–๎€ธ=(โˆ’1)๐‘ก๐‘ก๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธ+(โˆ’1)๐‘ก๐‘ก๐ท๐‘กโˆ’1๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธ+(โˆ’1)๐‘ก๐‘ฅ๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธโˆ’(โˆ’1)๐‘ก๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘—๎€ธ=(โˆ’1)๐‘ก๐‘ก๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธ+(โˆ’1)๐‘ก๐‘ก๐ท๐‘กโˆ’1๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธ+๐œ†๐‘–๐‘ฅ๐‘ (๐‘ฅ)๐ฒ๐‘‡๐‘–โˆ’๐‘ (๐‘ฅ)๐‘›๎“๐‘—=1๐œ†๐‘—๐‘๐‘–๐‘—๐ฒ๐‘‡๐‘—.(1.11)

We have ๎€œ๐‘๐‘Ž๐šฝ๐‘–(โˆ’1)๐‘ก๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐šฝ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ=๐œ†๐‘–๎€œ๐‘๐‘Ž๐‘ฅ๐‘ (๐‘ฅ)๐šฝ๐‘–๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ+(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ+(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐‘กโˆ’1๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธโˆ’๎€œ๐‘‘๐‘ฅ๐‘๐‘Ž๐‘ (๐‘ฅ)๐šฝ๐‘–๐‘›๎“๐‘—=1๐œ†๐‘—๐‘๐‘–๐‘—๐ฒ๐‘‡๐‘—๐‘‘๐‘ฅ.(1.12)

In addition, using the fact that ฮฆ๐‘– are weighted orthogonal to ๐ฒ1,๐ฒ2,โ€ฆ,๐ฒ๐‘› and ๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๎€œ๐‘‘๐‘ฅ=๐‘๐‘Ž๐‘ฅ๐‘ (๐‘ฅ)๐šฝ๐‘–๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ,(1.13) we know that the last term of (1.12) is equal to zero. Thus, we have ๎€œ๐‘๐‘Ž๐šฝ๐‘–(โˆ’1)๐‘ก๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐šฝ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ=๐œ†๐‘–๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅ+(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธ+๐‘‘๐‘ฅ(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐‘กโˆ’1๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ.(1.14) Set ๐ผ๐‘–=(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ,๐ผ=๐‘›๎“๐‘–=1๐ผ๐‘–,๐ฝ๐‘–=(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐‘กโˆ’1๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ,๐ฝ=๐‘›๎“๐‘–=1๐ฝ๐‘–.(1.15) From (1.14), we have ๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž๐šฝ๐‘–(โˆ’1)๐‘ก๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐šฝ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ=๐‘›๎“๐‘–=1๐œ†๐‘–๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅ+๐ผ+๐ฝ.(1.16) By using (1.10) and (1.16), one can give ๐œ†๐‘›๐‘›+1๎“๐‘–=1๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅโ‰ค๐‘›๎“๐‘–=1๐œ†๐‘–๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅ+๐ผ+๐ฝ.(1.17) Substituting ๐œ†๐‘› for ๐œ†๐‘– in (1.17), we get ๎€ท๐œ†๐‘›+1โˆ’๐œ†๐‘›๎€ธ๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅโ‰ค๐ผ+๐ฝ.(1.18)

In order to get the estimations of the eigenvalues, we only need to show the estimates about ๐ผ,๐ฝ, and โˆ‘๐‘›๐‘–=1โˆซ๐‘๐‘Ž๐‘ (๐‘ฅ)|ฮฆ๐‘–|2๐‘‘๐‘ฅ.

2. Lemmas

Lemma 2.1. Suppose that the eigenfunctions ๐ฒ๐‘– of (1.4)-(1.5) correspond to the eigenvalues ๐œ†๐‘–. Then one has (1)โˆซ๐‘๐‘Ž|๐ท๐‘๐ฒ๐‘–|2๐‘‘๐‘ฅโ‰ค๐œˆ1โˆ’1/(๐‘+1)(โˆซ๐‘๐‘Ž|๐ท๐‘+1๐ฒ๐‘–|2๐‘‘๐‘ฅ)๐‘/(๐‘+1),๐‘=1,2,โ€ฆ,๐‘กโˆ’1; (2)โˆซ๐‘๐‘Ž|๐ท๐ฒ๐‘–|2๐‘‘๐‘ฅโ‰ค๐œˆ1โˆ’(1โˆ’(1/๐‘ก))(๐œ†๐‘–/๐œ‡1)1/๐‘ก.

Proof. (1) By induction. If ๐‘=1, using integration by parts and the Schwarz inequality, we have ๎€œ๐‘๐‘Ž||๐ท๐ฒ๐‘–||2||||๎€œ๐‘‘๐‘ฅโ‰ค๐‘๐‘Ž||๐ท๐ฒ๐‘–||2||||=||||๎€œ๐‘‘๐‘ฅ๐‘๐‘Ž๐ท๐ฒ๐‘–๐ท๐ฒ๐‘‡๐‘–||||=||||๎€œ๐‘‘๐‘ฅ๐‘๐‘Ž๐ฒ๐‘–๐ท2๐ฒ๐‘‡๐‘–||||โ‰ค๎‚ต๎€œ๐‘‘๐‘ฅ๐‘๐‘Ž||๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ1/2๎‚ต๎€œ๐‘๐‘Ž||๐ท2๐ฒ๐‘‡๐‘–||2๎‚ถ๐‘‘๐‘ฅ1/2โ‰ค๐œˆ1โˆ’1/2๎‚ต๎€œ๐‘๐‘Ž||๐ท2๐ฒ๐‘‡๐‘–||2๎‚ถ๐‘‘๐‘ฅ1/2.(2.1) Therefore, when ๐‘=1, (1) is true.
If for ๐‘=๐‘˜, (1) is true, that is, ๎€œ๐‘๐‘Ž||๐ท๐‘˜๐ฒ๐‘–||2๐‘‘๐‘ฅโ‰ค๐œˆ1โˆ’1/(๐‘˜+1)๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘˜+1๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ๐‘˜/(๐‘˜+1).(2.2) For ๐‘=๐‘˜+1, using integration by parts, the Schwarz inequality and the result when ๐‘=๐‘˜, one can give ๎€œ๐‘๐‘Ž||๐ท๐‘˜+1๐ฒ๐‘–||2||||๎€œ๐‘‘๐‘ฅโ‰ค๐‘๐‘Ž||๐ท๐‘˜+1๐ฒ๐‘–||2||||=||||๎€œ๐‘‘๐‘ฅ๐‘๐‘Ž๐ท๐‘˜๐ฒ๐‘–โ‹…๐ท๐‘˜+2๐ฒ๐‘‡๐‘–||||โ‰ค๎‚ต๎€œ๐‘‘๐‘ฅ๐‘๐‘Ž||๐ท๐‘˜๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ1/2๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘˜+2๐ฒ๐‘‡๐‘–||2๎‚ถ๐‘‘๐‘ฅ1/2โ‰ค๐œˆ1โˆ’1/(2(๐‘˜+1))๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘˜+2๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ1/2๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘˜+1๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ๐‘˜/(2(๐‘˜+1)).(2.3) By further calculating, one can give ๎€œ๐‘๐‘Ž||๐ท๐‘˜+1๐ฒ๐‘–||2๐‘‘๐‘ฅโ‰ค๐œˆ1โˆ’1/((๐‘˜+1)+1)๎‚ต๎€œ๐‘๐‘Ž||๐ท(๐‘˜+1)+1๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ(๐‘˜+1)/((๐‘˜+1)+1).(2.4) Therefore, when ๐‘=๐‘˜+1, (1) is true.
(2) Using (1) and the inductive method, we have ๎€œ๐‘๐‘Ž||๐ท๐‘๐ฒ๐‘–||2๐‘‘๐‘ฅโ‰ค๐œˆ1โˆ’1/(๐‘+1)๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘+1๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ๐‘/(๐‘+1)โ‰ค๐œˆ1โˆ’2/(๐‘+2)๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘+2๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ๐‘/(๐‘+2)โ‰คโ‹ฏโ‰ค๐œˆ1โˆ’(1โˆ’(๐‘/๐‘ก))๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘ก๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ๐‘/๐‘ก.(2.5) From (1.8) and (2.5), we get ๎€œ๐‘๐‘Ž||๐ท๐‘๐ฒ๐‘–||2๐‘‘๐‘ฅโ‰ค๐œˆ1โˆ’(1โˆ’(๐‘/๐‘ก))๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘ก๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ๐‘/๐‘กโ‰ค๐œˆ1โˆ’(1โˆ’(๐‘/๐‘ก))๎‚ต๐œ†๐‘–๐œ‡1๎‚ถ๐‘/๐‘ก,๐‘=1,2,โ€ฆ,๐‘ก.(2.6) Taking ๐‘=1, we have ๎€œ๐‘๐‘Ž||๐ท๐ฒ๐‘–||2๐‘‘๐‘ฅโ‰ค๐œˆ1โˆ’(1โˆ’(1/๐‘ก))๎‚ต๐œ†๐‘–๐œ‡1๎‚ถ1/๐‘ก.(2.7) So Lemma 2.1 is true.

Lemma 2.2. Let ๐œ†1,๐œ†2,โ€ฆ,๐œ†๐‘› be the eigenvalues of (1.4)-(1.5). Then one has ๐ผ+๐ฝโ‰ค๐‘ก(2๐‘กโˆ’1)๐œ‡1โˆ’(1โˆ’(1/๐‘ก))๐œˆ1โˆ’1/๐‘ก๐œ‡2๐‘›๎“๐‘–=1๐œ†๐‘–1โˆ’(1/๐‘ก).(2.8)

Proof. Since ๐ผ๐‘–=(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ=(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๎ƒฉ๐‘ฅ๐ฒ๐‘–โˆ’๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๐ฒ๐‘—๎ƒช๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ=(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๐‘ฅ๐ฒ๐‘–๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธโˆ’๐‘‘๐‘ฅ(โˆ’1)๐‘ก๐‘ก๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๎€œ๐‘๐‘Ž๐ฒ๐‘—๐ท๐‘ก๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ=๐‘ก2๎€œ๐‘๐‘Ž๐ท๐‘กโˆ’1๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€œ๐‘‘๐‘ฅ+๐‘ก๐‘๐‘Ž๐‘ฅ๐ท๐‘ก๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅโˆ’๐‘ก๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๎€œ๐‘๐‘Ž๐ท๐‘ก๐ฒ๐‘—๎€ท๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๎€ธ๐ฝ๐‘‘๐‘ฅ,(2.9)๐‘–=(โˆ’1)๐‘ก๐‘ก๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐‘กโˆ’1๎€ท๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธ๎€œ๐‘‘๐‘ฅ=โˆ’๐‘ก(๐‘กโˆ’1)๐‘๐‘Ž๐ท๐‘กโˆ’2๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€œ๐‘‘๐‘ฅโˆ’๐‘ก๐‘๐‘Ž๐‘ฅ๐ท๐‘กโˆ’1๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ+๐‘ก๐‘›๎“๐‘—=1๐‘๐‘–๐‘—๎€œ๐‘๐‘Ž๐ท๐‘กโˆ’1๐ฒ๐‘—๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ,(2.10) we have ๐ผ+๐ฝ=๐‘›๎“๐‘–=1๎€ท๐ผ๐‘–+๐ฝ๐‘–๎€ธ=๐‘›๎“๐‘–=1๐‘ก๎€œ๐‘๐‘Ž๎€ท๐‘ก๐ท๐‘กโˆ’1๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–โˆ’(๐‘กโˆ’1)๐ท๐‘กโˆ’2๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅโˆ’๐‘ก๐‘›๎“๐‘–,๐‘—=1๐‘๐‘–๐‘—๎€œ๐‘๐‘Ž๎€ท๐ท๐‘ก๐ฒ๐‘—๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–โˆ’๐ท๐‘กโˆ’1๐ฒ๐‘—๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ.(2.11) By ๐‘Ž๐‘–๐‘—(๐‘ฅ)=๐‘Ž๐‘—๐‘–(๐‘ฅ), the last term of (2.11) is zero. Then we can get ๐ผ+๐ฝ=๐‘›๎“๐‘–=1๐‘ก๎€œ๐‘๐‘Ž๎€ท๐‘ก๐ท๐‘กโˆ’1๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–โˆ’(๐‘กโˆ’1)๐ท๐‘กโˆ’2๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–๎€ธ๐‘‘๐‘ฅ.(2.12) Using (2ยฐ), Lemma 2.1, (1) and (2.6), we have ๎€œ๐‘๐‘Ž๐ท๐‘กโˆ’1๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘กโˆ’1๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅโ‰ค๐œ‡2๎€œ๐‘๐‘Ž||๐ท๐‘กโˆ’1๐ฒ๐‘–||2๐‘‘๐‘ฅโ‰ค๐œ‡2๐œˆ1โˆ’1/๐‘ก๎‚ต๐œ†๐‘–๐œ‡1๎‚ถ1โˆ’(1/๐‘ก).(2.13) Using (2ยฐ), the Schwarz inequality, Lemma 2.1โ€‰โ€‰(1), and (2.6), one can give ||||โˆ’๎€œ๐‘๐‘Ž๐ท๐‘กโˆ’2๐ฒ๐‘–๐€(๐‘ฅ)๐ท๐‘ก๐ฒ๐‘‡๐‘–||||๐‘‘๐‘ฅโ‰ค๐œ‡2๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘กโˆ’2๐ฒ๐‘–||2๎‚ถ๐‘‘๐‘ฅ1/2๎‚ต๎€œ๐‘๐‘Ž||๐ท๐‘ก๐ฒ๐‘‡๐‘–||2๎‚ถ๐‘‘๐‘ฅ1/2โ‰ค๐œ‡2๐œˆ1โˆ’1/๐‘ก๎‚ต๐œ†๐‘–๐œ‡1๎‚ถ1โˆ’(1/๐‘ก).(2.14) Therefore, we obtain ๐ผ+๐ฝโ‰ค๐‘ก(2๐‘กโˆ’1)๐œ‡1โˆ’(1โˆ’(1/๐‘ก))๐œˆ1โˆ’1/๐‘ก๐œ‡2๐‘›๎“๐‘–=1๐œ†๐‘–1โˆ’(1/๐‘ก).(2.15)

Lemma 2.3. If ฮฆ๐‘– and ๐œ†๐‘–(๐‘–=1,2,โ€ฆ,๐‘›) as above, then one has ๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐œ‡๐‘‘๐‘ฅโ‰ฅ11/๐‘ก๐œˆ12โˆ’(1/๐‘ก)๐‘›24๐œˆ22๎ƒฉ๐‘›๎“๐‘–=1๐œ†๐‘–1/๐‘ก๎ƒชโˆ’1.(2.16)

Proof. By the definition of ฮฆ๐‘–, one has ๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ=๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž๐‘ฅ๐ฒ๐‘–๐ท๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅโˆ’๐‘›๎“๐‘–,๐‘—=1๐‘๐‘–๐‘—๎€œ๐‘๐‘Ž๐ฒ๐‘—๐ท๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ.(2.17) Using ๐‘๐‘–๐‘—=๐‘๐‘—๐‘– and โˆซ๐‘๐‘Ž๐ฒ๐‘—๐ท๐ฒ๐‘‡๐‘–โˆซ๐‘‘๐‘ฅ=โˆ’๐‘๐‘Ž๐ฒ๐‘–๐ท๐ฒ๐‘‡๐‘—๐‘‘๐‘ฅ, it is easy to see that the last term of (2.17) is zero. Then we have ๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ=๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž๐‘ฅ๐ฒ๐‘–๐ท๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ.(2.18) Using integration by parts, one can give ๎€œ๐‘๐‘Ž๐‘ฅ๐ฒ๐‘–๐ท๐ฒ๐‘‡๐‘–๎€œ๐‘‘๐‘ฅ=โˆ’๐‘๐‘Ž||๐ฒ๐‘–||2๎€œ๐‘‘๐‘ฅโˆ’๐‘๐‘Ž๐‘ฅ๐ฒ๐‘–๐ท๐ฒ๐‘‡๐‘–๐‘‘๐‘ฅ,(2.19)๎€œ๐‘๐‘Ž๐‘ฅ๐ฒ๐‘–๐ท๐ฒ๐‘‡๐‘–1๐‘‘๐‘ฅ=โˆ’2๎€œ๐‘๐‘Ž||๐ฒ๐‘–||2๐‘‘๐‘ฅ.(2.20) By 1/๐œˆ2โ‰คโˆซ๐‘๐‘Ž|๐ฒ๐‘–|2๐‘‘๐‘ฅโ‰ค1/๐œˆ1, we have ||||๎€œ๐‘๐‘Ž๐‘ฅ๐ฒ๐‘–๐ท๐ฒ๐‘‡๐‘–||||=1๐‘‘๐‘ฅ2๎€œ๐‘๐‘Ž||๐ฒ๐‘–||21๐‘‘๐‘ฅโ‰ฅ2๐œˆ2.(2.21) From (2.18) and (2.21), we can get ๐‘›๎“๐‘–=1||||๎€œ๐‘๐‘Ž๐šฝ๐‘–๐ท๐ฒ๐‘‡๐‘–||||โ‰ฅ๐‘›๐‘‘๐‘ฅ2๐œˆ2.(2.22) Using the Schwarz inequality, Lemma 2.1โ€‰โ€‰(2), and (3ยฐ), we have ๐‘›24๐œˆ22โ‰ค๎ƒฉ๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅ๎ƒช๎ƒฉ๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž||๐ท๐ฒ๐‘–||2๎ƒชโ‰ค๎ƒฉ๐‘ (๐‘ฅ)๐‘‘๐‘ฅ๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž๐‘ ||๐šฝ(๐‘ฅ)๐‘–||2๎ƒช๐œˆ๐‘‘๐‘ฅ1โˆ’(2โˆ’(1/๐‘ก))๐œ‡1๐‘›โˆ’1/๐‘ก๎“๐‘–=1๐œ†๐‘–1/๐‘ก.(2.23) By further calculating, we can easily get Lemma 2.3.

3. Main Results

Theorem 3.1. If ๐œ†๐‘–(๐‘–=1,2,โ€ฆ,๐‘›+1) are the eigenvalues of (1.4)-(1.5), then (1)๐œ†๐‘›+1โ‰ค๐œ†๐‘›+4๐‘ก(2๐‘กโˆ’1)๐œ‡2๐œˆ22๐œ‡1๐œˆ21๐‘›2๐‘›๎“๐‘–=1๐œ†๐‘–๐‘›1โˆ’(1/๐‘ก)๎“๐‘–=1๐œ†๐‘–1/๐‘ก;(3.1)(2)๐œ†๐‘›+1โ‰ค๎ƒฉ1+4๐‘ก(2๐‘กโˆ’1)๐œ‡2๐œˆ22๐œ‡1๐œˆ21๎ƒช๐œ†๐‘›.(3.2)

Proof. From (1.18), we can get ๎€ท๐œ†๐‘›+1โˆ’๐œ†๐‘›๎€ธ๎ƒฉโ‰ค(๐ผ+๐ฝ)๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๎ƒช๐‘‘๐‘ฅโˆ’1.(3.3) Using Lemmas 2.2 and 2.3, we can easily get (3.1). In (3.1), Replacing ๐œ†๐‘– with ๐œ†๐‘›, by further calculating, we can get (3.2).

Theorem 3.2. For ๐‘›โ‰ฅ1, one has ๐‘›๎“๐‘–=1๐œ†๐‘–1/๐‘ก๐œ†๐‘›+1โˆ’๐œ†๐‘–โ‰ฅ๐œ‡1๐œˆ21๐‘›24๐‘ก(2๐‘กโˆ’1)๐œ‡2๐œˆ22๎ƒฉ๐‘›๎“๐‘–=1๐œ†๐‘–1โˆ’(1/๐‘ก)๎ƒชโˆ’1.(3.4)

Proof. Choosing the parameter ๐œŽ>๐œ†๐‘›, using (1.17), one can give ๐œ†๐‘›๐‘›+1๎“๐‘–=1๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅโ‰ค๐œŽ๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅ+๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž๎€ท๐œ†๐‘–๎€ธ||๐šฝโˆ’๐œŽ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅ+๐ผ+๐ฝ.(3.5) By (2.22) and the Young inequality, we obtain ๐‘›2๐œˆ2โ‰ค๐›ฟ2๐‘›๎“๐‘–=1๎€ท๐œŽโˆ’๐œ†๐‘–๎€ธ๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||21๐‘‘๐‘ฅ+2๐›ฟ๐‘›๎“๐‘–=1๎€ท๐œŽโˆ’๐œ†๐‘–๎€ธโˆ’1๎€œ๐‘๐‘Ž||๐ท๐ฒ๐‘–||2๐‘ (๐‘ฅ)๐‘‘๐‘ฅ,(3.6) where ๐›ฟ>0 is a constant to be determined. Set ๐‘‰=๐‘›๎“๐‘–=1๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅ,๐‘‡=๐‘›๎“๐‘–=1๎€ท๐œŽโˆ’๐œ†๐‘–๎€ธ๎€œ๐‘๐‘Ž||๐šฝ๐‘ (๐‘ฅ)๐‘–||2๐‘‘๐‘ฅ.(3.7) Using Lemma 2.1, (3.5), and (3.6), we can get the following results, respectively, ๎€ท๐œ†๐‘›+1๎€ธ๐‘›โˆ’๐œŽ๐‘‰+๐‘‡โ‰ค๐ผ+๐ฝ,(3.8)๐œˆ21โ‰ค๐›ฟ๐‘‡+๐›ฟ๐œ‡1โˆ’1/๐‘ก๐œˆ1๐‘›โˆ’(2โˆ’(1/๐‘ก))๎“๐‘–=1๎€ท๐œŽโˆ’๐œ†๐‘–๎€ธโˆ’1๐œ†๐‘–1/๐‘ก.(3.9) In order to get the minimum of the right of (3.9), we can take ๐›ฟ=๐‘‡โˆ’1/2๎ƒฉ๐œ‡1โˆ’1/๐‘ก๐œˆ1๐‘›โˆ’(2โˆ’(1/๐‘ก))๎“๐‘–=1๎€ท๐œŽโˆ’๐œ†๐‘–๎€ธโˆ’1๐œ†๐‘–1/๐‘ก๎ƒช1/2.(3.10) By (3.9), and (3.10), we can easily get ๐œ‡๐‘‡โ‰ฅ11/๐‘ก๐œˆ12โˆ’(1/๐‘ก)๐‘›24๐œˆ22๎ƒฉ๐‘›๎“๐‘–=1๐œ†๐‘–1/๐‘ก๐œŽโˆ’๐œ†๐‘–๎ƒชโˆ’1.(3.11) Using Lemma 2.2, (3.8), and (3.11), we have ๎€ท๐œ†๐‘›+1๎€ธ๐œ‡โˆ’๐œŽ๐‘‰+11/๐‘ก๐œˆ12โˆ’(1/๐‘ก)๐‘›24๐œˆ22๎ƒฉ๐‘›๎“๐‘–=1๐œ†๐‘–1/๐‘ก๐œŽโˆ’๐œ†๐‘–๎ƒชโˆ’1โ‰ค๐‘ก(2๐‘กโˆ’1)๐œ‡1โˆ’(1โˆ’(1/๐‘ก))๐œˆ1โˆ’1/๐‘ก๐œ‡2๐‘›๎“๐‘–=1๐œ†๐‘–1โˆ’(1/๐‘ก),(3.12) that is, ๎€ท๐œ†๐‘›+1๎€ธโˆ’๐œŽ๐‘‰โ‰ค๐‘ก(2๐‘กโˆ’1)๐œ‡1โˆ’(1โˆ’(1/๐‘ก))๐œˆ1โˆ’1/๐‘ก๐œ‡2๐‘›๎“๐‘–=1๐œ†๐‘–1โˆ’(1/๐‘ก)โˆ’๐œ‡11/๐‘ก๐œˆ12โˆ’(1/๐‘ก)๐‘›24๐œˆ22๎ƒฉ๐‘›๎“๐‘–=1๐œ†๐‘–1/๐‘ก๐œŽโˆ’๐œ†๐‘–๎ƒชโˆ’1.(3.13) Let the right term of (3.13) be ๐‘“(๐œŽ). It is easy to see that lim๐œŽโ†’+โˆž๐‘“(๐œŽ)=โˆ’โˆž,lim๐œŽโ†’๐œ†+๐‘›๐‘“(๐œŽ)=๐‘ก(2๐‘กโˆ’1)๐œ‡1โˆ’(1โˆ’(1/๐‘ก))๐œˆ1โˆ’1/๐‘ก๐œ‡2๐‘›๎“๐‘–=1๐œ†๐‘–1โˆ’(1/๐‘ก)>0.(3.14) Hence, there is ๐œŽ0โˆˆ(๐œ†๐‘›,+โˆž), such that ๐‘›๎“๐‘–=1๐œ†๐‘–1/๐‘ก๐œŽ0โˆ’๐œ†๐‘–=๐œ‡1๐œˆ21๐‘›24๐‘ก(2๐‘กโˆ’1)๐œ‡2๐œˆ22๎ƒฉ๐‘›๎“๐‘–=1๐œ†๐‘–1โˆ’(1/๐‘ก)๎ƒชโˆ’1.(3.15) On the other hand, letting ๐‘”(๐œŽ)=๐‘›๎“๐‘–=1๐œ†๐‘–1/๐‘ก๐œŽโˆ’๐œ†๐‘–,(3.16) we have ๐‘”๎…ž(๐œŽ)=โˆ’๐‘›๎“๐‘–=1๐œ†๐‘–1/๐‘ก๎€ท๐œŽโˆ’๐œ†๐‘–๎€ธ2โ‰ค0.(3.17) It implies that ๐‘”(๐œŽ) is the monotone decreasing and continuous function, and its value range is (0,+โˆž). Therefore, there exits exactly one ๐œŽ0 to satisfy (3.15). From (3.13), we know that ๐œŽ0>๐œ†๐‘›+1. Replacing ๐œŽ0 with ๐œ†๐‘›+1 in (3.15), we can get the result.

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