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

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 (𝑛+1)th 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 𝜆_𝑛+1 in terms of 𝜆₁,𝜆₂,…,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].

Let [𝑎,𝑏]⊂𝑅¹ be a bounded domain and 𝑡≥2 be an integer. The following eigenvalue problems are studied: ((−−11))^𝑡_𝑡𝐷^𝑡𝑎₁₁𝐷^𝑡𝑦₁+𝑎₁₂𝐷^𝑡𝑦₂+⋯+𝑎_1𝑛𝐷^𝑡𝑦_𝑛=𝜆𝑠(𝑥)𝑦₁,(−1)_𝑡𝐷^𝑡𝑎₂₁𝐷^𝑡𝑦₁+𝑎₂₂𝐷^𝑡𝑦₂⋮+⋯+𝑎_2𝑛𝐷^𝑡𝑦_𝑛=𝜆𝑠(𝑥)𝑦₂,𝐷_𝑘𝐷^𝑡𝑎_𝑛1𝐷^𝑡𝑦₁+𝑎_𝑛2𝐷^𝑡𝑦₂+⋯+𝑎_𝑛𝑛𝐷^𝑡𝑦_𝑛=𝜆𝑠(𝑥)𝑦_𝑛,𝑦_𝑖(𝑎)=𝐷^𝑘𝑦_𝑖(𝑏)=0(𝑖=1,2,…,𝑛,𝑘=0,1,2,…,𝑡−1),(1.1) where 𝐷=𝑑/𝑑𝑥,𝐷^𝑘=𝑑^𝑘/𝑑𝑥^𝑘, 𝑎_𝑖𝑗(𝑥)(𝑖,𝑗=1,2,…,𝑛) and 𝑠(𝑥) satisfies the following conditions: (1°)𝑎_𝑖𝑗(𝑥)∈𝐶^𝑡[𝑎,𝑏],𝑎_𝑖𝑗(𝑥)=𝑎_𝑗𝑖(𝑥),𝑖,𝑗=1,2,…,𝑛; (2°) for the arbitrary 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)∈𝑅^𝑛, we have 𝜇₁||𝜉||²≤^𝑛_𝑖_,𝑗=1𝑎_𝑖𝑗(𝑥)𝜉_𝑖𝜉_𝑗≤𝜇₂||𝜉||²,∀𝑥∈[𝑎,𝑏],(1.2) where 𝜇₂≥𝜇₁>0, 𝜇₁,𝜇₂ are both constants;(3°)𝑠(𝑥)∈𝐶[𝑎,𝑏], and there are constants 𝜈₁≤𝜈₂, such that 0<𝜈₁≤𝑠(𝑥)≤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 𝐲^𝑇=⎛⎜⎜⎜⎜⎜⎜⎝𝑦¹𝑦₂𝑦⋮_𝑛⎞⎟⎟⎟⎟⎟⎟⎠,𝐷_𝑡𝐲^𝑇=⎛⎜⎜⎜⎜⎜⎜⎝𝐷^𝑡𝑦₁𝐷^𝑡𝑦₂𝐷⋮_𝑡𝑦_𝑛⎞⎟⎟⎟⎟⎟⎟⎠,𝐀(𝑥)=⎛⎜⎜⎜⎜⎜⎜⎝𝑎𝑎₁₁(𝑥)𝑎₁₂(𝑥)⋯𝑎_1𝑛(𝑥)₂₁𝑎(𝑥)𝑎₂₂(⋮𝑥)⋯𝑎_2𝑛(𝑥)_𝑛1(𝑥)𝑎_𝑛2(𝑥)⋯𝑎_𝑛𝑛(𝑥)⎞⎟⎟⎟⎟⎟⎟⎠.(1.3)

By virtue of 𝑎_𝑖𝑗(𝑥)=𝑎_𝑗𝑖(𝑥), therefore 𝐀^𝑇(𝑥)=𝐀(𝑥), (1.1) can be changed into the following form: 𝐲(−1)^𝑡𝐷^𝑡𝐀(𝑥)𝐷^𝑡𝐲^𝑇=𝜆𝑠(𝑥)𝐲^𝑇,(1.4)_(𝑘)(𝑎)=𝐲^(𝑘)(𝑏)=0,𝑘=0,1,2,…,𝑡−1.(1.5) Obviously, (1.4)-(1.5) is equivalent to (1.1).

Suppose that 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑛≤⋯ are eigenvalues of (1.4)-(1.5), 𝐲₁,𝐲₂,…,𝐲_𝑛,… are the corresponding eigenfunctions and satisfy the following weighted orthogonal conditions:^𝑏𝑎𝑠(𝑥)𝐲_𝑖𝐲^𝑇𝑗𝑑𝑥=𝛿_𝑖𝑗=10,,𝑖𝑖≠=𝑗𝑗,,𝑖,𝑗=1,2,….(1.6)

Multiplying 𝐲_𝑖 in sides of (1.4), by using (1.5) and integration by parts, we have 𝜆_𝑖=^𝑏𝑎𝐷^𝑡𝐲_𝑖𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥,𝑖=1,2,….(1.7) From (2°), we have ^𝑏𝑎||𝐷^𝑡𝐲_𝑖||²𝑑𝑥=^𝑏𝑎𝐷^𝑡𝐲_𝑖𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥≤𝜆^𝑖𝜇₁,𝑖=1,2,….(1.8)

For fixed 𝑛, let 𝚽_𝑖=𝑥𝐲_𝑖−^𝑛_𝑗=1𝑏_𝑖𝑗𝐲_𝑗,𝑖=1,2,…,𝑛,(1.9) where 𝑏_𝑖𝑗=∫^𝑏𝑎𝑥𝑠(𝑥)𝐲_𝑖𝐲^𝑇𝑗𝑑𝑥. Obviously, 𝑏_𝑖𝑗=𝑏_𝑗𝑖, and Φ_𝑖 are weighted orthogonal to 𝐲₁,𝐲₂,…,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 𝜆_𝑛+1≤(−1)^𝑡∫^𝑏𝑎𝚽_𝑖𝐷^𝑡𝐀(𝑥)𝐷^𝑡𝚽^𝑇𝑖𝑑𝑥∫^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥.(1.10) It is easy to see that (−1)^𝑡𝐷^𝑡𝐀(𝑥)𝐷^𝑡𝚽^𝑇𝑖=+(−(1−)1^𝑡𝑡)𝐷^𝑡_𝑡𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖+(−1)^𝑡𝑡𝐷^𝑡−1𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑥𝐷^𝑡𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖−(−1)^𝑡𝑛_𝑗=1=(−1)_𝑡𝑏_𝑖𝑗𝐷^𝑡𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑗+𝜆_𝑖𝑡𝐷^𝑡𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖+(−1)^𝑡𝑡𝐷^𝑡−1𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑥𝑠(𝑥)𝐲^𝑇𝑖−𝑠(𝑥)^𝑛_𝑗=1𝜆_𝑗𝑏_𝑖𝑗𝐲^𝑇𝑗.(1.11)

We have ^𝑏𝑎𝚽_𝑖(−1)^𝑡𝐷^𝑡𝐀(𝑥)𝐷^𝑡𝚽^𝑇𝑖𝑑𝑥=𝜆_𝑖^𝑏𝑎+(−1𝑥)_𝑡𝑠(𝑥)𝚽_𝑖𝐲^𝑇𝑖𝑑𝑥+(−1)^𝑡𝑡^𝑏𝑎𝚽_𝑖𝐷^𝑡𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥−𝑡^𝑏𝑎𝚽_𝑖𝐷^𝑡−1𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥_𝑏𝑎𝑠(𝑥)𝚽_𝑖𝑛_𝑗=1𝜆_𝑗𝑏_𝑖𝑗𝐲^𝑇𝑗𝑑𝑥.(1.12)

In addition, using the fact that Φ_𝑖 are weighted orthogonal to 𝐲₁,𝐲₂,…,𝐲_𝑛 and ^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥=^𝑏𝑎𝑥𝑠(𝑥)𝚽_𝑖𝐲^𝑇𝑖𝑑𝑥,(1.13) we know that the last term of (1.12) is equal to zero. Thus, we have ^𝑏𝑎𝚽_𝑖(−1)^𝑡𝐷^𝑡𝐀(𝑥)𝐷^𝑡𝚽^𝑇𝑖𝑑𝑥=𝜆_𝑖^𝑏𝑎+(−1𝑠)_𝑡(𝑥)||𝚽_𝑖||²𝑑𝑥+(−1)_𝑡𝑡^𝑏𝑎𝚽_𝑖𝐷^𝑡𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥𝑡^𝑏𝑎𝚽_𝑖𝐷^𝑡−1𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥.(1.14) Set 𝐼_𝑖=(−1)^𝑡𝑡^𝑏𝑎𝚽_𝑖𝐷^𝑡𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥,𝐼=^𝑛_𝑖=1𝐽_𝑖𝐼_𝑖,=(−1)^𝑡𝑡^𝑏𝑎𝚽_𝑖𝐷^𝑡−1𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥,𝐽=^𝑛_𝑖=1𝐽_𝑖.(1.15) From (1.14), we have ^𝑛_𝑖=1^𝑏𝑎𝚽_𝑖(−1)^𝑡𝐷^𝑡𝐀(𝑥)𝐷^𝑡𝚽^𝑇𝑖𝑑𝑥=^𝑛_𝑖=1𝜆_𝑖^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥+𝐼+𝐽.(1.16) By using (1.10) and (1.16), one can give 𝜆_𝑛𝑛+1_𝑖=1^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥≤^𝑛_𝑖=1𝜆_𝑖^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥+𝐼+𝐽.(1.17) Substituting 𝜆_𝑛 for 𝜆_𝑖 in (1.17), we get 𝜆_𝑛+1−𝜆_𝑛^𝑛_𝑖=1^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥≤𝐼+𝐽.(1.18)

In order to get the estimations of the eigenvalues, we only need to show the estimates about 𝐼,𝐽, and ∑^𝑛𝑖=1∫^𝑏𝑎𝑠(𝑥)|Φ_𝑖|²𝑑𝑥.

2. Lemmas

Lemma 2.1. Suppose that the eigenfunctions 𝐲_𝑖 of (1.4)-(1.5) correspond to the eigenvalues 𝜆_𝑖. Then one has (1)∫^𝑏𝑎|𝐷^𝑝𝐲_𝑖|²𝑑𝑥≤𝜈^{1−1/(𝑝+1)}(∫^𝑏𝑎|𝐷^𝑝+1𝐲_𝑖|²𝑑𝑥)^{𝑝/(𝑝+1)},𝑝=1,2,…,𝑡−1; (2)∫^𝑏𝑎|𝐷𝐲_𝑖|²𝑑𝑥≤𝜈^{1−(1−(1/𝑡))}(𝜆_𝑖/𝜇₁)^1/𝑡.

Proof. (1) By induction. If 𝑝=1, using integration by parts and the Schwarz inequality, we have ^𝑏𝑎||𝐷𝐲_𝑖||²𝑑𝑥≤≤||||^𝑏𝑎||𝐷𝐲_𝑖||²𝑑𝑥||||=||||^𝑏𝑎𝐷𝐲_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥||||=||||^𝑏𝑎𝐲_𝑖𝐷²𝐲^𝑇𝑖𝑑𝑥_𝑏𝑎||||||𝐲_𝑖||²𝑑𝑥^1/2^𝑏𝑎||𝐷²𝐲^𝑇𝑖||²𝑑𝑥^1/2≤𝜈^1−1/2^𝑏𝑎||𝐷²𝐲^𝑇𝑖||²𝑑𝑥^1/2.(2.1) Therefore, when 𝑝=1, (1) is true.
If for 𝑝=𝑘, (1) is true, that is, ^𝑏𝑎||𝐷^𝑘𝐲_𝑖||²𝑑𝑥≤𝜈^{1−1/(𝑘+1)}^𝑏𝑎||𝐷^𝑘+1𝐲_𝑖||²𝑑𝑥^{𝑘/(𝑘+1)}.(2.2) For 𝑝=𝑘+1, using integration by parts, the Schwarz inequality and the result when 𝑝=𝑘, one can give ^𝑏𝑎||𝐷^𝑘+1𝐲_𝑖||²𝑑𝑥≤≤||||^𝑏𝑎||𝐷^𝑘+1𝐲_𝑖||²𝑑𝑥||||=||||^𝑏𝑎𝐷^𝑘𝐲_𝑖⋅𝐷^𝑘+2𝐲^𝑇𝑖𝑑𝑥_𝑏𝑎||||||𝐷^𝑘𝐲_𝑖||²𝑑𝑥^1/2^𝑏𝑎||𝐷^𝑘+2𝐲^𝑇𝑖||²𝑑𝑥^1/2≤𝜈^{1−1/(2(𝑘+1))}^𝑏𝑎||𝐷^𝑘+2𝐲_𝑖||²𝑑𝑥^1/2^𝑏𝑎||𝐷^𝑘+1𝐲_𝑖||²𝑑𝑥^{𝑘/(2(𝑘+1))}.(2.3) By further calculating, one can give ^𝑏𝑎||𝐷^𝑘+1𝐲_𝑖||²𝑑𝑥≤𝜈^{1−1/((𝑘+1)+1)}^𝑏𝑎||𝐷^(𝑘+1)+1𝐲_𝑖||²𝑑𝑥^{(𝑘+1)/((𝑘+1)+1)}.(2.4) Therefore, when 𝑝=𝑘+1, (1) is true.
(2) Using (1) and the inductive method, we have ^𝑏𝑎||𝐷^𝑝𝐲_𝑖||²𝑑𝑥≤𝜈^{1−1/(𝑝+1)}^𝑏𝑎||𝐷^𝑝+1𝐲_𝑖||²𝑑𝑥^{𝑝/(𝑝+1)}≤𝜈^{1−2/(𝑝+2)}^𝑏𝑎||𝐷^𝑝+2𝐲_𝑖||²𝑑𝑥^{𝑝/(𝑝+2)}≤⋯≤𝜈^{1−(1−(𝑝/𝑡))}^𝑏𝑎||𝐷^𝑡𝐲_𝑖||²𝑑𝑥^𝑝/𝑡.(2.5) From (1.8) and (2.5), we get ^𝑏𝑎||𝐷^𝑝𝐲_𝑖||²𝑑𝑥≤𝜈^{1−(1−(𝑝/𝑡))}^𝑏𝑎||𝐷^𝑡𝐲_𝑖||²𝑑𝑥^𝑝/𝑡≤𝜈^{1−(1−(𝑝/𝑡))}𝜇𝜆^𝑖₁^𝑝/𝑡,𝑝=1,2,…,𝑡.(2.6) Taking 𝑝=1, we have ^𝑏𝑎||𝐷𝐲_𝑖||²𝑑𝑥≤𝜈^{1−(1−(1/𝑡))}𝜇𝜆^𝑖₁^1/𝑡.(2.7) So Lemma 2.1 is true.

Lemma 2.2. Let 𝜆₁,𝜆₂,…,𝜆_𝑛 be the eigenvalues of (1.4)-(1.5). Then one has 𝐼+𝐽≤𝑡(2𝑡−1)𝜇^{1−(1−(1/𝑡))}𝜈^1−1/𝑡𝜇_2𝑛_𝑖=1𝜆^{𝑖1−(1/𝑡)}.(2.8)

Proof. Since 𝐼_𝑖==((−−11))^𝑡_𝑡𝑡^𝑏𝑎𝚽_𝑖𝐷^𝑡𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥𝑡^𝑏𝑎𝑥𝐲^𝑖−^𝑛_𝑗=1𝑏_𝑖𝑗𝐲_𝑗𝐷^𝑡=(−1)_𝑡𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥−(−1𝑡)_𝑡^𝑏𝑎𝑥𝐲_𝑖𝐷^𝑡𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥𝑡^𝑛_𝑗=1𝑏_𝑖𝑗^𝑏𝑎=𝑡₂𝐲_𝑗𝐷^𝑡𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥^𝑏𝑎−𝑡_𝑛𝐷^𝑡−1𝐲_𝑖𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥+𝑡^𝑏𝑎𝑥𝐷^𝑡𝐲_𝑖𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥_𝑗=1𝑏_𝑖𝑗^𝑏𝑎𝐽𝐷^𝑡𝐲_𝑗𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥,(2.9)_𝑖==−(−𝑡1(𝑡)^𝑡−𝑡1^𝑏𝑎)𝚽_𝑖𝑏𝑎𝐷^𝑡−1𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥+𝑡_𝑛𝐷^𝑡−2𝐲_𝑖𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥−𝑡^𝑏𝑎𝑥𝐷^𝑡−1𝐲_𝑖𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥_𝑗=1𝑏_𝑖𝑗^𝑏𝑎𝐷^𝑡−1𝐲_𝑗𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥,(2.10) we have 𝐼+𝐽=^𝑛_𝑖=1𝐼_𝑖=_𝑛+𝐽_𝑖_𝑖=1−𝑡𝑡_𝑛^𝑏𝑎𝑡𝐷^𝑡−1𝐲_𝑖𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖−(𝑡−1)𝐷^𝑡−2𝐲_𝑖𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥_𝑖_,𝑗=1𝑏_𝑖𝑗^𝑏𝑎𝐷^𝑡𝐲_𝑗𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖−𝐷^𝑡−1𝐲_𝑗𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥.(2.11) By 𝑎_𝑖𝑗(𝑥)=𝑎_𝑗𝑖(𝑥), the last term of (2.11) is zero. Then we can get 𝐼+𝐽=^𝑛_𝑖=1𝑡^𝑏𝑎𝑡𝐷^𝑡−1𝐲_𝑖𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖−(𝑡−1)𝐷^𝑡−2𝐲_𝑖𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥.(2.12) Using (2°), Lemma 2.1, (1) and (2.6), we have ^𝑏𝑎𝐷^𝑡−1𝐲_𝑖𝐀(𝑥)𝐷^𝑡−1𝐲^𝑇𝑖𝑑𝑥≤𝜇₂^𝑏𝑎||𝐷^𝑡−1𝐲_𝑖||²𝑑𝑥≤𝜇₂𝜈^1−1/𝑡𝜇𝜆^𝑖₁^1−(1/𝑡).(2.13) Using (2°), the Schwarz inequality, Lemma 2.1  (1), and (2.6), one can give ||||−^𝑏𝑎𝐷^𝑡−2𝐲_𝑖𝐀(𝑥)𝐷^𝑡𝐲^𝑇𝑖𝑑𝑥||||≤𝜇₂^𝑏𝑎||𝐷^𝑡−2𝐲_𝑖||²𝑑𝑥^1/2^𝑏𝑎||𝐷^𝑡𝐲^𝑇𝑖||²𝑑𝑥^1/2≤𝜇₂𝜈^1−1/𝑡𝜇𝜆^𝑖₁^1−(1/𝑡).(2.14) Therefore, we obtain 𝐼+𝐽≤𝑡(2𝑡−1)𝜇^{1−(1−(1/𝑡))}𝜈^1−1/𝑡𝜇_2𝑛_𝑖=1𝜆^{𝑖1−(1/𝑡)}.(2.15)

Lemma 2.3. If Φ_𝑖 and 𝜆_𝑖(𝑖=1,2,…,𝑛) as above, then one has ^𝑛_𝑖=1^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥≥𝜇^11/𝑡𝜈^{12−(1/𝑡)}𝑛²4𝜈²²^𝑛_𝑖=1𝜆^𝑖1/𝑡⁻¹.(2.16)

Proof. By the definition of Φ_𝑖, one has ^𝑛_𝑖=1^𝑏𝑎𝚽_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥=^𝑛_𝑖=1^𝑏𝑎𝑥𝐲_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥−^𝑛_𝑖_,𝑗=1𝑏_𝑖𝑗^𝑏𝑎𝐲_𝑗𝐷𝐲^𝑇𝑖𝑑𝑥.(2.17) Using 𝑏_𝑖𝑗=𝑏_𝑗𝑖 and ∫^𝑏𝑎𝐲_𝑗𝐷𝐲^𝑇𝑖𝑑𝑥=−∫^𝑏𝑎𝐲_𝑖𝐷𝐲^𝑇𝑗𝑑𝑥, it is easy to see that the last term of (2.17) is zero. Then we have ^𝑛_𝑖=1^𝑏𝑎𝚽_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥=^𝑛_𝑖=1^𝑏𝑎𝑥𝐲_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥.(2.18) Using integration by parts, one can give ^𝑏𝑎𝑥𝐲_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥=−^𝑏𝑎||𝐲_𝑖||²𝑑𝑥−^𝑏𝑎𝑥𝐲_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥,(2.19)^𝑏𝑎𝑥𝐲_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥=−12^𝑏𝑎||𝐲_𝑖||²𝑑𝑥.(2.20) By 1/𝜈₂≤∫^𝑏𝑎|𝐲_𝑖|²𝑑𝑥≤1/𝜈₁, we have ||||^𝑏𝑎𝑥𝐲_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥||||=12^𝑏𝑎||𝐲_𝑖||²𝑑𝑥≥12𝜈₂.(2.21) From (2.18) and (2.21), we can get ^𝑛_𝑖=1||||^𝑏𝑎𝚽_𝑖𝐷𝐲^𝑇𝑖𝑑𝑥||||≥𝑛2𝜈₂.(2.22) Using the Schwarz inequality, Lemma 2.1  (2), and (3°), we have 𝑛²4𝜈²²≤^𝑛_𝑖=1^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥^𝑛_𝑖=1^𝑏𝑎||𝐷𝐲_𝑖||²≤𝑠(𝑥)𝑑𝑥_𝑛_𝑖=1^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥𝜈^{1−(2−(1/𝑡))}𝜇^{1𝑛−1/𝑡}_𝑖=1𝜆^𝑖1/𝑡.(2.23) By further calculating, we can easily get Lemma 2.3.

3. Main Results

Theorem 3.1. If 𝜆_𝑖(𝑖=1,2,…,𝑛+1) are the eigenvalues of (1.4)-(1.5), then (1)𝜆_𝑛+1≤𝜆_𝑛+4𝑡(2𝑡−1)𝜇²𝜈²²𝜇₁𝜈²¹𝑛_2𝑛𝑖=1𝜆^{𝑖𝑛1−(1/𝑡)}_𝑖=1(2)𝜆_𝑛+1𝜆^𝑖1/𝑡;(3.1)≤1+4𝑡(2𝑡−1)𝜇²𝜈²²𝜇₁𝜈²¹𝜆^𝑛.(3.2)

Proof. From (1.18), we can get 𝜆_𝑛+1−𝜆_𝑛≤(𝐼+𝐽)^𝑛_𝑖=1^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥⁻¹.(3.3) Using Lemmas 2.2 and 2.3, we can easily get (3.1). In (3.1), Replacing 𝜆_𝑖 with 𝜆_𝑛, by further calculating, we can get (3.2).

Theorem 3.2. For 𝑛≥1, one has ^𝑛_𝑖=1𝜆^𝑖1/𝑡𝜆_𝑛+1−𝜆_𝑖≥𝜇¹𝜈²¹𝑛²4𝑡(2𝑡−1)𝜇₂𝜈²²^𝑛_𝑖=1𝜆^{𝑖1−(1/𝑡)}⁻¹.(3.4)

Proof. Choosing the parameter 𝜎>𝜆_𝑛, using (1.17), one can give 𝜆_𝑛𝑛+1_𝑖=1^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥≤𝜎^𝑛_𝑖=1^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥+^𝑛_𝑖=1^𝑏𝑎𝜆_𝑖−𝜎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥+𝐼+𝐽.(3.5) By (2.22) and the Young inequality, we obtain 𝑛2𝜈₂≤𝛿2^𝑛_𝑖=1𝜎−𝜆_𝑖^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥+12𝛿^𝑛_𝑖=1𝜎−𝜆_𝑖⁻¹^𝑏𝑎||𝐷𝐲_𝑖||²𝑠(𝑥)𝑑𝑥,(3.6) where 𝛿>0 is a constant to be determined. Set 𝑉=^𝑛_𝑖=1^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥,𝑇=^𝑛_𝑖=1𝜎−𝜆_𝑖^𝑏𝑎𝑠(𝑥)||𝚽_𝑖||²𝑑𝑥.(3.7) Using Lemma 2.1, (3.5), and (3.6), we can get the following results, respectively, 𝑛𝜆_𝑛+1−𝜎𝑉+𝑇≤𝐼+𝐽,(3.8)𝜈₂≤𝛿𝑇+1𝛿𝜇^1−1/𝑡𝜈^{1𝑛−(2−(1/𝑡))}_𝑖=1𝜎−𝜆_𝑖⁻¹𝜆^𝑖1/𝑡.(3.9) In order to get the minimum of the right of (3.9), we can take 𝛿=𝑇^−1/2𝜇^1−1/𝑡𝜈^{1𝑛−(2−(1/𝑡))}_𝑖=1𝜎−𝜆_𝑖⁻¹𝜆^𝑖1/𝑡^1/2.(3.10) By (3.9), and (3.10), we can easily get 𝑇≥𝜇^11/𝑡𝜈^{12−(1/𝑡)}𝑛²4𝜈²²^𝑛_𝑖=1𝜆^𝑖1/𝑡𝜎−𝜆_𝑖⁻¹.(3.11) Using Lemma 2.2, (3.8), and (3.11), we have 𝜆_𝑛+1−𝜎𝑉+𝜇^11/𝑡𝜈^{12−(1/𝑡)}𝑛²4𝜈²²^𝑛_𝑖=1𝜆^𝑖1/𝑡𝜎−𝜆_𝑖⁻¹≤𝑡(2𝑡−1)𝜇^{1−(1−(1/𝑡))}𝜈^1−1/𝑡𝜇_2𝑛_𝑖=1𝜆^{𝑖1−(1/𝑡)},(3.12) that is, 𝜆_𝑛+1−𝜎𝑉≤𝑡(2𝑡−1)𝜇^{1−(1−(1/𝑡))}𝜈^1−1/𝑡𝜇^2𝑛_𝑖=1𝜆^{𝑖1−(1/𝑡)}−𝜇^11/𝑡𝜈^{12−(1/𝑡)}𝑛²4𝜈²²^𝑛_𝑖=1𝜆^𝑖1/𝑡𝜎−𝜆_𝑖⁻¹.(3.13) Let the right term of (3.13) be 𝑓(𝜎). It is easy to see that lim_{𝜎→𝜆+𝑛}lim_𝜎→+∞𝑓(𝜎)=−∞,𝑓(𝜎)=𝑡(2𝑡−1)𝜇^{1−(1−(1/𝑡))}𝜈^1−1/𝑡𝜇_2𝑛_𝑖=1𝜆^{𝑖1−(1/𝑡)}>0.(3.14) Hence, there is 𝜎₀∈(𝜆_𝑛,+∞), such that ^𝑛_𝑖=1𝜆^𝑖1/𝑡𝜎₀−𝜆_𝑖=𝜇¹𝜈²¹𝑛²4𝑡(2𝑡−1)𝜇₂𝜈²²^𝑛_𝑖=1𝜆^{𝑖1−(1/𝑡)}⁻¹.(3.15) On the other hand, letting 𝑔(𝜎)=^𝑛_𝑖=1𝜆^𝑖1/𝑡𝜎−𝜆_𝑖,(3.16) we have 𝑔^(𝜎)=−^𝑛_𝑖=1𝜆^𝑖1/𝑡𝜎−𝜆_𝑖²≤0.(3.17) It implies that 𝑔(𝜎) 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>𝜆_𝑛+1. Replacing 𝜎₀ with 𝜆_𝑛+1 in (3.15), we can get the result.