Discrete Dynamics in Nature and SocietyVolume 2012 (2012), Article ID 860598, 16 pagesdoi:10.1155/2012/860598

Research Article

Lyapunov-Type Inequalities for the Quasilinear Difference Systems

Qi-Ming Zhang¹ and X. H. Tang²

¹College of Science, Hunan University of Technology, Zhuzhou, Hunan 412000, China²School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received 22 October 2011; Accepted 4 December 2011

Academic Editor: Zengji Du

Copyright © 2012 Qi-Ming Zhang and X. H. Tang. 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.

We establish several Lyapunov-type inequalities for quasilinear difference systems, which generalize or improve all related existing ones. Applying these results, we also obtain some lower bounds for the first eigencurve in the generalized spectra.

1. Introduction

In 1964, Atkinson [1] investigated the following boundary value problem: with Dirichlet boundary condition: and he proved that boundary value problem (1.1) with (1.2) has exactly real and simple eigenvalues, which can be arranged in the increasing order where with , , and for all . Here and in the sequel, .

In 1983, Cheng [2] proved that if the second-order difference equation has a real solution such that then one has the following inequality where for all , and and the constant 4 in (1.6) cannot be replaced by a larger number. Inequality (1.6) is a discrete analogy of the following so-called Lyapunov inequality: if Hill's equation has a real solution such that where is a real-valued continuous function defined on , with . Equation (1.8) was first established by Liapounoff [3] in 1907.

In 2008, Ünal et al. [4] established the following Lyapunov-type inequality: if the following second-order half-linear difference equation: has a solution satisfying where and in the sequel .

Applying inequality (1.11) to (1.4) (i.e., (1.12) with , and ), we can obtain the following Lyapunov-type inequality: which was also obtained in [5]. When is odd, (1.14) is the same as (1.6). However, (1.14) is worse than (1.6) when is even. For more discrete cases and continuous cases for Lyapunov-type inequalities, we refer the reader to [5–18].

For a single -Laplacian equation (1.12), there are many papers which deal with various dynamics behavior of its solutions in the literatures. However, we are not aware of similar works for -Laplacian systems. We consider here the following quasilinear difference system of resonant type and the quasilinear difference system involving the -Laplacian

For the sake of convenience, we give the following hypotheses (H1) and (H2) for system (1.15) and hypothesis (H3) for system (1.16):(H1) and are real-valued functions and and for all ;(H2), satisfy and ;(H3) and are real-valued functions and for . Furthermore, and satisfy .

System (1.15) and (1.16) are the discrete analogies of the following two quasilinear differential systems: respectively. Recently, Nápoli and Pinasco [19], Cakmak and Tiryaki [20, 21], and Tang and He [22] established some Lyapunov-type inequalities for systems (1.17) and (1.18). Motivated by the above-mentioned papers, the purpose of this paper is to establish some Lyapunov-type inequalities for systems (1.15) and (1.16). As a byproduct, we derive a better Lyapunov-type inequality than (1.11) for the second-order half-linear difference equation (1.12). In particular, (1.19) produces a new Lyapunov-type inequality for Hill's equation (1.4) when and . It is easy to see that (1.20) is better than (1.6).

This paper is organized as follows. Section 2 gives some Lyapunov-type inequalities for system (1.15), and Lyapunov-type inequalities for system (1.16) are established in Section 3. In Section 4, we apply our Lyapunov-type inequalities to obtain lower bounds for the first eigencurve in the generalized spectra.

2. Lyapunov-Type Inequalities for System (1.15)

In this section, we establish some Lyapunov-type inequalities for system (1.15).

Denote

Theorem 2.1. Let with . Suppose that hypotheses (H1) and (H2) are satisfied. If system (1.15) has a solution satisfying the boundary value conditions: then one has the following inequality: where and in the sequel for .

Proof. By (1.15) and (2.3), we obtain It follows from (2.1), (2.3), and the Hölder inequality that From (2.7) and (2.8), we have Now, it follows from (2.3), (2.5), (2.9), (H2), and the Hölder inequality that where Similar to the proof of (2.9), from (2.2) and (2.3), we have It follows from (2.3), (2.6), (2.13), (H2), and the Hölder inequality that where Next, we prove that If (2.16) is not true, then From (2.5) and (2.17), we have It follows from (H1) that Combining (2.7) with (2.19), we obtain that for , which contradicts (2.3). Therefore, (2.16) holds. Similarly, we have From (2.10), (2.11), (2.14), (2.16), (2.20), and (H2), we have It follows from (2.12), (2.15), and (2.21) that (2.4) holds.

Corollary 2.2. Let with . Suppose that hypothesis (H1) and (H2) are satisfied. If system (1.15) has a solution satisfying (2.3), then one has the following inequality:

Proof. Since it follows from (2.4) and (H2) that (2.22) holds.

Corollary 2.3. Let with . Suppose that hypotheses (H1) and (H2) are satisfied. If system (1.15) has a solution satisfying (2.3), then one has the following inequality:

Proof. Since it follows from (2.22) and (H2) that (2.24) holds.

When , , , and , system (1.15) reduces to the second-order half-linear difference equation (1.12). Hence, we can directly derive the following Lyapunov-type inequality for (1.12) from (2.10) and (2.16).

Theorem 2.4. Let with . Suppose that and . If (1.12) has a solution satisfying (1.13), then one has the following inequality: Since it follows from Theorem 2.4 that the following corollary holds.

Corollary 2.5. Let with . Suppose that and . If (1.12) has a solution satisfying (1.13), then one has the following inequality:

Remark 2.6. It is easy to see that Lyapunov-type inequalities (2.26) and (2.28) are better than (1.11).

3. Lyapunov-Type Inequalities for System (1.16)

In this section, we establish some Lyapunov-type inequalities for system (1.16). Denote

Theorem 3.1. Let with . Suppose that hypothesis (H3) is satisfied. If system (1.16) has a solution satisfying the boundary value conditions: then one has the following inequality:

Proof. By (1.16), (H3), and (3.3), we obtain It follows from (3.1), (3.3), and the Hölder inequality that Similarly, it follows from (3.2), (3.3), and the Hölder inequality that From (3.6) and (3.7), we have Now, it follows from (3.3), (3.5), (3.8), (H3), and the generalized Hölder inequality that where Next, we prove that If (3.11) is not true, then there exists such that From (3.5), (3.12), and the generalized Hölder inequality, we have It follows from the fact that that Combining (3.6) with (3.14), we obtain that for , which contradicts (3.3). Therefore, (3.11) holds. From (3.9), (3.11), and (H3), we have It follows from (3.10) and (3.15) that (3.4) holds.

Corollary 3.2. Let with . Suppose that hypothesis (H3) is satisfied. If system (1.16) has a solution satisfying (3.3), then one has the following inequality:

Proof. Since it follows from (3.4) and (H3) that (3.16) holds.

Corollary 3.3. Let with . Suppose that hypothesis (H3) is satisfied. If system (1.16) has a solution satisfying (3.3), then one has the following inequality where .

Proof. Since it follows from (3.16) and (H3) that (3.18) holds.

4. Some Applications

In this section, we apply our Lyapunov-type inequalities to obtain lower bounds for the first eigencurve in the generalized spectra.

Let with . We consider here a quasilinear difference system of the form: where , , and are the same as those in (1.16), and satisfies Dirichlet boundary conditions:

We define the generalized spectrum S of a nonlinear difference system as the set of vector such that the eigenvalue problem (4.1) with (4.2) admits a nontrivial solution.

Eigenvalue problem or boundary value problem (4.1) with (4.2) is a generalization of the following -Laplacian difference equation with Dirichlet boundary condition: where , , and . When , Atkinson [1, Theorems 4.3.1 and 4.3.5] investigated the existence of eigenvalues for (4.3) with (4.4), see also [23].

Let and for . Then we can apply Theorem 3.1 to boundary value problem (4.1) with (4.2) and obtain a lower bound for the first eigencurve in the generalized spectra.

Theorem 4.1. Let with . Assume that , satisfy , and that for all . Then there exists a function such that for every generalized eigenvalue , of boundary value problem (4.1) with (4.2), where is given by:

Proof. For the eigenvalue , (4.1) with (4.2) has a nontrivial solution . That is system (1.16) with and has a solution satisfying (3.3), it follows from (3.4) that for all, , and that Hence, we have This completes the proof of Theorem 4.1.

When , boundary value problem (4.1) with (4.2) reduces to the simpler form: with Dirichlet boundary conditions: where , satisfy , and for all .

Applying Theorem 4.1 to system (4.8) with (4.9) and system (4.3) with (4.4), respectively, we have the following two corollaries immediately.

Corollary 4.2. Let with . Assume that , satisfy , and that for all . Then there exists a function such that for every generalized eigenvalue of system (4.8) with (4.9), where is given by: where denote and denote .

Corollary 4.3. Let with . Assume that and for all . Then for every eigenvalue of system (4.3) with (4.4), one has

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