Lyapunov-Type Inequalities for the Quasilinear Difference Systems

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:Δ(𝑟(𝑛)Δ𝑢(𝑛))=𝜆𝑞(𝑛)𝑢(𝑛+1) (1.1) with Dirichlet boundary condition: 𝑢(𝑎)=𝑢(𝑏)=0,𝑢(𝑛)≢0,∀𝑛∈ℤ[𝑎,𝑏], (1.2) and he proved that boundary value problem (1.1) with (1.2) has exactly 𝑏−𝑎−1 real and simple eigenvalues, which can be arranged in the increasing order𝜆₁<𝜆₂<⋯<𝜆_{𝑏−𝑎−1}, (1.3) where 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2, 𝜆∈ℝ, 𝑟(𝑛)>0 and 𝑞(𝑛)>0 for all 𝑛∈ℤ. Here and in the sequel, ℤ[𝑎,𝑏]={𝑎,𝑎+1,𝑎+2,…,𝑏−1,𝑏}.

In 1983, Cheng [2] proved that if the second-order difference equationΔ²𝑢(𝑛)+𝑞(𝑛)𝑢(𝑛+1)=0 (1.4) has a real solution 𝑢(𝑛) such that𝑢(𝑎)=𝑢(𝑏)=0,𝑢(𝑛)≢0,𝑛∈ℤ[𝑎,𝑏], (1.5) then one has the following inequality𝔉(𝑏−𝑎)^𝑏−2_𝑛=𝑎𝑞(𝑛)≥4, (1.6) where 𝑞(𝑛)≥0 for all 𝑛∈ℤ, and𝔉(𝑚)=𝑚²−1𝑚,if𝑚−1iseven,𝑚,if𝑚−1isodd, (1.7) 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:(𝑏−𝑎)^𝑏_𝑎𝑞(𝑡)𝑑𝑡>4, (1.8) if Hill's equation𝑢^′′(𝑡)+𝑞(𝑡)𝑢(𝑡)=0 (1.9) has a real solution 𝑢(𝑡) such that𝑢(𝑎)=𝑢(𝑏)=0,𝑢(𝑡)≠0,∀𝑡∈(𝑎,𝑏), (1.10) where 𝑞(𝑡) is a real-valued continuous function defined on ℝ, 𝑎,𝑏∈ℝ with 𝑎<𝑏. Equation (1.8) was first established by Liapounoff [3] in 1907.

In 2008, Ünal et al. [4] established the following Lyapunov-type inequality:^𝑏−1_𝑛=𝑎1[𝑟(𝑛)]^{1/(𝑝−1)1−(1/𝑝)𝑏−2}_𝑛=𝑎𝑞⁺(𝑛)^1/𝑝≥2, (1.11) if the following second-order half-linear difference equation:Δ𝑟(𝑛)|Δ𝑢(𝑛)|^𝑝−2Δ𝑢(𝑛)+𝑞(𝑛)|𝑢(𝑛+1)|^𝑝−2𝑢(𝑛+1)=0 (1.12) has a solution 𝑢(𝑛) satisfying𝑢(𝑎)=𝑢(𝑏)=0,𝑢(𝑛)≢0,∀𝑛∈ℤ[𝑎,𝑏], (1.13) where and in the sequel 𝑞⁺(𝑛)=max{𝑞(𝑛),0}.

Applying inequality (1.11) to (1.4) (i.e., (1.12) with 𝑝=2,𝑟(𝑛)=1, and 𝑞(𝑛)≥0), we can obtain the following Lyapunov-type inequality:(𝑏−𝑎)^𝑏−2_𝑛=𝑎𝑞(𝑛)≥4, (1.14) which was also obtained in [5]. When 𝑏−𝑎−1 is odd, (1.14) is the same as (1.6). However, (1.14) is worse than (1.6) when 𝑏−𝑎−1 is even. For more discrete cases and continuous cases for Lyapunov-type inequalities, we refer the reader to [5–18].

For a single 𝑝-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 𝑝-Laplacian systems. We consider here the following quasilinear difference system of resonant type−Δ𝑟₁(𝑛)|Δ𝑢(𝑛)|^𝑝1−2Δ𝑢(𝑛)=𝑓₁(𝑛)|𝑢(𝑛+1)|^𝛼1−2|𝑣(𝑛+1)|^𝛼2𝑢(),𝑟𝑛+1−Δ₂(𝑛)|Δ𝑣(𝑛)|^𝑝2−2Δ𝑣(𝑛)=𝑓₂(𝑛)|𝑢(𝑛+1)|^𝛽1|𝑣(𝑛+1)|^𝛽2−2𝑣(𝑛+1), (1.15) and the quasilinear difference system involving the (𝑝₁,𝑝₂,…,𝑝_𝑛)-Laplacian −Δ𝑟₁(𝑛)Δ𝑢₁(𝑛)^𝑝1−2Δ𝑢₁(𝑛)=𝑓₁(𝑛)𝑢₁(𝑛+1)^𝛼1−2𝑢₂(𝑛+1)^𝛼2⋯𝑢_𝑚(𝑛+1)^𝛼𝑚𝑢₁(),𝑟𝑛+1−Δ₂(𝑛)Δ𝑢₂(𝑛)^𝑝2−2Δ𝑢₂(𝑛)=𝑓₂(𝑛)𝑢₁(𝑛+1)^𝛼1𝑢₂(𝑛+1)^𝛼2−2⋯𝑢_𝑚(𝑛+1)^𝛼𝑚𝑢₂(),⋮𝑟𝑛+1−Δ_𝑚(𝑛)Δ𝑢_𝑚(𝑛)^𝑝𝑚−2Δ𝑢_𝑚(𝑛)=𝑓_𝑚(𝑛)𝑢₁(𝑛+1)^𝛼1𝑢₂(𝑛+1)^𝛼2⋯𝑢_𝑚(𝑛+1)^𝛼𝑚−2𝑢_𝑚(𝑛+1). (1.16)

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)𝑟₁(𝑛),𝑟₂(𝑛),𝑓₁(𝑛) and 𝑓₂(𝑛) are real-valued functions and 𝑟₁(𝑛)>0 and 𝑟₂(𝑛)>0 for all 𝑛∈ℤ;(H2)1<𝑝₁,𝑝₂<∞, 𝛼₁,𝛼₂,𝛽₁,𝛽₂>0 satisfy 𝛼₁/𝑝₁+𝛼₂/𝑝₂=1 and 𝛽₁/𝑝₁+𝛽₂/𝑝₂=1;(H3)𝑟_𝑖(𝑛) and 𝑓_𝑖(𝑛) are real-valued functions and 𝑟_𝑖(𝑛)>0 for 𝑖=1,2,…,𝑚. Furthermore, 1<𝑝_𝑖<∞ and 𝛼_𝑖>0 satisfy ∑^𝑚_𝑖=1(𝛼_𝑖/𝑝_𝑖)=1.

System (1.15) and (1.16) are the discrete analogies of the following two quasilinear differential systems:−𝑟₁(𝑡)𝑢^′(𝑡)^𝑝−2𝑢^′(𝑡)^′=𝑓₁(𝑡)|𝑢(𝑡)|^𝛼1−2|𝑣(𝑡)|^𝛼2𝑢(𝑡),−𝑟₂(𝑡)𝑣^′(𝑡)^𝑞−2𝑣^′(𝑡)^′=𝑓₂(𝑡)|𝑢(𝑡)|^𝛽1|𝑣(𝑡)|^𝛽2−2𝑣(𝑡),−𝑟₁(𝑡)𝑢^′₁(𝑡)^𝑝1−2𝑢^′₁(𝑡)^′=𝑓₁(𝑡)𝑢₁(𝑡)^𝛼1−2𝑢₂(𝑡)^𝛼2⋯𝑢_𝑛(𝑡)^𝛼𝑛𝑢₁(𝑡),−𝑟₂(𝑡)𝑢^′₂(𝑡)^𝑝2−2𝑢^′₂(𝑡)^′=𝑓₂(𝑡)𝑢₁(𝑡)^𝛼1𝑢₂(𝑡)^𝛼2−2⋯𝑢_𝑛(𝑡)^𝛼𝑛𝑢₂(𝑡),⋮−𝑟_𝑛(𝑡)𝑢^′_𝑛(𝑡)^𝑝𝑛−2𝑢^′_𝑛(𝑡)^′=𝑓_𝑛(𝑡)𝑢₁(𝑡)^𝛼1𝑢₂(𝑡)^𝛼2⋯𝑢_𝑛(𝑡)^𝛼𝑛−2𝑢_𝑛(𝑡), (1.17)(1.18) 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)^𝑏−2_𝑛=𝑎∑^𝑛_𝑠=𝑎[𝑟(𝑠)]^{1/(1−𝑝)𝑝−1}∑^𝑏−1_𝑠=𝑛+1[𝑟(𝑠)]^{1/(1−𝑝)𝑝−1}∑^𝑛_𝑠=𝑎[𝑟(𝑠)]^{1/(1−𝑝)𝑝−1}+∑^𝑏−1_𝑠=𝑛+1[𝑟(𝑠)]^{1/(1−𝑝)𝑝−1}𝑞⁺(𝑛)≥1 (1.19) for the second-order half-linear difference equation (1.12). In particular, (1.19) produces a new Lyapunov-type inequality1𝑏−𝑎^𝑏−2_𝑛=𝑎(𝑛+1−𝑎)(𝑏−𝑛−1)𝑞⁺(𝑛)≥1 (1.20) for Hill's equation (1.4) when 𝑝=2 and 𝑟(𝑡)=1. 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).

Denote𝜁₁(𝑛)∶=^𝑛_𝜏=𝑎[𝑟₁(𝜏)]^{1/(1−𝑝1)𝑝1−1},𝜂₁(𝑛)∶=^𝑏−1_𝜏=𝑛+1𝑟₁(𝜏)^{1/(1−𝑝1)𝑝1−1},𝜁₂(𝑛)∶=^𝑛_𝜏=𝑎𝑟₂(𝜏)^{1/(1−𝑝2)𝑝2−1},𝜂₂(𝑛)∶=^𝑏−1_𝜏=𝑛+1𝑟₂(𝜏)^{1/(1−𝑝2)𝑝2−1}. (2.1)(2.2)

Theorem 2.1. Let 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Suppose that hypotheses (H1) and (H2) are satisfied. If system (1.15) has a solution (𝑢(𝑛),𝑣(𝑛)) satisfying the boundary value conditions: 𝑢(𝑎)=𝑢(𝑏)=0=𝑣(𝑎)=𝑣(𝑏),𝑢(𝑛)≢0,𝑣(𝑛)≢0,∀𝑛∈ℤ[𝑎,𝑏], (2.3) then one has the following inequality: ^𝑏−2_𝑛=𝑎𝜁₁(𝑛)𝜂₁(𝑛)𝜁₁(𝑛)+𝜂₁(𝑛)𝑓⁺₁(𝑛)^{𝛼1𝛽1/𝑝21𝑏−2}_𝑛=𝑎𝜁₁(𝑛)𝜂₁(𝑛)𝜁₁(𝑛)+𝜂₁(𝑛)𝑓⁺₂(𝑛)^{𝛽1𝛼2/𝑝1𝑝2}×^𝑏−2_𝑛=𝑎𝜁₂(𝑛)𝜂₂(𝑛)𝜁₂(𝑛)+𝜂₂(𝑛)𝑓⁺₁(𝑛)^{𝛽1𝛼2/𝑝1𝑝2𝑏−2}_𝑛=𝑎𝜁₂(𝑛)𝜂₂(𝑛)𝜁₂(𝑛)+𝜂₂(𝑛)𝑓⁺₂(𝑛)^{𝛼2𝛽2/𝑝22}≥1, (2.4) where and in the sequel 𝑓⁺_𝑖(𝑛)=max{𝑓_𝑖(𝑛),0} for 𝑖=1,2.

Proof. By (1.15) and (2.3), we obtain ^𝑏−1_𝑛=𝑎𝑟₁(𝑛)|Δ𝑢(𝑛)|^𝑝1=^𝑏−2_𝑛=𝑎𝑓₁(𝑛)|𝑢(𝑛+1)|^𝛼1|𝑣(𝑛+1)|^𝛼2,_{𝑏−1𝑛=𝑎}𝑟₂(𝑛)|Δ𝑣(𝑛)|^𝑝2=^𝑏−2_𝑛=𝑎𝑓₂(𝑛)|𝑢(𝑛+1)|^𝛽1|𝑣(𝑛+1)|^𝛽2. (2.5)(2.6) It follows from (2.1), (2.3), and the Hölder inequality that |𝑢(𝑛+1)|^𝑝1=^𝑛_𝜏=𝑎Δ𝑢(𝜏)^𝑝1≤^𝑛_𝜏=𝑎|Δ𝑢(𝜏)|^𝑝1≤^𝑛_𝜏=𝑎𝑟₁(𝜏)^{1/(1−𝑝1)𝑝1𝑛−1}_𝜏=𝑎𝑟₁(𝜏)|Δ𝑢(𝜏)|^𝑝1=𝜁₁(𝑛)^𝑛_𝜏=𝑎𝑟₁(𝜏)|Δ𝑢(𝜏)|^𝑝1|𝑢(,𝑎≤𝑛≤𝑏−1,𝑛+1)|^𝑝1=^𝑏−1_𝜏=𝑛+1Δ𝑢(𝜏)^𝑝1≤^𝑏−1_𝜏=𝑛+1|Δ𝑢(𝜏)|^𝑝1≤^𝑏−1_𝜏=𝑛+1𝑟₁(𝜏)^{1/(1−𝑝1)𝑝1−1𝑏−1}_𝜏=𝑛+1𝑟₁(𝜏)|Δ𝑢(𝜏)|^𝑝1=𝜂₁(𝑛)^𝑏−1_𝜏=𝑛+1𝑟₁(𝜏)|Δ𝑢(𝜏)|^𝑝1,𝑎≤𝑛≤𝑏−1. (2.7)(2.8) From (2.7) and (2.8), we have |𝑢(𝑛+1)|^𝑝1≤𝜁₁(𝑛)𝜂₁(𝑛)𝜁₁(𝑛)+𝜂₁(𝑛)^𝑏−1_𝜏=𝑎𝑟₁(𝜏)|Δ𝑢(𝜏)|^𝑝1,𝑎≤𝑛≤𝑏−1. (2.9) Now, it follows from (2.3), (2.5), (2.9), (H2), and the Hölder inequality that ^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑢(𝑛+1)|^𝑝1≤^𝑏−2_𝑛=𝑎𝜁₁(𝑛)𝜂₁(𝑛)𝜁₁(𝑛)+𝜂₁(𝑛)𝑓⁺₁(𝑛)^𝑏−1_𝑛=𝑎𝑟₁(𝑛)|Δ𝑢(𝑛)|^𝑝1=𝑀₁₁^𝑏−2_𝑛=𝑎𝑓₁(𝑛)|𝑢(𝑛+1)|^𝛼1|𝑣(𝑛+1)|^𝛼2≤𝑀₁₁^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑢(𝑛+1)|^𝛼1|𝑣(𝑛+1)|^𝛼2≤𝑀₁₁^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑢(𝑛+1)|^{𝑝1𝛼1/𝑝1𝑏−2}_𝑛=𝑎𝑓⁺₁(𝑛)|𝑣(𝑛+1)|^{𝑝2𝛼2/𝑝2}, (2.10)^𝑏−2_𝑛=𝑎𝑓⁺₂(𝑛)|𝑢(𝑛+1)|^𝑝1≤^𝑏−2_𝑛=𝑎𝜁₁(𝑛)𝜂₁(𝑛)𝜁₁(𝑛)+𝜂₁(𝑛)𝑓⁺₂(𝑛)^𝑏−1_𝑛=𝑎𝑟₁(𝑛)|Δ𝑢(𝑛)|^𝑝1=𝑀₁₂^𝑏−2_𝑛=𝑎𝑓₁(𝑛)|𝑢(𝑛+1)|^𝛼1|𝑣(𝑛+1)|^𝛼2≤𝑀₁₂^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑢(𝑛+1)|^𝛼1|𝑣(𝑛+1)|^𝛼2≤𝑀₁₂^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑢(𝑛+1)|^{𝑝1𝛼1/𝑝1𝑏−2}_𝑛=𝑎𝑓⁺₁(𝑛)|𝑣(𝑛+1)|^{𝑝2𝛼2/𝑝2}, (2.11) where 𝑀₁₁=^𝑏−2_𝑛=𝑎𝜁₁(𝑛)𝜂₁(𝑛)𝜁₁(𝑛)+𝜂₁(𝑛)𝑓⁺₁(𝑛),𝑀₁₂=^𝑏−2_𝑛=𝑎𝜁₁(𝑛)𝜂₁(𝑛)𝜁₁(𝑛)+𝜂₁(𝑛)𝑓⁺₂(𝑛). (2.12) Similar to the proof of (2.9), from (2.2) and (2.3), we have |𝑣(𝑛+1)|^𝑝2≤𝜁₂(𝑛)𝜂₂(𝑛)𝜁₂(𝑛)+𝜂₂(𝑛)^𝑏−1_𝜏=𝑎𝑟₂(𝜏)|Δ𝑣(𝜏)|^𝑝2,𝑎≤𝑛≤𝑏−1. (2.13) It follows from (2.3), (2.6), (2.13), (H2), and the Hölder inequality that ^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑣(𝑛+1)|^𝑝2≤^𝑏−2_𝑛=𝑎𝜁₂(𝑛)𝜂₂(𝑛)𝜁₂(𝑛)+𝜂₂(𝑛)𝑓⁺₁(𝑛)^𝑏−1_𝑛=𝑎𝑟₂(𝑛)|Δ𝑣(𝑛)|^𝑝2=𝑀₂₁^𝑏−2_𝑛=𝑎𝑓₂(𝑛)|𝑢(𝑛+1)|^𝛽1|𝑣(𝑛+1)|^𝛽2≤𝑀₂₁^𝑏−2_𝑛=𝑎𝑓⁺₂(𝑛)|𝑢(𝑛+1)|^𝛽1|𝑣(𝑛+1)|^𝛽2≤𝑀₂₁^𝑏−2_𝑛=𝑎𝑓⁺₂(𝑛)|𝑢(𝑛+1)|^{𝑝1𝛽1/𝑝1𝑏−2}_𝑛=𝑎𝑓⁺₂(𝑛)|𝑣(𝑛+1)|^{𝑝2𝛽2/𝑝2},_{𝑏−2𝑛=𝑎}𝑓⁺₂(𝑛)|𝑣(𝑛+1)|^𝑝2≤^𝑏−2_𝑛=𝑎𝜁₂(𝑛)𝜂₂(𝑛)𝜁₂(𝑛)+𝜂₂(𝑛)𝑓⁺₂(𝑛)^𝑏−1_𝑛=𝑎𝑟₂(𝑛)|Δ𝑣(𝑛)|^𝑝2=𝑀₂₂^𝑏−2_𝑛=𝑎𝑓₂(𝑛)|𝑢(𝑛+1)|^𝛽1|𝑣(𝑛+1)|^𝛽2≤𝑀₂₂^𝑏−2_𝑛=𝑎𝑓⁺₂(𝑛)|𝑢(𝑛+1)|^𝛽1|𝑣(𝑛+1)|^𝛽2≤𝑀₂₂^𝑏−2_𝑛=𝑎𝑓⁺₂(𝑛)|𝑢(𝑛+1)|^{𝑝1𝛽1/𝑝1𝑏−2}_𝑛=𝑎𝑓⁺₂(𝑛)|𝑣(𝑛+1)|^{𝑝2𝛽2/𝑝2}, (2.14) where 𝑀₂₁=^𝑏−2_𝑛=𝑎𝜁₂(𝑛)𝜂₂(𝑛)𝜁₂(𝑛)+𝜂₂(𝑛)𝑓⁺₁(𝑛),𝑀₂₂=^𝑏−2_𝑛=𝑎𝜁₂(𝑛)𝜂₂(𝑛)𝜁₂(𝑛)+𝜂₂(𝑛)𝑓⁺₂(𝑛). (2.15)
Next, we prove that ^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑢(𝑛+1)|^𝑝1>0. (2.16) If (2.16) is not true, then ^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑢(𝑛+1)|^𝑝1=0. (2.17) From (2.5) and (2.17), we have 0≤^𝑏−1_𝑛=𝑎𝑟₁(𝑛)|Δ𝑢(𝑛)|^𝑝1=^𝑏−2_𝑛=𝑎𝑓₁(𝑛)|𝑢(𝑛+1)|^𝛼1|𝑣(𝑛+1)|^𝛼2≤^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑢(𝑛+1)|^𝛼1|𝑣(𝑛+1)|^𝛼2=0. (2.18) It follows from (H1) that Δ𝑢(𝑛)≡0,𝑎≤𝑛≤𝑏−1. (2.19) Combining (2.7) with (2.19), we obtain that 𝑢(𝑛)≡0 for 𝑎≤𝑛≤𝑏, which contradicts (2.3). Therefore, (2.16) holds. Similarly, we have ^𝑏−2_𝑛=𝑎𝑓⁺₂(𝑛)|𝑢(𝑛+1)|^𝑝1>0,^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)|𝑣(𝑛+1)|^𝑝2>0,^𝑏−2_𝑛=𝑎𝑓⁺₂(𝑛)|𝑣(𝑛+1)|^𝑝2>0. (2.20) From (2.10), (2.11), (2.14), (2.16), (2.20), and (H2), we have 𝑀^{𝛼1𝛽1/𝑝21}₁₁𝑀^{𝛽1𝛼2/𝑝1𝑝2}₁₂𝑀^{𝛽1𝛼2/𝑝1𝑝2}₂₁𝑀^{𝛼2𝛽2/𝑝22}₂₂≥1. (2.21) It follows from (2.12), (2.15), and (2.21) that (2.4) holds.

Corollary 2.2. Let 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Suppose that hypothesis (H1) and (H2) are satisfied. If system (1.15) has a solution (𝑢(𝑛),𝑣(𝑛)) satisfying (2.3), then one has the following inequality: ^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)𝜁₁(𝑛)𝜂₁(𝑛)^{1/2𝛼1𝛽1/𝑝21𝑏−2}_𝑛=𝑎𝑓⁺₂(𝑛)𝜁₁(𝑛)𝜂₁(𝑛)^{1/2𝛽1𝛼2/𝑝1𝑝2}×^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)𝜁₂(𝑛)𝜂₂(𝑛)^{1/2𝛽1𝛼2/𝑝1𝑝2𝑏−2}_𝑛=𝑎𝑓⁺₂(𝑛)𝜁₂(𝑛)𝜂₂(𝑛)^{1/2𝛼2𝛽2/𝑝22}≥2^{(𝑝2𝛽1+𝑝1𝛼2)/𝑝1𝑝2}. (2.22)

Proof. Since 𝜁_𝑖(𝑛)+𝜂_𝑖(𝑛)≥2𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)^1/2,𝑖=1,2, (2.23) it follows from (2.4) and (H2) that (2.22) holds.

Corollary 2.3. Let 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Suppose that hypotheses (H1) and (H2) are satisfied. If system (1.15) has a solution (𝑢(𝑛),𝑣(𝑛)) satisfying (2.3), then one has the following inequality: ^𝑏−1_𝑛=𝑎𝑟₁(𝑛)^{1/(1−𝑝1)𝛽1(𝑝1−1)/𝑝1𝑏−1}_𝑛=𝑎𝑟₂(𝑛)^{1/(1−𝑝2)𝛼2(𝑝2−1)/𝑝2}×^𝑏−2_𝑛=𝑎𝑓⁺₁(𝑛)^{𝛽1/𝑝1𝑏−2}_𝑛=𝑎𝑓⁺₂(𝑛)^𝛼2/𝑝2≥2^𝛽1+𝛼2. (2.24)

Proof. Since 𝜁₁(𝑛)𝜂₁(𝑛)^1/2=^𝑛_𝜏=𝑎𝑟₁(𝜏)^{1/(1−𝑝1)𝑏−1}_𝜏=𝑛+1𝑟₁(𝜏)^{1/(1−𝑝1)(𝑝1−1)/2}≤12^{𝑝1−1𝑏−1}_𝜏=𝑎𝑟₁(𝜏)^{1/(1−𝑝1)𝑝1−1},𝜁₂(𝑛)𝜂₂(𝑛)^1/2=^𝑛_𝜏=𝑎𝑟₂(𝜏)^{1/(1−𝑝2)𝑏−1}_𝜏=𝑛+1𝑟₂(𝜏)^{1/(1−𝑝2)(𝑝2−1)/2}≤12^{𝑝2−1𝑏−1}_𝜏=𝑎𝑟₂(𝜏)^{1/(1−𝑝2)𝑝2−1}, (2.25) it follows from (2.22) and (H2) that (2.24) holds.

When 𝛼₁=𝛽₂=𝑝₁=𝑝₂=𝑝, 𝛼₂=𝛽₁=0, 𝑟₁(𝑡)=𝑟₂(𝑡)=𝑟(𝑡), and 𝑓₁(𝑡)=𝑓₂(𝑡)=𝑞(𝑡), 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 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Suppose that 𝑝>1 and 𝑟(𝑛)>0. If (1.12) has a solution 𝑢(𝑛) satisfying (1.13), then one has the following inequality: ^𝑏−2_𝑛=𝑎∑^𝑛_𝜏=𝑎[𝑟(𝜏)]^{1/(1−𝑝)𝑝−1}∑^𝑏−1_𝜏=𝑛+1[𝑟(𝜏)]^{1/(1−𝑝)𝑝−1}∑^𝑛_𝜏=𝑎[𝑟(𝜏)]^{1/(1−𝑝)𝑝−1}+∑^𝑏−1_𝜏=𝑛+1[𝑟(𝜏)]^{1/(1−𝑝)𝑝−1}𝑞⁺(𝑛)≥1. (2.26) Since ^𝑛_𝜏=𝑎[𝑟(𝜏)]^{1/(1−𝑝)𝑝−1}+^𝑏−1_𝜏=𝑛+1[𝑟(𝜏)]^{1/(1−𝑝)𝑝−1}≥2^𝑛_𝜏=𝑎[𝑟(𝜏)]^{1/(1−𝑝)𝑏−1}_𝜏=𝑛+1[𝑟(𝜏)]^{1/(1−𝑝)(𝑝−1)/2}, (2.27) it follows from Theorem 2.4 that the following corollary holds.

Corollary 2.5. Let 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Suppose that 𝑝>1 and 𝑟(𝑛)>0. If (1.12) has a solution 𝑢(𝑛) satisfying (1.13), then one has the following inequality: ^𝑏−2_𝑛=𝑎𝑞⁺(𝑛)^𝑛_𝜏=𝑎[𝑟(𝜏)]^{1/(1−𝑝)𝑏−1}_𝜏=𝑛+1[𝑟(𝜏)]^{1/(1−𝑝)(𝑝−1)/2}≥2. (2.28)

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). Denote𝜁_𝑖(𝑛)∶=^𝑛_𝜏=𝑎𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)𝑝𝑖−1}𝜂,𝑖=1,2,…,𝑚,_𝑖(𝑛)∶=^𝑏−1_𝜏=𝑛+1𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)𝑝𝑖−1},𝑖=1,2,…,𝑚. (3.1)(3.2)

Theorem 3.1. Let 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Suppose that hypothesis (H3) is satisfied. If system (1.16) has a solution (𝑢₁(𝑛),𝑢₂(𝑛),…,𝑢_𝑚(𝑛)) satisfying the boundary value conditions: 𝑢_𝑖(𝑎)=𝑢_𝑖(𝑏)=0,𝑢_𝑖(𝑛)≢0,∀𝑛∈ℤ[𝑎,𝑏],𝑖=1,2,…,𝑚, (3.3) then one has the following inequality: ^𝑚𝑚_𝑖=1𝑗=1^𝑏−2_𝑛=𝑎𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)𝜁_𝑖(𝑛)+𝜂_𝑖(𝑛)𝑓⁺_𝑗(𝑛)^{𝛼𝑖𝛼𝑗/𝑝𝑖𝑝𝑗}≥1. (3.4)

Proof. By (1.16), (H3), and (3.3), we obtain ^𝑏−1_𝑛=𝑎𝑟_𝑖(𝑛)Δ𝑢_𝑖(𝑛)^𝑝𝑖=^𝑏−2_𝑛=𝑎𝑓_𝑖(𝑛)^𝑚_𝑘=1𝑢_𝑘(𝑛+1)^𝛼𝑘,𝑖=1,2,…,𝑚. (3.5) It follows from (3.1), (3.3), and the Hölder inequality that 𝑢_𝑖(𝑛+1)^𝑝𝑖=^𝑛_𝜏=𝑎Δ𝑢_𝑖(𝜏)^𝑝𝑖≤^𝑛_𝜏=𝑎𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)𝑝𝑖𝑛−1}_𝜏=𝑎𝑟_𝑖(𝜏)Δ𝑢_𝑖(𝜏)^𝑝𝑖=𝜁_𝑖(𝑛)^𝑛_𝜏=𝑎𝑟_𝑖(𝜏)Δ𝑢_𝑖(𝜏)^𝑝𝑖,𝑎≤𝑛≤𝑏−1,𝑖=1,2,…,𝑚. (3.6) Similarly, it follows from (3.2), (3.3), and the Hölder inequality that 𝑢_𝑖(𝑛+1)^𝑝𝑖=^𝑏−1_𝜏=𝑛+1Δ𝑢_𝑖(𝜏)^𝑝𝑖≤^𝑏−1_𝜏=𝑛+1𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)𝑝𝑖−1𝑏−1}_𝜏=𝑛+1𝑟_𝑖(𝜏)Δ𝑢_𝑖(𝜏)^𝑝𝑖=𝜂_𝑖(𝑛)^𝑏−1_𝜏=𝑛+1𝑟_𝑖(𝜏)Δ𝑢_𝑖(𝜏)^𝑝𝑖,𝑎≤𝑛≤𝑏−1,𝑖=1,2,…,𝑚. (3.7) From (3.6) and (3.7), we have 𝑢_𝑖(𝑛+1)^𝑝𝑖≤𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)𝜁_𝑖(𝑛)+𝜂_𝑖(𝑛)^𝑏−1_𝜏=𝑎𝑟_𝑖(𝜏)Δ𝑢_𝑖(𝜏)^𝑝𝑖,𝑎≤𝑛≤𝑏−1,𝑖=1,2,…,𝑚. (3.8)
Now, it follows from (3.3), (3.5), (3.8), (H3), and the generalized Hölder inequality that ^𝑏−2_𝑛=𝑎𝑓⁺_𝑗(𝑛)𝑢_𝑖(𝑛+1)^𝑝𝑖≤^𝑏−2_𝑛=𝑎𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)𝜁_𝑖(𝑛)+𝜂_𝑖(𝑛)𝑓⁺_𝑗(𝑛)^𝑏−1_𝜏=𝑎𝑟_𝑖(𝜏)Δ𝑢_𝑖(𝜏)^𝑝𝑖=𝑀_𝑖𝑗^𝑏−2_𝑛=𝑎𝑓_𝑖(𝑛)^𝑚_𝑘=1|𝑢_𝑘(𝑛+1)|^𝛼𝑘≤𝑀_𝑖𝑗^𝑏−2_𝑛=𝑎𝑓⁺_𝑖(𝑛)^𝑚_𝑘=1𝑢_𝑘(𝑛+1)^𝛼𝑘≤𝑀^𝑚_{𝑖𝑗𝑘=1}^𝑏−2_𝑛=𝑎𝑓⁺_𝑖(𝑛)𝑢_𝑘(𝑛+1)^{𝑝𝑘𝛼𝑘/𝑝𝑘}, (3.9) where 𝑀_𝑖𝑗=^𝑏−2_𝑛=𝑎𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)𝜁_𝑖(𝑛)+𝜂_𝑖(𝑛)𝑓⁺_𝑗(𝑛),𝑖,𝑗=1,2,…,𝑚. (3.10)
Next, we prove that ^𝑏−2_𝑛=𝑎𝑓⁺_𝑖(𝑛)𝑢_𝑘(𝑛+1)^𝑝𝑘>0,𝑖,𝑘=1,2,…,𝑚. (3.11) If (3.11) is not true, then there exists 𝑖₀,𝑘₀∈{1,2,…,𝑚} such that ^𝑏−2_𝑛=𝑎𝑓⁺_𝑖0(𝑛)𝑢_𝑘0(𝑛+1)^𝑝𝑘0=0. (3.12) From (3.5), (3.12), and the generalized Hölder inequality, we have 0≤^𝑏−1_𝑛=𝑎𝑟_𝑖0(𝑛)Δ𝑢_𝑖0(𝑛)^𝑝𝑖0=^𝑏−2_𝑛=𝑎𝑓_𝑖0(𝑛)^𝑚_𝑘=1𝑢_𝑘(𝑛+1)^𝛼𝑘≤^𝑚_𝑘=1^𝑏−2_𝑛=𝑎𝑓⁺_𝑖0(𝑛)𝑢_𝑘(𝑛+1)^{𝑝𝑘𝛼𝑘/𝑝𝑘}=0. (3.13) It follows from the fact that 𝑟_𝑖0(𝑛)>0 that Δ𝑢_𝑖0(𝑛)≡0,𝑎≤𝑛≤𝑏−1. (3.14) Combining (3.6) with (3.14), we obtain that 𝑢_𝑖0(𝑛)≡0 for 𝑎≤𝑛≤𝑏, which contradicts (3.3). Therefore, (3.11) holds. From (3.9), (3.11), and (H3), we have ^𝑚𝑚_𝑖=1𝑗=1𝑀^{𝛼𝑖𝛼𝑗/𝑝𝑖𝑝𝑗}_𝑖𝑗≥1. (3.15) It follows from (3.10) and (3.15) that (3.4) holds.

Corollary 3.2. Let 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Suppose that hypothesis (H3) is satisfied. If system (1.16) has a solution (𝑢₁(𝑛),𝑢₂(𝑛),…,𝑢_𝑚(𝑛)) satisfying (3.3), then one has the following inequality: ^𝑚𝑚_𝑖=1𝑗=1^𝑏−2_𝑛=𝑎𝑓⁺_𝑗(𝑛)𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)^{1/2𝛼𝑖𝛼𝑗/𝑝𝑖𝑝𝑗}≥2. (3.16)

Proof. Since 𝜁_𝑖(𝑛)+𝜂_𝑖(𝑛)≥2𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)^1/2,𝑖=1,2,…,𝑚, (3.17) it follows from (3.4) and (H3) that (3.16) holds.

Corollary 3.3. Let 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Suppose that hypothesis (H3) is satisfied. If system (1.16) has a solution (𝑢₁(𝑛),𝑢₂(𝑛),…,𝑢_𝑚(𝑛)) satisfying (3.3), then one has the following inequality ^𝑚_𝑖=1^𝑏−1_𝑛=𝑎𝑟_𝑖(𝑛)^{1/(1−𝑝𝑖)𝛼𝑖(𝑝𝑖−1)/𝑝𝑖𝑚}_𝑗=1^𝑏−2_𝑛=𝑎𝑓⁺_𝑗(𝑛)^{𝛼𝑗/𝑝𝑗}≥2^𝒜, (3.18) where 𝒜=∑^𝑚_𝑖=1𝛼_𝑖.

Proof. Since 𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)^1/2=^𝑛_𝜏=𝑎𝑟_𝑖(𝑛)^{1/(1−𝑝𝑖)𝑏−1}_𝜏=𝑛+1𝑟_𝑖(𝜏)^{1/(1−𝑝𝑖)(𝑝𝑖−1)/2}≤12^{𝑝𝑖−1𝑏−1}_𝜏=𝑎[𝑟_𝑖(𝜏)]^{1/(1−𝑝𝑖)𝑝𝑖−1},𝑖=1,2,…,𝑚, (3.19) 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 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. We consider here a quasilinear difference system of the form:−ΔΔ𝑢₁(𝑛)^𝑝1−2Δ𝑢₁(𝑛)=𝜆₁𝛼₁𝑞(𝑛)𝑢₁(𝑛+1)^𝛼1−2𝑢₂(𝑛+1)^𝛼2⋯𝑢_𝑚(𝑛+1)^𝛼𝑚𝑢₁(𝑛+1),−ΔΔ𝑢₂(𝑛)^𝑝2−2Δ𝑢₂(𝑛)=𝜆₂𝛼₂𝑞(𝑛)𝑢₁(𝑛+1)^𝛼1𝑢₂(𝑛+1)^𝛼2−2⋯𝑢_𝑚(𝑛+1)^𝛼𝑚𝑢₂(),⋮𝑛+1−ΔΔ𝑢_𝑚(𝑛)^𝑝𝑚−2Δ𝑢_𝑚(𝑛)=𝜆_𝑚𝛼_𝑚𝑞(𝑛)𝑢₁(𝑛+1)^𝛼1𝑢₂(𝑛+1)^𝛼2⋯𝑢_𝑚(𝑛+1)^𝛼𝑚−2𝑢_𝑚(𝑛+1), (4.1) where 𝑞(𝑛)>0, 𝜆_𝑖∈ℝ, 𝑝_𝑖 and 𝛼_𝑖 are the same as those in (1.16), and 𝑢_𝑖 satisfies Dirichlet boundary conditions:𝑢_𝑖(𝑎)=𝑢_𝑖(𝑏)=0,𝑢_𝑖(𝑛)>0,∀𝑛∈ℤ[𝑎+1,𝑏−1],𝑖=1,2,…,𝑚. (4.2)

We define the generalized spectrum S of a nonlinear difference system as the set of vector (𝜆₁,𝜆₂,…,𝜆_𝑚)∈ℝ^𝑚 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 𝑝-Laplacian difference equation−Δ|Δ𝑢(𝑛)|^𝑝−2Δ𝑢(𝑛)=𝜆𝑝𝑞(𝑛)|𝑢(𝑛+1)|^𝑝−2𝑢(𝑛+1), (4.3) with Dirichlet boundary condition:𝑢(𝑎)=0=𝑢(𝑏),𝑢(𝑛)>0,∀𝑛∈ℤ[𝑎+1,𝑏−1], (4.4) where 𝑝>1, 𝜆∈ℝ, and 𝑞(𝑛)>0. When 𝑝=2, 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 𝑓_𝑖(𝑛)=𝜆_𝑖𝛼_𝑖𝑞(𝑛) and 𝑟_𝑖(𝑛)=1 for 𝑖=1,2,…,𝑚. 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 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Assume that 1<𝑝_𝑖<∞, 𝛼_𝑖>0 satisfy ∑^𝑚_𝑖=1(𝛼_𝑖/𝑝_𝑖)=1, and that 𝑞(𝑛)>0 for all 𝑛∈ℤ. Then there exists a function ℎ(𝜆₁,…,𝜆_𝑚−1) such that 𝜆_𝑚≥ℎ(𝜆₁,…,𝜆_𝑚−1) for every generalized eigenvalue (𝜆₁,𝜆₂,…,𝜆_𝑚) of boundary value problem (4.1) with (4.2), where ℎ(𝜆₁,…,𝜆_𝑚−1) is given by: ℎ𝜆₁,…,𝜆_𝑚−1=1𝛼_𝑚^𝑚−1_𝑗=1𝜆_𝑗𝛼_𝑗^{𝛼𝑗/𝑝𝑗𝑚}_𝑖=1^𝑏−2_𝑛=𝑎[(𝑛−𝑎+1)(𝑏−𝑛−1)]^𝑝𝑖−1(𝑛−𝑎+1)^𝑝𝑖−1+(𝑏−𝑛−1)^𝑝𝑖−1𝑞(𝑛)^{𝛼𝑖/𝑝𝑖−(𝑝𝑚/𝛼𝑚)}. (4.5)

Proof. For the eigenvalue (𝜆₁,𝜆₂,…,𝜆_𝑚), (4.1) with (4.2) has a nontrivial solution (𝑢₁(𝑛),𝑢₂(𝑛),…,𝑢_𝑚(𝑛)). That is system (1.16) with 𝑟_𝑖(𝑛)=1 and 𝑓_𝑖(𝑛)=𝜆_𝑖𝛼_𝑖𝑞(𝑛) has a solution (𝑢₁(𝑛),𝑢₂(𝑛),…,𝑢_𝑚(𝑛)) satisfying (3.3), it follows from (3.4) that 𝑓_𝑖(𝑛)=𝜆_𝑖𝛼_𝑖𝑞(𝑛)>0,for all𝑛∈ℤ, 𝑖=1,2,…,𝑚, and that 1≤^𝑚𝑚_𝑖=1𝑗=1^𝑏−2_𝑛=𝑎𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)𝜁_𝑖(𝑛)+𝜂_𝑖(𝑛)𝑓⁺_𝑗(𝑛)^{𝛼𝑖𝛼𝑗/𝑝𝑖𝑝𝑗}=^𝑚_𝑗=1𝜆_𝑗𝛼_𝑗^{𝛼𝑗/𝑝𝑗𝑚}_𝑖=1^𝑏−2_𝑛=𝑎𝜁_𝑖(𝑛)𝜂_𝑖(𝑛)𝜁_𝑖(𝑛)+𝜂_𝑖(𝑛)𝑞(𝑛)^{𝛼𝑖/𝑝𝑖}=^𝑚_𝑗=1𝜆_𝑗𝛼_𝑗^{𝛼𝑗/𝑝𝑗𝑚}_𝑖=1^𝑏−2_𝑛=𝑎[(𝑛−𝑎+1)(𝑏−𝑛−1)]^𝑝𝑖−1(𝑛−𝑎+1)^𝑝𝑖−1+(𝑏−𝑛−1)^𝑝𝑖−1𝑞(𝑛)^{𝛼𝑖/𝑝𝑖}. (4.6) Hence, we have 𝜆_𝑚≥1𝛼_𝑚^𝑚−1_𝑗=1𝜆_𝑗𝛼_𝑗^{𝛼𝑗/𝑝𝑗𝑚}_𝑖=1^𝑏−2_𝑛=𝑎[(𝑛−𝑎+1)(𝑏−𝑛−1)]^𝑝𝑖−1(𝑛−𝑎+1)^𝑝𝑖−1+(𝑏−𝑛−1)^𝑝𝑖−1𝑞(𝑛)^{𝛼𝑖/𝑝𝑖−(𝑝𝑚/𝛼𝑚)}. (4.7) This completes the proof of Theorem 4.1.

When 𝑚=2, boundary value problem (4.1) with (4.2) reduces to the simpler form:−ΔΔ𝑢₁(𝑛)^𝑝1−2Δ𝑢₁(𝑛)=𝜆₁𝛼₁𝑞(𝑛)𝑢₁(𝑛+1)^𝛼1−2𝑢₂(𝑛+1)^𝛼2𝑢₁(𝑛+1),−ΔΔ𝑢₂(𝑛)^𝑝2−2Δ𝑢₂(𝑛)=𝜆₂𝛼₂𝑞(𝑛)𝑢₁(𝑛+1)^𝛼1𝑢₂(𝑛+1)^𝛼2−2𝑢₂(𝑛+1), (4.8) with Dirichlet boundary conditions:𝑢_𝑖(𝑎)=𝑢_𝑖(𝑏)=0,𝑢_𝑖(𝑛)>0,∀𝑛∈ℤ[𝑎+1,𝑏−1],𝑖=1,2, (4.9) where 1<𝑝₁,𝑝₂<∞, 𝛼₁,𝛼₂>0 satisfy 𝛼₁/𝑝₁+𝛼₂/𝑝₂=1, and 𝑞(𝑛)>0 for all 𝑛∈ℤ.

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 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Assume that 1<𝑝₁,𝑝₂<∞, 𝛼₁,𝛼₂>0 satisfy 𝛼₁/𝑝₁+𝛼₂/𝑝₂=1, and that 𝑞(𝑛)>0 for all 𝑛∈ℤ. Then there exists a function ℎ(𝜆₁) such that 𝜆₂≥ℎ(𝜆₁) for every generalized eigenvalue (𝜆₁,𝜆₂) of system (4.8) with (4.9), where ℎ(𝜆₁) is given by: ℎ𝜆₁=1/𝛼₂𝜆₁𝛼₁∑^𝑏−2_𝑛=𝑎[𝒳𝒴]^𝑝1−1/𝒳^𝑝1−1+𝒴^𝑝1−1𝑞(𝑛)^{(𝑝2/𝛼2)−1}∑^𝑏−2_𝑛=𝑎[𝒳𝒴]^𝑝2−1/𝒳^𝑝2−1+𝒴^𝑝2−1𝑞(𝑛), (4.10) where 𝒳 denote (𝑛−𝑎+1) and 𝒴 denote (𝑏−𝑛−1).

Corollary 4.3. Let 𝑎,𝑏∈ℤ with 𝑎≤𝑏−2. Assume that 𝑝>1 and 𝑞(𝑛)>0 for all 𝑛∈ℤ. Then for every eigenvalue 𝜆 of system (4.3) with (4.4), one has 1𝜆≥𝑝^𝑏−2_𝑛=𝑎[(𝑛−𝑎+1)(𝑏−𝑛−1)]^𝑝−1(𝑛−𝑎+1)^𝑝−1+(𝑏−𝑛−1)^𝑝−1𝑞(𝑛)⁻¹. (4.11)