# Lyapunov-Type Inequalities for the Quasilinear Difference Systems

Authors: Qi-Ming Zhang; X. H. Tang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## 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.


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## Body

## 1. Introduction

In 1964, Atkinson [1] investigated the following boundary value problem:(1.1)Δ(r(n)Δu(n))=λq(n)u(n+1)
with Dirichlet boundary condition: (1.2)u(a)=u(b)=0,u(n)≢0,∀n∈Z[a,b],
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 order(1.3)λ1<λ2<⋯<λb-a-1,
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 equation(1.4)Δ2u(n)+q(n)u(n+1)=0
has a real solution u(n) such that(1.5)u(a)=u(b)=0,u(n)≢0,n∈Z[a,b],
then one has the following inequality(1.6)F(b-a)∑n=ab-2q(n)≥4,
where q(n)≥0 for all n∈ℤ, and(1.7)F(m)={m2-1m,ifm-1iseven,m,ifm-1isodd,
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:(1.8)(b-a)∫ab|q(t)|dt>4,
if Hill's equation(1.9)u′′(t)+q(t)u(t)=0
has a real solution u(t) such that(1.10)u(a)=u(b)=0,u(t)≠0,∀t∈(a,b),
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:(1.11)(∑n=ab-11[r(n)]1/(p-1))1-(1/p)(∑n=ab-2q+(n))1/p≥2,
if the following second-order half-linear difference equation:(1.12)Δ(r(n)|Δu(n)|p-2Δu(n))+q(n)|u(n+1)|p-2u(n+1)=0
has a solution u(n) satisfying(1.13)u(a)=u(b)=0,u(n)≢0,∀n∈Z[a,b],
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:(1.14)(b-a)∑n=ab-2q(n)≥4,
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 singlep-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(1.15)-Δ(r1(n)|Δu(n)|p1-2Δu(n))=f1(n)|u(n+1)|α1-2|v(n+1)|α2u(n+1),-Δ(r2(n)|Δv(n)|p2-2Δv(n))=f2(n)|u(n+1)|β1|v(n+1)|β2-2v(n+1),
and the quasilinear difference system involving the (p1,p2,…,pn)-Laplacian
(1.16)-Δ(r1(n)|Δu1(n)|p1-2Δu1(n))=f1(n)|u1(n+1)|α1-2|u2(n+1)|α2⋯|um(n+1)|αmu1(n+1),-Δ(r2(n)|Δu2(n)|p2-2Δu2(n))=f2(n)|u1(n+1)|α1|u2(n+1)|α2-2⋯|um(n+1)|αmu2(n+1),⋮-Δ(rm(n)|Δum(n)|pm-2Δum(n))=fm(n)|u1(n+1)|α1|u2(n+1)|α2⋯|um(n+1)|αm-2um(n+1).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)
ri(n) and fi(n) are real-valued functions and ri(n)>0 for i=1,2,…,m. Furthermore, 1<pi<∞ and αi>0 satisfy ∑i=1m(αi/pi)=1.System (1.15) and (1.16) are the discrete analogies of the following two quasilinear differential systems:(1.17)-(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),(1.18)-(r1(t)|u1′(t)|p1-2u1′(t))′=f1(t)|u1(t)|α1-2|u2(t)|α2⋯|un(t)|αnu1(t),-(r2(t)|u2′(t)|p2-2u2′(t))′=f2(t)|u1(t)|α1|u2(t)|α2-2⋯|un(t)|αnu2(t),⋮-(rn(t)|un′(t)|pn-2un′(t))′=fn(t)|u1(t)|α1|u2(t)|α2⋯|un(t)|αn-2un(t),
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)(1.19)∑n=ab-2[(∑s=an[r(s)]1/(1-p))p-1(∑s=n+1b-1[r(s)]1/(1-p))p-1(∑s=an[r(s)]1/(1-p))p-1+(∑s=n+1b-1[r(s)]1/(1-p))p-1q+(n)]≥1
for the second-order half-linear difference equation (1.12). In particular, (1.19) produces a new Lyapunov-type inequality(1.20)1b-a∑n=ab-2(n+1-a)(b-n-1)q+(n)≥1
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. Section2 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(2.1)ζ1(n):=(∑τ=an[r1(τ)]1/(1-p1))p1-1,η1(n):=(∑τ=n+1b-1[r1(τ)]1/(1-p1))p1-1,(2.2)ζ2(n):=(∑τ=an[r2(τ)]1/(1-p2))p2-1,η2(n):=(∑τ=n+1b-1[r2(τ)]1/(1-p2))p2-1.Theorem 2.1.
Leta,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:
(2.3)u(a)=u(b)=0=v(a)=v(b),u(n)≢0,v(n)≢0,∀n∈Z[a,b],
then one has the following inequality:
(2.4)(∑n=ab-2ζ1(n)η1(n)ζ1(n)+η1(n)f1+(n))α1β1/p12(∑n=ab-2ζ1(n)η1(n)ζ1(n)+η1(n)f2+(n))β1α2/p1p2×(∑n=ab-2ζ2(n)η2(n)ζ2(n)+η2(n)f1+(n))β1α2/p1p2(∑n=ab-2ζ2(n)η2(n)ζ2(n)+η2(n)f2+(n))α2β2/p22≥1,
where and in the sequel fi+(n)=max{fi(n),0} for i=1,2.Proof.
By (1.15) and (2.3), we obtain
(2.5)∑n=ab-1r1(n)|Δu(n)|p1=∑n=ab-2f1(n)|u(n+1)|α1|v(n+1)|α2,(2.6)∑n=ab-1r2(n)|Δv(n)|p2=∑n=ab-2f2(n)|u(n+1)|β1|v(n+1)|β2.
It follows from (2.1), (2.3), and the Hölder inequality that
(2.7)|u(n+1)|p1=|∑τ=anΔu(τ)|p1≤(∑τ=an|Δu(τ)|)p1≤(∑τ=an[r1(τ)]1/(1-p1))p1-1∑τ=anr1(τ)|Δu(τ)|p1=ζ1(n)∑τ=anr1(τ)|Δu(τ)|p1,a≤n≤b-1,(2.8)|u(n+1)|p1=|∑τ=n+1b-1Δu(τ)|p1≤(∑τ=n+1b-1|Δu(τ)|)p1≤(∑τ=n+1b-1[r1(τ)]1/(1-p1))p1-1∑τ=n+1b-1r1(τ)|Δu(τ)|p1=η1(n)∑τ=n+1b-1r1(τ)|Δu(τ)|p1,a≤n≤b-1.
From (2.7) and (2.8), we have
(2.9)|u(n+1)|p1≤ζ1(n)η1(n)ζ1(n)+η1(n)∑τ=ab-1r1(τ)|Δu(τ)|p1,a≤n≤b-1.
Now, it follows from (2.3), (2.5), (2.9), (H2), and the Hölder inequality that
(2.10)∑n=ab-2f1+(n)|u(n+1)|p1≤∑n=ab-2[ζ1(n)η1(n)ζ1(n)+η1(n)f1+(n)]∑n=ab-1r1(n)|Δu(n)|p1=M11∑n=ab-2f1(n)|u(n+1)|α1|v(n+1)|α2≤M11∑n=ab-2f1+(n)|u(n+1)|α1|v(n+1)|α2≤M11(∑n=ab-2f1+(n)|u(n+1)|p1)α1/p1(∑n=ab-2f1+(n)|v(n+1)|p2)α2/p2,(2.11)∑n=ab-2f2+(n)|u(n+1)|p1≤∑n=ab-2[ζ1(n)η1(n)ζ1(n)+η1(n)f2+(n)]∑n=ab-1r1(n)|Δu(n)|p1=M12∑n=ab-2f1(n)|u(n+1)|α1|v(n+1)|α2≤M12∑n=ab-2f1+(n)|u(n+1)|α1|v(n+1)|α2≤M12(∑n=ab-2f1+(n)|u(n+1)|p1)α1/p1(∑n=ab-2f1+(n)|v(n+1)|p2)α2/p2,
where
(2.12)M11=∑n=ab-2[ζ1(n)η1(n)ζ1(n)+η1(n)f1+(n)],M12=∑n=ab-2[ζ1(n)η1(n)ζ1(n)+η1(n)f2+(n)].
Similar to the proof of (2.9), from (2.2) and (2.3), we have
(2.13)|v(n+1)|p2≤ζ2(n)η2(n)ζ2(n)+η2(n)∑τ=ab-1r2(τ)|Δv(τ)|p2,a≤n≤b-1.
It follows from (2.3), (2.6), (2.13), (H2), and the Hölder inequality that
(2.14)∑n=ab-2f1+(n)|v(n+1)|p2≤∑n=ab-2[ζ2(n)η2(n)ζ2(n)+η2(n)f1+(n)]∑n=ab-1r2(n)|Δv(n)|p2=M21∑n=ab-2f2(n)|u(n+1)|β1|v(n+1)|β2≤M21∑n=ab-2f2+(n)|u(n+1)|β1|v(n+1)|β2≤M21(∑n=ab-2f2+(n)|u(n+1)|p1)β1/p1(∑n=ab-2f2+(n)|v(n+1)|p2)β2/p2,∑n=ab-2f2+(n)|v(n+1)|p2≤∑n=ab-2[ζ2(n)η2(n)ζ2(n)+η2(n)f2+(n)]∑n=ab-1r2(n)|Δv(n)|p2=M22∑n=ab-2f2(n)|u(n+1)|β1|v(n+1)|β2≤M22∑n=ab-2f2+(n)|u(n+1)|β1|v(n+1)|β2≤M22(∑n=ab-2f2+(n)|u(n+1)|p1)β1/p1(∑n=ab-2f2+(n)|v(n+1)|p2)β2/p2,
where
(2.15)M21=∑n=ab-2[ζ2(n)η2(n)ζ2(n)+η2(n)f1+(n)],M22=∑n=ab-2[ζ2(n)η2(n)ζ2(n)+η2(n)f2+(n)].
Next, we prove that(2.16)∑n=ab-2f1+(n)|u(n+1)|p1>0.
If (2.16) is not true, then
(2.17)∑n=ab-2f1+(n)|u(n+1)|p1=0.
From (2.5) and (2.17), we have
(2.18)0≤∑n=ab-1r1(n)|Δu(n)|p1=∑n=ab-2f1(n)|u(n+1)|α1|v(n+1)|α2≤∑n=ab-2f1+(n)|u(n+1)|α1|v(n+1)|α2=0.
It follows from (H1) that
(2.19)Δu(n)≡0,a≤n≤b-1.
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
(2.20)∑n=ab-2f2+(n)|u(n+1)|p1>0,∑n=ab-2f1+(n)|v(n+1)|p2>0,∑n=ab-2f2+(n)|v(n+1)|p2>0.
From (2.10), (2.11), (2.14), (2.16), (2.20), and (H2), we have
(2.21)M11α1β1/p12M12β1α2/p1p2M21β1α2/p1p2M22α2β2/p22≥1.
It follows from (2.12), (2.15), and (2.21) that (2.4) holds.Corollary 2.2.
Leta,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:
(2.22)(∑n=ab-2f1+(n)[ζ1(n)η1(n)]1/2)α1β1/p12(∑n=ab-2f2+(n)[ζ1(n)η1(n)]1/2)β1α2/p1p2×(∑n=ab-2f1+(n)[ζ2(n)η2(n)]1/2)β1α2/p1p2(∑n=ab-2f2+(n)[ζ2(n)η2(n)]1/2)α2β2/p22≥2(p2β1+p1α2)/p1p2.Proof.
Since(2.23)ζi(n)+ηi(n)≥2[ζi(n)ηi(n)]1/2,i=1,2,
it follows from (2.4) and (H2) that (2.22) holds.Corollary 2.3.
Leta,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:
(2.24)(∑n=ab-1[r1(n)]1/(1-p1))β1(p1-1)/p1(∑n=ab-1[r2(n)]1/(1-p2))α2(p2-1)/p2×(∑n=ab-2f1+(n))β1/p1(∑n=ab-2f2+(n))α2/p2≥2β1+α2.Proof.
Since(2.25)[ζ1(n)η1(n)]1/2=(∑τ=an[r1(τ)]1/(1-p1)∑τ=n+1b-1[r1(τ)]1/(1-p1))(p1-1)/2≤12p1-1(∑τ=ab-1[r1(τ)]1/(1-p1))p1-1,[ζ2(n)η2(n)]1/2=(∑τ=an[r2(τ)]1/(1-p2)∑τ=n+1b-1[r2(τ)]1/(1-p2))(p2-1)/2≤12p2-1(∑τ=ab-1[r2(τ)]1/(1-p2))p2-1,
it follows from (2.22) and (H2) that (2.24) holds.Whenα1=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.
Leta,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:
(2.26)∑n=ab-2[(∑τ=an[r(τ)]1/(1-p))p-1(∑τ=n+1b-1[r(τ)]1/(1-p))p-1(∑τ=an[r(τ)]1/(1-p))p-1+(∑τ=n+1b-1[r(τ)]1/(1-p))p-1q+(n)]≥1.
Since
(2.27)(∑τ=an[r(τ)]1/(1-p))p-1+(∑τ=n+1b-1[r(τ)]1/(1-p))p-1≥2(∑τ=an[r(τ)]1/(1-p)∑τ=n+1b-1[r(τ)]1/(1-p))(p-1)/2,
it follows from Theorem 2.4 that the following corollary holds.Corollary 2.5.
Leta,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:
(2.28)∑n=ab-2[q+(n)(∑τ=an[r(τ)]1/(1-p)∑τ=n+1b-1[r(τ)]1/(1-p))(p-1)/2]≥2.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(3.1)ζi(n):=(∑τ=an[ri(τ)]1/(1-pi))pi-1,i=1,2,…,m,(3.2)ηi(n):=(∑τ=n+1b-1[ri(τ)]1/(1-pi))pi-1,i=1,2,…,m.Theorem 3.1.
Leta,b∈ℤ with  a≤b-2. Suppose that hypothesis (H3) is satisfied. If system (1.16) has a solution  (u1(n),u2(n),…,um(n)) satisfying the boundary value conditions:
(3.3)ui(a)=ui(b)=0,ui(n)≢0,∀n∈Z[a,b],i=1,2,…,m,
then one has the following inequality:
(3.4)∏i=1m∏j=1m(∑n=ab-2[ζi(n)ηi(n)ζi(n)+ηi(n)fj+(n)])αiαj/pipj≥1.Proof.
By (1.16), (H3), and (3.3), we obtain
(3.5)∑n=ab-1ri(n)|Δui(n)|pi=∑n=ab-2[fi(n)∏k=1m|uk(n+1)|αk],i=1,2,…,m.
It follows from (3.1), (3.3), and the Hölder inequality that
(3.6)|ui(n+1)|pi=|∑τ=anΔui(τ)|pi≤(∑τ=an[ri(τ)]1/(1-pi))pi-1∑τ=anri(τ)|Δui(τ)|pi=ζi(n)∑τ=anri(τ)|Δui(τ)|pi,a≤n≤b-1,i=1,2,…,m.
Similarly, it follows from (3.2), (3.3), and the Hölder inequality that
(3.7)|ui(n+1)|pi=|∑τ=n+1b-1Δui(τ)|pi≤(∑τ=n+1b-1[ri(τ)]1/(1-pi))pi-1∑τ=n+1b-1ri(τ)|Δui(τ)|pi=ηi(n)∑τ=n+1b-1ri(τ)|Δui(τ)|pi,a≤n≤b-1,i=1,2,…,m.
From (3.6) and (3.7), we have
(3.8)|ui(n+1)|pi≤ζi(n)ηi(n)ζi(n)+ηi(n)∑τ=ab-1ri(τ)|Δui(τ)|pi,a≤n≤b-1,i=1,2,…,m.
Now, it follows from (3.3), (3.5), (3.8), (H3), and the generalized Hölder inequality that
(3.9)∑n=ab-2fj+(n)|ui(n+1)|pi≤∑n=ab-2[ζi(n)ηi(n)ζi(n)+ηi(n)fj+(n)]∑τ=ab-1ri(τ)|Δui(τ)|pi=Mij∑n=ab-2[fi(n)∏k=1m|uk(n+1)|αk]≤Mij∑n=ab-2[fi+(n)∏k=1m|uk(n+1)|αk]≤Mij∏k=1m[∑n=ab-2fi+(n)|uk(n+1)|pk]αk/pk,
where
(3.10)Mij=∑n=ab-2[ζi(n)ηi(n)ζi(n)+ηi(n)fj+(n)],i,j=1,2,…,m.
Next, we prove that(3.11)∑n=ab-2fi+(n)|uk(n+1)|pk>0,i,k=1,2,…,m.
If (3.11) is not true, then there exists i0,k0∈{1,2,…,m} such that
(3.12)∑n=ab-2fi0+(n)|uk0(n+1)|pk0=0.
From (3.5), (3.12), and the generalized Hölder inequality, we have
(3.13)0≤∑n=ab-1ri0(n)|Δui0(n)|pi0=∑n=ab-2fi0(n)∏k=1m|uk(n+1)|αk≤∏k=1m[∑n=ab-2fi0+(n)|uk(n+1)|pk]αk/pk=0.
It follows from the fact that ri0(n)>0 that
(3.14)Δui0(n)≡0,a≤n≤b-1.
Combining (3.6) with (3.14), we obtain that ui0(n)≡0 for a≤n≤b, which contradicts (3.3). Therefore, (3.11) holds. From (3.9), (3.11), and (H3), we have
(3.15)∏i=1m∏j=1mMijαiαj/pipj≥1.
It follows from (3.10) and (3.15) that (3.4) holds.Corollary 3.2.
Leta,b∈ℤ with  a≤b-2. Suppose that hypothesis (H3) is satisfied. If system (1.16) has a solution  (u1(n),u2(n),…,um(n)) satisfying (3.3), then one has the following inequality:
(3.16)∏i=1m∏j=1m(∑n=ab-2fj+(n)[ζi(n)ηi(n)]1/2)αiαj/pipj≥2.Proof.
Since(3.17)ζi(n)+ηi(n)≥2[ζi(n)ηi(n)]1/2,i=1,2,…,m,
it follows from (3.4) and (H3) that (3.16) holds.Corollary 3.3.
Leta,b∈ℤ with  a≤b-2. Suppose that hypothesis (H3) is satisfied. If system (1.16) has a solution  (u1(n),u2(n),…,um(n)) satisfying (3.3), then one has the following inequality
(3.18)∏i=1m(∑n=ab-1[ri(n)]1/(1-pi))αi(pi-1)/pi∏j=1m(∑n=ab-2fj+(n))αj/pj≥2A,
where 𝒜=∑i=1mαi.Proof.
Since(3.19)[ζi(n)ηi(n)]1/2=(∑τ=an[ri(n)]1/(1-pi)∑τ=n+1b-1[ri(τ)]1/(1-pi))(pi-1)/2≤12pi-1(∑τ=ab-1[ri(τ)]1/(1-pi))pi-1,i=1,2,…,m,
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.Leta,b∈ℤ with a≤b-2. We consider here a quasilinear difference system of the form:(4.1)-Δ(|Δu1(n)|p1-2Δu1(n))=λ1α1q(n)|u1(n+1)|α1-2|u2(n+1)|α2⋯|um(n+1)|αmu1(n+1),-Δ(|Δu2(n)|p2-2Δu2(n))=λ2α2q(n)|u1(n+1)|α1|u2(n+1)|α2-2⋯|um(n+1)|αmu2(n+1),⋮-Δ(|Δum(n)|pm-2Δum(n))=λmαmq(n)|u1(n+1)|α1|u2(n+1)|α2⋯|um(n+1)|αm-2um(n+1),
where q(n)>0, λi∈ℝ, pi and αi are the same as those in (1.16), and ui satisfies Dirichlet boundary conditions:(4.2)ui(a)=ui(b)=0,ui(n)>0,∀n∈Z[a+1,b-1],i=1,2,…,m.We define the generalized spectrum S of a nonlinear difference system as the set of vector(λ1,λ2,…,λm)∈ℝm 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(4.3)-Δ(|Δu(n)|p-2Δu(n))=λpq(n)|u(n+1)|p-2u(n+1),
with Dirichlet boundary condition:(4.4)u(a)=0=u(b),u(n)>0,∀n∈Z[a+1,b-1],
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].Letfi(n)=λiαiq(n) and ri(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.
Leta,b∈ℤ with  a≤b-2. Assume that  1<pi<∞,  αi>0 satisfy  ∑i=1m(αi/pi)=1, and that  q(n)>0 for all  n∈ℤ. Then there exists a function  h(λ1,…,λm-1) such that  λm≥h(λ1,…,λm-1) for every generalized eigenvalue  (λ1,λ2,…,λm) of boundary value problem (4.1) with (4.2), where  h(λ1,…,λm-1) is given by:
(4.5)h(λ1,…,λm-1)=1αm[∏j=1m-1(λjαj)αj/pj∏i=1m(∑n=ab-2[(n-a+1)(b-n-1)]pi-1(n-a+1)pi-1+(b-n-1)pi-1q(n))αi/pi]-(pm/αm).Proof.
For the eigenvalue(λ1,λ2,…,λm), (4.1) with (4.2) has a nontrivial solution (u1(n),u2(n),…,um(n)). That is system (1.16) with ri(n)=1 and fi(n)=λiαiq(n) has a solution (u1(n),u2(n),…,um(n)) satisfying (3.3), it follows from (3.4) that fi(n)=λiαiq(n)>0,for alln∈ℤ, i=1,2,…,m, and that
(4.6)1≤∏i=1m∏j=1m(∑n=ab-2ζi(n)ηi(n)ζi(n)+ηi(n)fj+(n))αiαj/pipj=∏j=1m(λjαj)αj/pj∏i=1m(∑n=ab-2ζi(n)ηi(n)ζi(n)+ηi(n)q(n))αi/pi=∏j=1m(λjαj)αj/pj∏i=1m(∑n=ab-2[(n-a+1)(b-n-1)]pi-1(n-a+1)pi-1+(b-n-1)pi-1q(n))αi/pi.
Hence, we have
(4.7)λm≥1αm[∏j=1m-1(λjαj)αj/pj∏i=1m(∑n=ab-2[(n-a+1)(b-n-1)]pi-1(n-a+1)pi-1+(b-n-1)pi-1q(n))αi/pi]-(pm/αm).
This completes the proof of Theorem 4.1.Whenm=2, boundary value problem (4.1) with (4.2) reduces to the simpler form:(4.8)-Δ(|Δu1(n)|p1-2Δu1(n))=λ1α1q(n)|u1(n+1)|α1-2|u2(n+1)|α2u1(n+1),-Δ(|Δu2(n)|p2-2Δu2(n))=λ2α2q(n)|u1(n+1)|α1|u2(n+1)|α2-2u2(n+1),
with Dirichlet boundary conditions:(4.9)ui(a)=ui(b)=0,ui(n)>0,∀n∈Z[a+1,b-1],i=1,2,
where 1<p1,p2<∞, α1,α2>0 satisfy α1/p1+α2/p2=1, and q(n)>0 for all n∈ℤ.Applying Theorem4.1 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.
Leta,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:
(4.10)h(λ1)=1/α2(λ1α1∑n=ab-2([XY]p1-1/(Xp1-1+Yp1-1))q(n))(p2/α2)-1∑n=ab-2([XY]p2-1/(Xp2-1+Yp2-1))q(n),
where 𝒳 denote (n-a+1) and 𝒴 denote (b-n-1).Corollary 4.3.
Leta,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
(4.11)λ≥1p[∑n=ab-2[(n-a+1)(b-n-1)]p-1(n-a+1)p-1+(b-n-1)p-1q(n)]-1.



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*Source: 860598-2012-02-16.xml*