# Lyapunov-Type Inequality for a Class of Discrete Systems with Antiperiodic Boundary Conditions

Authors: Xin-Ge Liu; Mei-Lan Tang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A class of higher-order 3-dimensional discrete systems with antiperiodic
boundary conditions is investigated. Based on the existence of the positive solution of linear
homogeneous system, several new Lyapunov-type inequalities are established.


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## Body

## 1. Introduction

Lyapunov-type inequalities have been proved to be very useful in oscillation theory, disconjugacy, eigenvalue problems, and numerous other applications in the theory of differential and difference equations [1–3]. In recent years, there are many literatures which improved and extended the classical Lyapunov inequality including continuous and discrete cases [4–6]. Guseinov and Kaymakçalan [7] considered the following discrete Hamiltonian system:
(1)Δx(t)=a(t)x(t+1)+b(t)u(t),Δu(t)=-c(t)x(t+1)-a(t)u(t),
where Δ denotes the forward difference operator, with the coefficients a(t) satisfying the condition 1-a(t)≠0,t∈Z. They [7] presented some Lyapunov-type inequalities for discrete linear scalar Hamiltonian systems when the coefficient c(t) is not necessarily nonnegative value. Applying these inequalities, they [7] obtained some stability criteria for discrete Hamiltonian systems.For simplicity, the following assumptions are introduced:(2)1-α(n)>0,∀n∈Z,(3)x(a)=0,orx(a)x(a+1)<0,x(b)=0,orx(b)x(b+1)<0,maxa≤n≤b|x(n)|>0,a,b∈Z.Recently, Zhang and Tang [8] also considered the discrete linear Hamiltonian system:
(4)Δx(n)=α(n)x(n+1)+β(n)y(n),Δy(n)=-γ(n)x(n+1)-α(n)y(n),
where α(n),β(n), and γ(n) are real-valued functions defined on Z and Δ denotes the forward difference operator defined by Δx(n)=x(n+1)-x(n), β(n)≥0. They [8] obtained the following interesting Lyapunov-type inequality.Theorem A.
Suppose that (2) holds, and let a,b∈Z with a<b-1. Assume (4) has a real solution (x(n),y(n)) such that (3) holds. Then one has the following inequality:
(5)∑n=ab-1‍|α(n)|+[∑n=ab‍β(n)∑n=ab-1‍γ+(n)]1/2≥2.
In 2012, the following assumptions are introduced in [9].(H1)
r1(n),r2(n),f1(n), and f2(n) are real-valued functions, and r1(n)>0, and r2(n)>0.(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(n)>0 satisfy ∑i=1m‍(αi/pi)=1.
Under the boundary value conditions, Zhang and Tang [9] considered the following quasilinear difference systems with hypotheses (H1) and (H2):
(6)-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)|p1-2Δv(n))=f2(n)|u(n+1)|β1|v(n+1)|β2-2v(n+1),
and the quasilinear difference systems involving the (p1,p2,…,pm)-Laplacian:
(7)-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).
Some Lyapunov-type inequalities are established in [9].
Recently, antiperiodic problems have received considerable attention as antiperiodic boundary conditions appear in numerous situations [10–12]. For the sake of convenience, in this paper, one will only consider the following higher-order 3-dimensional discrete system:
(8)|Δmx(n)|p1-2Δmx(n)+f1(n)ψq1,1(x(n))ψq1,2(y(n))ψq1,3(z(n))=0,|Δmy(n)|p2-2Δmy(n)+f2(n)ψq2,1(x(n))ψq2,2(y(n))ψq2,3(z(n))=0,|Δmz(n)|p3-2Δmz(n)+f3(n)ψq3,1(x(n))ψq3,2(y(n))ψq3,3(z(n))=0,
where 1<pk<+∞ for k=1,2,3; qi,j are nonnegative constants for i,j=1,2,3; ψq(u)=|u|q-1u for q>0 with ψ0(u)=sign(u)=±1 for q=0.
Obviously, the results obtained in [9] required that α1/p1+α2/p2=1 and β1/p1+β2/p2=1 or ∑i=1m‍(αi/pi)=1. The order of the quasilinear difference systems considered in [9] is less than 3. In this paper, one will remove the unreasonably severe constraints α1/p1+α2/p2=1 and β1/p1+β2/p2=1 or ∑i=1m‍(αi/pi)=1 in [9]. one will introduce the antiperiodic boundary conditions instead of boundary conditions in [9]. In this paper, one will establish some new Lyapunov-type inequalities for higher-order 3-dimensional discrete system (8) by a method different from that in [9] under the following antiperiodic boundary conditions:
(9)Δix(a)+Δix(b)=Δiy(a)+Δiy(b)Δix(a)+Δix(b)=Δiz(a)+Δiz(b)=0,ccccccccccccccccfccccci=0,1,…,m-1.
The similar results for higher-order m-dimensional discrete system are easy to obtain.
Throughout this paper,pi>1 and pi′ is a conjugate exponent; that is, 1/pi+1/pi′=1,i=1,2,3.


## 2. Main Results

Theorem 1.
Leta<b, and assume that there exists a positive solution (e1,e2,e3) of the following linear homogeneous system:
(10)(q1,1+1-p1)e1+q2,1e2+q3,1e3=0,q1,2e1+(q2,2+1-p2)e2+q3,2e3=0,q1,3e1+q2,3e2+(q3,3+1-p3)e3=0.
If (x(n),y(n),z(n)) is a nonzero solution of (8) satisfying the antiperiodic boundary conditions (9), then
(11)∏k=13‍(∑n=ab-1‍|fk(n)|pk/(pk-1))(1-1/pk)ek≥(b-a)∑i=13∑j=13(qi,j/pj)ei(2b-a)m∑i=13(pi-1)ei.Proof.
Let(x(n),y(n), and z(n)) be a nonzero solution of (8). By the antiperiodic boundary conditions (9), x(a)+x(b)=0. For n∈Z[a,b], we have
(12)x(n)=12∑k=an-1‍[x(k+1)-x(k)]-12∑k=nb-1‍[x(k+1)-x(k)]x(n)=12∑k=an-1‍Δx(k)-12∑k=nb-1‍Δx(k).
Using discrete Hölder inequality gives
(13)|x(n)|≤12∑k=ab-1‍|Δx(k)|≤12(b-a)1/p1′(∑k=ab-1‍|Δx(k)|p1)1/p1.
Similarly,(14)|Δix(n)|≤12∑k=ab-1‍|Δi+1x(k)|≤12(b-a)1/p1′(∑k=ab-1‍|Δi+1x(k)|p1)1/p1.
Then(15)|Δix(n)|p1≤(12)p1(b-a)p1/p1′(∑k=ab-1‍|Δi+1x(k)|p1).
Summing (15) from a to b-1, we have
(16)∑n=ab-1‍|Δix(n)|p1≤(b-a)(12)p1(b-a)p1p1′(∑k=ab-1‍|Δi+1x(k)|p1);
that is,
(17)(∑k=ab-1‍|Δix(k)|p1)1/p1≤b-a2(∑k=ab-1‍|Δi+1x(k)|p1)1/p1.
So(18)|x(n)|≤12(b-a)1/p1′(∑k=ab-1‍|Δx(k)|p1)1/p1≤12(b-a)1/p1′(b-a2)m-1(∑k=ab-1‍|Δmx(k)|p1)1/p1.
Similarly,
(19)|y(n)|≤12(b-a)1/p2′(b-a2)m-1(∑k=ab-1‍|Δmy(k)|p2)1/p2,(20)|z(n)|≤12(b-a)1/p3′(b-a2)m-1(∑k=ab-1‍|Δmz(k)|p3)1/p3.
Multiplying the first equation of (8) by Δmx(n) and using inequalities (18)–(20), we have
(21)|Δmx(n)|p1=|-f1(n)ψq1,1(x(n))ψq1,2(y(n))ψq1,3(z(n))Δmx(n)|=|f1(n)||x(n)|q1,1|y(n)|q1,2|z(n)|q1,3|Δmx(n)|≤[(∑k=ab-1‍|Δmz(k)|p3)1/p312(b-a)1/p1′(b-a2)m-1×(∑k=ab-1‍|Δmx(k)|p1)1/p1]q1,1×[(∑k=ab-1‍|Δmz(k)|p3)1/p312(b-a)1/p2′(b-a2)m-122222×(∑k=ab-1‍|Δmy(k)|p2)1/p2]q1,2×[(∑k=ab-1‍|Δmz(k)|p3)1/p312(b-a)1/p3′(b-a2)m-122221×(∑k=ab-1‍|Δmz(k)|p3)1/p3]q1,3×|f1(n)||Δmx(n)|.
Then(22)∑n=ab-1‍|Δmx(n)|p1≤(b-a)-∑j=13(q1,j/pj)(b-a2)m(∑j=13q1,j)×(∑k=ab-1‍|Δmx(k)|p1)q1,1/p1(∑k=ab-1‍|Δmy(k)|p2)q1,2/p2×(∑k=ab-1‍|Δmz(k)|p3)q1,3/p3∑n=ab-1‍|f1(n)||Δmx(n)|≤(b-a)-∑j=13(q1,j/pj)(b-a2)m(∑j=13q1,j)×(∑k=ab-1‍|Δmx(k)|p1)q1,1/p1(∑k=ab-1‍|Δmy(k)|p2)q1,2/p2×(∑k=ab-1‍|Δmz(k)|p3)q1,3/p3(∑n=ab-1‍|f1(n)|p1′)1/p1′×(∑n=ab-1‍|Δmx(n)|p1)1/p1.
So(23)(∑k=ab-1‍|Δmx(k)|p1)(q1,1+1)/p1-1(∑k=ab-1‍|Δmy(k)|p2)q1,2/p2×(∑k=ab-1‍|Δmz(k)|p3)q1,3/p3(∑n=ab-1‍|f1(n)|p1′)1/p1′≥(b-a)∑j=13(q1,j/pj)(2b-a)m(∑j=13q1,j).
For the second and third equations of (8), we also have
(24)(∑k=ab-1‍|Δmx(k)|p1)q2,1/p1(∑k=ab-1‍|Δmy(k)|p2)(q2,2+1)/p2-1×(∑k=ab-1‍|Δmz(k)|p3)q2,3/p3(∑n=ab-1‍|f2(n)|p2′)1/p2′≥(b-a)∑j=13(q2,j/pj)(2b-a)m(∑j=13q2,j),(25)(∑k=ab-1‍|Δmx(k)|p1)q3,1/p1(∑k=ab-1‍|Δmy(k)|p2)q3,2/p2×(∑k=ab-1‍|Δmz(k)|p3)(q3,3+1)/p3-1(∑n=ab-1‍|f3(n)|p3′)1/p3′≥(b-a)∑j=13(q3,j/pj)(2b-a)m(∑j=13q3,j).
Raising both sides of inequalities (23)–(25) to the powers e1,e2,ande3, respectively, and multiplying the resulting inequalities give
(26)(∑k=ab-1‍|Δmx(k)|p1)(∑i=13qi,1ei)/p1+(1-p1)e1/p1×(∑k=ab-1‍|Δmy(k)|p2)(∑i=13qi,2ei)/p2+(1-p2)e2/p2×(∑k=ab-1‍|Δmz(k)|p3)(∑i=13qi,3ei)/p3+(1-p3)e3/p3×∏k=13‍(∑n=ab-1‍|fk(n)|pk′)ek/pk′≥(b-a)∑i=13∑j=13((qi,j/pj)ei)(2b-a)m(∑i=13∑j=13qi,jei).
Since(e1,e2,e3) is a positive solution of the linear homogeneous system (10), then
(27)∏k=13‍(∑n=ab-1‍|fk(n)|pk′)ek/pk′≥(b-a)∑i=13∑j=13((qi,j/pj)ei)(2b-a)m(∑i=13∑j=13qi,jei).
Summing both sides of linear homogeneous system (10) yields
(28)∑i=13‍∑j=13‍qi,jei=∑i=13‍(pi-1)ei.
Noting that1/pk+1/pk′=1,k=1,2,3, we have
(29)∏k=13‍(∑n=ab-1‍|fk(n)|pk/(pk-1))(1-1/pk)ek≥(b-a)∑i=13∑j=13((qi,j/pj)ei)(2b-a)m∑i=13((pi-1)ei).Corollary 2.
Leta<b and assume
(30)(q1,1+1-p1)+q2,1+q3,1=0,q1,2+(q2,2+1-p2)+q3,2=0,q1,3+q2,3+(q3,3+1-p3)=0.
If (x(n),y(n),z(n)) is a nonzero solution of (8) satisfying the antiperiodic boundary conditions (9), then
(31)∏k=13‍(∑n=ab-1‍|fk(n)|pk/(pk-1))(1-1/pk)≥(b-a)∑i=13∑j=13(qi,j/pj)(2b-a)m∑i=13(pi-1).



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