# Periodic Solutions for a System of  Difference Equations

Authors: Shugui Kang; Bao Shi
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

This paper deals with the second-order nonlinear systems of difference equations, we obtain the existence theorems of periodic solutions. The theorems are proved by using critical point theory.


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## Body

## 1. Introduction

Letℕ,ℤ,ℝ  be the set of all natural numbers, integers,
and real numbers, respectively. For a,b∈ℤ,
note that ℤ[a,b]={a,a+1,…,b}, where a≤b.In this paper, we consider the existence of periodic
solutions for the system of difference equations of the form(1.1)Δ(pn1(Δx(n−1)1)δ)+qn1(xn1)δ=f1(n,Xn),Δ(pn2(Δx(n−1)2)δ)+qn2(xn2)δ=f2(n,Xn),⋮Δ(pnk(Δx(n−1)k)δ)+qnk(xnk)δ=fk(n,Xn),which can be recorded
as(1.2)Δ(P¯n(ΔXn−1T)δ)+Q¯n(XnT)δ=f(n,Xn),n∈ℤ,where k is a positive integer,(1.3)P¯n=(pn10⋯00pn2⋯0⋯00⋯pnk),Q¯n=(qn10⋯00qn2⋯0⋯00⋯qnk),and P¯n+ω=P¯n>0 (i.e., pn1>0,pn2>0,…,pnk>0), Q¯n+ω=Q¯n, f=(f1,f2,…,fk)T, fi=fi(n,Xn)=fi(n,xn1,xn2,…,xnk), f(n+ω,U)=f(n,U) for any (n,U)∈ℤ×ℝk, ω>0 is a positive integer, (−1)δ=−1, δ is the ratio of odd positive integers, ΔXnT=Xn+1T−XnT=(x(n+1)1−xn1,x(n+1)2−xn2,…,x(n+1)k−xnk)T,Δ2Xn−1T=Δ(ΔXn−1T)=ΔXnT−ΔXn−1T.
For U=(u1,u2,…,uk)∈ℝk,
define Uδ=(u1δ,u2δ,…,ukδ). |U|=(|u1|,|u2|,…,|uk|), |U|δ=(|u1|δ,|u2|δ,…,|uk|δ). A sequence X={Xn}n∈ℤ is a ω-periodic solution of (1.2) if substitution of
it into (1.2) yields an identity for all n∈ℤ.In [1, 2], the
qualitative behavior of linear difference equations(1.4)Δ(pnΔxn)+qnxn=0has been investigated. In [3],
the nonlinear difference equation(1.5)Δ(pnΔxn−1)+qnxn=f(n,xn)has been considered. In [4], by
critical point method, the existence of periodic and subharmonic solutions of
equation(1.6)Δ2xn−1+f(n,xn)=0,n∈ℤhas been studied. Other
interesting results can been found in [5–8]. In [9], the authors consider the
existence of periodic solutions for second-order nonlinear difference
equation(1.7)Δ(pn(Δxn−1)δ)+qnxnδ=f(n,xn),n∈ℤ,using critical point theory,
obtaining some new results. It is a discrete analogues of differential
equation(1.8)(p(t)ϕ(u′))′+f(t,u)=0.They do have physical
applications in the study of nuclear physics, gas aerodynamics, and so on
(see [10, 11]). In this paper, we obtain some new results of existence of
periodic solution for the second-order nonlinear system of difference equations
by using critical point theory. We remark, however, the result in [9] is only
good for (1.7) which is much less general than our results in what follows.


## 2. Some Basic Lemmas

LetE be a real Hilbert space, I∈C1(E,ℝ) mean that I is continuously Fréchet differentiable
functional defined on E. I is said to be satisfying Palais-Smale
condition (P-S condition) if any bounded sequence {I(un)}
and I'(un)→0(n→∞) possess a convergent subsequence in E.
Let Bρ be the open ball in E with radius ρ and centered at θ,
and let ∂Bρ denote its boundary, θ is null element of E.Lemma 2.1 (see [12]).
LetE be a real Hilbert space, and assume that I∈C1(E,ℝ) satisfies the P-S condition and the following
conditions:
(I1)
there exist constantsρ>0 and a>0 such that I(x)≥a for all x∈∂Bρ,
where Bρ={x∈E:∥x∥<ρ};(I2)
I(0)≤0 and there exists x0∉Bρ such that I(x0)≤0.Thenc=infh∈Γsups∈[0,1]I(h(s)) is a positive critical value of I,
where(2.1)Γ={h∈C([0,1],X):h(0)=θ,h(1)=x0}.LetΩ∗ be the set of sequences(2.2)X={Xn}n∈ℤ={…,X−n,…,X−1,X0,X1,…,Xn,…},where Xn=(xn1,xn2,…,xnk)∈ℝk,
that is,(2.3)Ω∗={X={Xn}n∈ℤ:Xn∈ℝk,n∈ℤ}.For any X,Y∈Ω∗, a,b∈ℝ, aX+bY is defined by(2.4)aX+bY={aXn+bYn}n=−∞+∞,then Ω∗ is a vector space. For given positive integer ω, Eω is defined as a subspace of Ω∗ by(2.5)Eω={X={Xn}∈Ω∗:Xn+ω=Xn,n∈ℤ}.Obviously, Eω is isomorphic to ℝkω,
for any X,Y∈Eω,
defined inner product(2.6)〈X,Y〉=∑i=1ω〈Xi,Yi〉,by which the norm ∥⋅∥ can be induced by(2.7)∥X∥=(∑i=1ω∥Xi∥2)1/2,X∈Eω.where ∥Xi∥=(∑j=1k|xij|2)1/2. It is obvious that Eω with the inner product defined by (2.6) is a
finite-dimensional Hilbert space and linearly homeomorphic to ℝkω.
Define the functional J on Eω as follows:(2.8)J(X)=1δ+1∑n=1ω〈Pn,(ΔXn−1)δ+1〉−1δ+1∑n=1ω〈Qn,Xnδ+1〉+∑n=1ωF(n,Xn),X∈Eω,where F(n,Xn) such that ∇UF(n,U)=f(n,U),
that is,(2.9)fi(n,U)=fi(n,u1,u2,…,uk)=∂∂uiF(n,u1,u2,…,uk)for any (n,U)∈ℤ[1,ω]×ℝk, Pn=(pn1,pn2,…,pnk), Qn=(qn1,qn2,…,qnk).
Clearly J∈C1(Eω,ℝ),
and for any X={Xn}n∈ℤ∈Eω,
by X0=Xω and X1=Xω+1,
we have(2.10)∂J(X)∂xnl=−Δ(pnl(Δx(n−1)l)δ)−qnl(xnl)δ+fl(n,Xn),l∈ℤ[1,k],n∈ℤ[1,ω].Thus X={Xn}n∈ℤ is a critical point of J on Eω (J'(X)=0) if and only if(2.11)Δ(pnl(Δx(n−1)l)δ)+qnl(xnl)δ=fl(n,Xn),l∈ℤ[1,k],n∈ℤ[1,ω].That is,(2.12)Δ(P¯n(ΔXn−1T)δ)+Q¯n(XnT)δ=f(n,Xn),n∈ℤ.By the periodicity of Xn and f(n,Xn) in the first variable n,
we know that if X={Xn}n∈ℤ∈Eω is a critical point of the real functional J defined by (2.8), then it is a periodic solution
of (1.2).ForX={Xn}n∈ℤ∈Eω, Xn=(xn1,xn2,…,xnk)∈ℝk, r>1,
denote(2.13)∥X∥r=(∑i=1ω∥Xi∥r)1/r,∥Xn∥r=(∑i=1k∥xni∥r)1/r.Clearly, ∥X∥2=∥X∥, ∥Xn∥2=∥Xn∥.
Because of ∥⋅∥r1 and ∥⋅∥r2 being equivalent when r1,r2>1,
so there exist constants c1,c2,c3,c4,ℏ1,ℏ2,ℏ3, and ℏ4 such that c2≥c1>0, c4≥c3>0, ℏ2≥ℏ1>0, and ℏ4≥ℏ3>0,(2.14)c1∥X∥≤∥X∥δ+1≤c2∥X∥,c3∥X∥≤∥X∥β≤c4∥X∥,ℏ1∥Xn∥≤∥Xn∥δ+1≤ℏ2∥Xn∥,ℏ3∥Xn∥≤∥Xn∥β≤ℏ4∥Xn∥,for all X∈Eω, δ>0, and β>1.Lemma 2.2.
Suppose
that(F1)
there exist constantsa1>0, a2>0, β>δ+1 such that(2.15)F(n,U)≤−a1∥U∥β+a2for any (n,U)∈ℤ[1,ω]×Rk;(F2)
(2.16)qni≤0,n∈ℤ,i∈ℤ[1,k].
Then(2.17)J(X)=1δ+1∑n=1ω〈Pn,(ΔXn−1)δ+1〉−1δ+1∑n=1ω〈Qn,Xnδ+1〉+∑n=1ωF(n,Xn)satisfies P-S condition.Proof.
For
any sequence{X(l)}={…,X−n(l),…,X−1(l),X0(l),X1(l),…,Xn(l),…}∈Eω, J(X(l)) is bounded and J'(X(l))→0(l→∞). Then there exists a positive constant M>0,
such that |J(X(l))|≤M.
From (F1), we have(2.18)−M≤J(X(l))=1δ+1∑n=1ω[〈Pn,(Xn(l)−Xn−1(l))δ+1〉−〈Qn,(Xn(l))δ+1〉]+∑n=1ωF(n,Xn(l))≤1δ+1∑n=1ω2δ+1〈Pn,(|Xn(l)|δ+1+|Xn−1(l)|δ+1)〉−1δ+1∑n=1ω〈Qn,|Xn(l)|δ+1〉+∑n=1ωF(n,Xn(l))≤2δ+1δ+1∑n=1ω〈Pn+Pn+1,|Xn(l)|δ+1〉−1δ+1∑n=1ω〈Qn,|Xn(l)|δ+1〉+∑n=1ωF(n,Xn(l))≤2δ+1δ+1∑n=1ω〈Pn+Pn+1,|Xn(l)|δ+1〉−1δ+1∑n=1ω〈Qn,|Xn(l)|δ+1〉−a1∑n=1ω∥Xn(l)∥β+a2ω=1δ+1∑n=1ω〈2δ+1(Pn+Pn+1)−Qn,|Xn(l)|δ+1〉−a1∥X(l)∥ββ+a2ω. Set(2.19)A0=maxn∈ℤ[1,ω],i∈ℤ[1,k]{2δ+1(pni+p(n+1)i)−qni}.Then A0>0, and(2.20)−M≤J(X(l))≤A0δ+1∑n=1ω∥Xn(l)∥δ+1δ+1−a1∥X(l)∥ββ+a2ω≤A0ℏ2δ+1δ+1∑n=1ω∥Xn(l)∥δ+1−a1∥X(l)∥ββ+a2ω=A0ℏ2δ+1δ+1∥X(l)∥δ+1δ+1−a1∥X(l)∥ββ+a2ω. Because of β>δ+1,
and (β−δ−1)/β+(δ+1)/β=1,
in view of Hölder inequality,  we have(2.21)∑n=1ω∥Xn(l)∥δ+1≤ω(β−δ−1)/β(∑n=1ω∥Xn(l)∥β)(δ+1)/β.Thus(2.22)∥X(l)∥ββ≥ω(δ+1−β)/(δ+1)∥X(l)∥δ+1β.Then we have(2.23)−M≤J(X(l))≤A0ℏ2δ+1δ+1∥X(l)∥δ+1δ+1−a1∥X(l)∥ββ+a2ω≤A0ℏ2δ+1δ+1∥X(l)∥δ+1δ+1−a1ω(δ+1−β)/(δ+1)∥X(l)∥δ+1β+a2ω. Thus, for any l∈ℕ,(2.24)a1ω(δ+1−β)/(δ+1)∥X(l)∥δ+1β−A0ℏ2δ+1δ+1∥X(l)∥δ+1δ+1≤M+a2ω.Because of β>δ+1,
it is easily seen that the inequality (2.24) implies that {X(l)} is a bounded sequence in Eω.
Thus {X(l)} possesses convergent subsequences. The proof
is complete.


## 3. Main Result

Theorem 3.1.
Suppose that condition (F1) holds, and
(F3)
for eachn∈ℤ,(3.1)lim∥U∥→0F(n,U)∥U∥δ+1=0;(F4)
for anyi∈ℤ[1,k],n∈ℤ[1,ω],(3.2)qni<0;(F5)
F(n,θ)=0.
Then (1.2) has at least two nontrivial ω-periodic solutions.Proof.
By
Lemma2.2, J satisfies P-S condition. Next, we will verify
the conditions (I1) and (I2) of Lemma 2.1. By (F3), there exists ρ>0,
such that(3.3)|F(n,U)|≤−qmaxℏ1δ+12(δ+1)∥U∥δ+1for any ∥U∥<ρ and n∈ℤ[1,ω],
where qmax=maxn∈ℤ[1,ω],i∈ℤ[1,k]qni<0.
Thus(3.4)J(X)≥−1δ+1∑n=1ω〈Qn,Xnδ+1〉+∑n=1ωF(n,Xn)≥−qmaxδ+1∑n=1ω∥Xn∥δ+1δ+1+qmaxℏ1δ+12(δ+1)∑n=1ω∥Xn∥δ+1≥−qmaxℏ1δ+1δ+1∑n=1ω∥Xn∥δ+1+qmaxℏ1δ+12(δ+1)∑n=1ω∥Xn∥δ+1=−qmaxℏ1δ+12(δ+1)∥X∥δ+1δ+1≥−qmaxℏ1δ+1c1δ+12(δ+1)∥X∥δ+1 for any X∈Eω with ∥X∥≤ρ.
We choose a=−ℏ1δ+1c1δ+1(qmax/2(δ+1))ρδ+1,
then we have(3.5)J(X)|∂Bρ≥a>0,that is, the condition (I1) of Lemma 2.1 holds.
Obviously,J(0)=0.
For any given V∈Eω with ∥V∥=1 and constant α>0,(3.6)J(αV)=1δ+1∑n=1ω〈Pn,(αVn−αVn−1)δ+1〉−1δ+1∑n=1ω〈Qn,(αVn)δ+1〉+∑n=1ωF(n,αVn)=1δ+1∑n=1ω{pn1(αvn1−αv(n−1)1)δ+1+pn2(αvn2−αv(n−1)2)δ+1+⋯+pnk(αvnk−αv(n−1)k)δ+1}−1δ+1∑n=1ω{qn1(αvn1)δ+1+qn2(αvn2)δ+1+⋯+qnk(αvnk)δ+1}+∑n=1ωF(n,αVn)≤1δ+1∑n=1ω{pn1(2α)δ+1+pn2(2α)δ+1+⋯+pnk(2α)δ+1}−1δ+1∑n=1ω{qn1αδ+1+qn2αδ+1+⋯+qnkαδ+1}−a1∑n=1ω∥αVn∥β+a2ω≤1δ+1∑n=1ω{2δ+1∥Pn∥1+∥Qn∥1}αδ+1−a1αβ∑n=1ω∥Vn∥β+a2ω≤1δ+1∑n=1ω{2δ+1∥Pn∥1+∥Qn∥1}αδ+1−a1αβ∥V∥ββ+a2ω≤1δ+1∑n=1ω{2δ+1∥Pn∥1+∥Qn∥1}αδ+1−a1c3βαβ+a2ω→−∞,(α→+∞). Thus we can choose a sufficiently
large α such that α>ρ,
and X¯=αV∈Eω, J(X¯)<0.
According to Lemma 2.1, there exists at least one critical value c≥a>0.
We suppose that X∗ is a critical point corresponding to c,
then J(X∗)=c and J′(X∗)=0.
By similar argument of Lemma2.2, we know that J(X) is bounded from above, so there exists X∗∗∈Eω such that J(X)≤J(X∗∗)=cmax for any X∈Eω.
Obviously, X∗∗≠0.
If X∗∗≠X∗,
then the proof is complete. Otherwise, X∗∗=X∗, c=cmax.
In view of Lemma 2.1,(3.7)c=infh∈Γsups∈[0,1]J(h(s)),where Γ={h∈C([0,1],Eω):h(0)=θ,h(1)=X¯}.
Then cmax=maxs∈[0,1]J(h(s)) for any h∈Γ holds. In view of the continuity of J(h(s)) in s, J(θ)≤0, and J(X¯)<0,
we know that there exists some s0∈(0,1) such that J(h(s0))=cmax.
If we choose h1,h2∈Γ such that(3.8){h1(s):s∈(0,1)}∩{h2(s):s∈(0,1)}=ϕ,then there  exist 					s1,s2∈(0,1) such that J(h1(s1))=J(h2(s2))=cmax.
Then J possesses two different critical points Y^=h1(s1) and
Ž=h2(s2) in Eω,
hence, we obtain at least two nontrivial critical points which correspond to
the critical value cmax.
Thus (1.2) possesses at least two nontrivial ω-periodic solutions. The proof is complete.



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