# Periodic Solutions with Prescribed Minimal Period for2n th-Order Nonlinear Discrete Systems

Authors: Haiping Shi; Peifang Luo; Zan Huang
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of2nth-order nonlinear discrete system are obtained. The main approach used in our paper is variational technique and the linking theorem. The problem is to solve the existence of periodic solutions with prescribed minimal period of 2nth-order discrete systems.


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## Body

## 1. Introduction

In the following and in the sequel, we denote byℕ, ℤ, and ℝ the sets of all natural numbers, integers, and real numbers, respectively. The symbol ∗ is defined by the transpose of a vector. Let ⋅ be the greatest integer function. For any integers a and b with a≤b, we let ℤa,b denote the discrete interval a,a+1,…,b and ℤa=a,a+1,…,.Now, we concern with the following 2nth-order nonlinear discrete system:(1)Δ2nxk−n=−1ngk,xk,n∈ℤ3,k∈ℤ,where Δ is the forward difference operator defined by Δxk=xk+1−xk, Δ0xk=xk, and Δjxk=Δj−1Δxk for j≥2, gt,x∈C1ℝ2,ℝ, and gt+M,x=gt,x for a given integer M≥3.We may think of (1) as a discrete analogue of the following 2nth-order differential equation:(2)d2nxtdt2n=−1ngt,xt,t∈ℝ.Equations similar in structure to (2) have been studied by many authors [1–6]. The difficulty of this paper comparing with system (2) is that there are few known techniques for studying the existence of periodic solutions with minimal period of (1).Denote(3)ω=2πM.Throughout this paper, we suppose that there is a functionGt,x∈C2ℝ2,ℝ with Gt+M,x=Gt,x, G−t,−x=Gt,x, Gt,x≥0, and(4)∂Gt,x∂x=gt,x,∀t∈ℝ.Bin [7] in 2013 considered the second-order discrete Hamiltonian systems:(5)Δxk−1+W′k,xk=0,xk+M=xk.By using Morse theory, some new results concerning the existence of nontrivial periodic solution are obtained.Using the critical point method, Liu et al. [8] studied the following forward and backward difference equation:(6)Δnrk−nΔnxk−n=−1nfk,xk+1,xk,xk−1,n∈ℤ3,k∈ℤ.Some new criteria for the existence and multiplicity of periodic and subharmonic solutions are established.In 2016, Leng [9] established some new criteria for the existence and multiplicity of periodic and subharmonic solutions to the 2nth-order difference equation with ϕc-Laplacian(7)Δnrk−nϕcΔnxk−n=−1nfk,xk+1,xk,xk−1,k∈ℤ,using the linking theorem in combination with variational technique.By establishing a new proper variational framework and using the critical point theory, He [10] established some new existence criteria to guarantee that the 2nth-order nonlinear difference equation containing both advance and retardation with p-Laplacian(8)Δnrk−nφpΔnxk−1+qnφpxk=fk,xk+n,…,xk,…,xk−n,n∈ℤ3,k∈ℤ,has infinitely many homoclinic orbits.Lin and Zhou [11] concerned with the existence and multiplicity of periodic and subharmonic solutions of the following 2nth-order difference equation(9)Δnrk−nϕΔnxk−1=−1nfk,xk+n,…,xk,…,xk−n,k∈ℤ.By using the critical point theory, some new sufficient conditions are obtained and some previous results are generalized.Xia [12] in 2018 considered the existence of periodic solutions for a higher order nonlinear difference equation(10)∑i=0nriXk−i+Xk+i+fk,Xk+Γ,…,Xk,…,Xk−Γ=0,n∈ℕ,k∈ℤ,by making use of the critical point theory. Some existence criteria are established. Some results are generalized.In 1978, Rabinowitz [13] proposed a conjecture that the Hamiltonian system has a nonconstant periodic solution with prescribed minimal period under some given conditions. From then on, there has been much progress [14–16] on Rabinowitz’s conjecture under various conditions. Ambrosetti and Mancini [14] assumed that the dual functional is bounded from below and the Hamiltonian system has a minimum to which correspond a solution with given minimal period. Ekeland and Hofer [16] proved that if the Hamiltonian system is flat near an equilibrium and superquadratic near infinity, it has a periodic solution with minimal period. The estimate of number of periodic solutions was established in [15]. In contrast to differential equations, the research on periodic solutions with prescribed minimal period of higher order discrete systems is fresh and there are very few literature (see [7–12, 17–29]) on it. Comparing this paper with references [8–11], the advantages and differences of this paper are that the existence of periodic solutions with prescribed minimal period of (1) is obtained in this paper; however, only periodic solutions are obtained in the references [8–11]. Given integer T≥2, Long [23] considered the following T-cycle discrete Hamiltonian systems:(11)JΔxk=∇Hx,Lxk.By making use of minimax theory and geometrical index theory, some results on the existence and multiplicity of subharmonic solutions with prescribed minimal period to the abovementioned discrete Hamiltonian systems are obtained. Yu et al. [27] in 2004 obtained some sufficient conditions on the existence of subharmonic solutions with prescribed minimal period for the second-order difference equation by using variational methods. Therefore, there is still spacious room to explore the periodic solutions with prescribed minimal period of higher order discrete systems. Motivated by the papers [9, 12], for a given integer s with s>1, the aim of this paper is to obtain some new results for the existence of at least two nontrivial periodic solutions with minimal period sM to a 2nth-order discrete system by using critical point method.Here, we give the existence results of at least two nontrivial periodic solutions with minimal periodsM as follows.Theorem 1.
Suppose thatGt,x satisfies the following assumptions:G1 There are constants δ1>0 and ρ∈0,4sin2π/sMn/2 such that(12)Gt,x≤ρx2,∀t∈ℤ,x≤δ1.G2 There are constants τ>0 and 〉∈4n/2,+∞ such that(13)Gt,x≥x2−τ,∀t,x∈ℝ2.G3 There are three constants κ>0 and C>D>0 such that(14)∂2Gt,x∂x2ξ,ξ≤Cξ2,∀t,x∈ℝ2,ξ∈ℝ,∂2Gt,x∂x2ξ,ξ≥Dξ2,∀x≤κ,s∈ℝ,ξ∈ℝ.G4 If ϖ is a rational number, x is a solution of (1) with a minimal period ϖM, and gt,x also has a minimal period ϖM, then ϖ must be an integer.G5 Denote by ms the least prime factor of s:(15)4sin2ωms2sn>C,4sin2ω2sn<D,∑k=1sMg2k,0<4πκ24sin2ωms/2sn−CD−4sin2ω/2snω.Then, (1) possesses at least two nontrivial periodic solutions with minimal period sM.Corollary 1.
Suppose thatGt,x satisfies G1−G4 and(16)4sin2ω2n>C,4sin2ω2sn<D.
If(17)∑k=1sMg2k,0<4πκ24sin2ω/2n−CDω,then there is S>0 such that for any prime integer s>S, (1) possesses at least two nontrivial periodic solutions with minimal period sM.Theorem 2.
Suppose thatGt,x satisfies G3−G5 and the following assumptions:G6limx⟶+∞Gt,x/x2=0 uniformly for t∈ℝ.G7 There are constants δ2>0 and θ>2 such that(18)0<θGt,x≤xgt,x,∀t∈ℤ,x≤δ2.Then, (1) has at least two nontrivial periodic solutions with minimal period sM.Theorem 3.
Suppose thatGt,x satisfies G1−G4 and the following assumptions. If(19)gk,0=0,∀k∈ℤ,4sin2ωms2sn>C,4sin2ω2sn<D,then (1) has at least two nontrivial periodic solutions with minimal period sM.Corollary 2.
Suppose thatGt,x satisfies G1−G4 and the following assumptions. If(20)gk,0=0,∀k∈ℤ,4sin2ω2n>C,4sin2ω2sn<D.then there is S>0 such that for any prime integer s>S, (1) possesses at least two nontrivial periodic solutions with minimal period sM.The remainder of this paper is organized as follows. In Section2, we build the variational functional and gather some basic notations that are necessary in the proofs of our main theorems. In Section 3, we state some useful lemmas. In Section 4, the main results will be proved.Regarding the basis for variational methods, we refer the reader to [30]. Regarding the basic knowledge of integral inequalities and extended hypergeometric functions, the reader is referred to [31].


## 2. Preliminaries

In this section, we shall establish the variational framework associated with (1) and gather some basic notations that are necessary in the proofs of our main theorems.Let the vector spaceB be defined by(21)B=x|xk+sM=xk,and for any x∈B, define the inner product(22)x,y=∑i=1sMxiyi,and the norm(23)x=∑i=1sMxi21/2.For anyx∈B, let I be the functional defined by(24)Ix=12∑k=1sMΔnxk−12−∑k=1sMGk,xk.Then,I is continuously differentiable and(25)∂I∂uk=−1nΔ2nxk−n−gk,xk,∀k∈ℤ1,sM.Thus,x is a critical point of Ix on B if and only if(26)Δ2nxk−n=−1ngk,xk,∀k∈ℤ1,sM.Therefore, we reduce the problem of findingsM-periodic solutions of (1) to that of seeking critical points of the functional I on B.Denote(27)L=2−10⋯0−1−12−1⋯000−12⋯00⋯⋯⋯⋯⋯⋯000⋯2−1−100⋯−12sM×sM.It is clear that the eigenvalues ofL are(28)λi=4sin2iπsM,i=0,1,2,…,sM−1.Furthermore,L is positively semidefinite and all of eigenvalues of L are positive except for 0, and(29)minλ1,λ2,…,λsM−1=4sin2πsM,maxλ1,λ2,…,λsM−1≤4.Obviously, 0 is an eigenvalue ofL and 1,1,…,1∗ is an eigenvector associated to 0. Let A=c,c,…,c∗∈B|c∈ℝ, then A is an invariant subspace of B. Denote B by B=A⊕B.The eigenvectors ofL corresponding to λi are defined by(30)αi=cos2iπsM,cos2iπ⋅2sM,…,cos2iπ⋅sMsM∗,i=1,2,…,sM−12,βi=sin2iπsM,sin2iπ⋅2sM,…,sin2iπ⋅sMsM∗,i=1,2,…,sM−12.Set(31)Y=spanαi,i=1,2,…,sM−12,Z=spanβi,i=1,2,…,sM−12.IfsM is odd, then B=A⊕Y⊕Z. For any x∈B and i∈ℤ,(32)xi=p+−1iq+∑k=1sM−1/2pkcosωksi+qksinωksi,where p,q,pk, and qk are constants.IfsM is even, then 4 is the eigenvalue of L. Let η denote the eigenvector corresponding to 4, and W=span η. We have B=A⊕Y⊕Z⊕W. For any x∈B and i∈ℤ,(33)xi=p+∑k=1sM−1/2pkcosωksi+qksinωksi,where p,pk, and qk are constants.Suppose thatB is a real Banach space and I∈C1B,ℝ. As usual, I is said to satisfy the Palais–Smale condition if every sequence xj⊂B, such that Ixj is bounded and I′xj⟶0,j⟶∞, has a convergent subsequence. The sequence xj is called a Palais–Smale sequence.


## 3. Some Lemmas

To apply critical point theory to study the existence of periodic solutions with minimal periodsM of (1), some lemmas should be stated in this section which will be used in proofs of our main results.Below, we denote byBrx the open ball centered at x∈B with radius r>0, B¯rx as its closure, and ∂Brx as its boundary.Lemma 1 (linking theorem [30]).
LetB be a real Banach space, B=B1⊕B2, where B1 is finite dimensional. Suppose that I∈C1B,ℝ satisfies the Palais–Smale condition and the following:I1 There are positive constants c and r such that I|∂Br0∩B2≥cI2 There is μ∈∂B10∩B2 and a positive constant c^≥r such that I|∂Ω≤0, where Ω=B¯c^0∩B1⊕tμ|0<t<c^
Then,I possesses a critical value c0≥c, where(34)c0=infd∈Υsupx∈ΩIdx,and Υ=d∈CΩ¯,B|d|∂Ω=id, where id denotes the identity operator.
Set(35)B˜=x∈B|x−k=xk,∀k∈ℤ.
We haveB˜=Z, then(36)xk=∑i=1sM−1/2qisinωisk,∀k∈ℤ.Lemma 2.
Suppose thatGt,x satisfies G1−G5. Then,(37)Ix=12∑k=1sMΔnxk−12−∑k=1sMGk,xk,is bounded from above in B.Proof.
For anyx∈B, by G2, we have(38)Ix=12∑k=1sMΔnxk−1,Δnxk−1−∑k=1sMGk,xk=12∑k=1sMΔnxk,Δnxk−∑k=1sMGk,xk=12y∗Ly−∑k=1sMGk,xk≤12×4y2−∑k=1sMϱx2−τ=2y2−ϱx2+sMτ,where y=Δn−1x1,Δn−1x2,…,Δn−1xsM∗. Since(39)y2=∑k=1sMΔn−2xk+1−Δn−2xk2≤4∑k=1sMΔn−2xk2≤⋯≤4n−1x2,then(40)Ix≤124n−2ϱx2+sMτ≤sMτ.
The proof is finished.Lemma 3.
Suppose thatGt,x satisfies G1−G5. Then,(41)Ix=12∑k=1sMΔnxk−12−∑k=1sMGk,xk,satisfies the Palais–Smale condition.Proof.
Assume thatIxj is a bounded sequence from the lower bound. Then, there is a positive constant c1>0 such that(42)−c1≤Ixj,∀j∈ℕ.
The proof of Lemma2 implies that(43)−c1≤Ixj≤4n−2ϱxj2+sMτ,∀j∈ℕ.
Therefore,(44)xj2≤c1+sMτ2ϱ−4n.
It comes fromϱ>4n/2 that we can find a positive constant c2 such that for any j∈ℕ, xj≤c2. As a consequence of this, we know that the sequence xjj∈ℕ is a bounded in the finite-dimensional space B. Thus, it has a convergent subsequence. The Palais–Smale condition is verified.Lemma 4.
Suppose thatx is a critical point of Ix on B˜. Then, x is a critical point of Ix on B.The proof of Lemma4 is similar to the proof of Lemma 2.2 in [12]. For simplicity, we omit its proof.Let(45)Θϖ=−s24sin2ωϖ/2sn−C∑k=1sMg2k,0.Lemma 5.
Suppose thatGt,x satisfies G1−G5 and Ix<Θms. If x is a critical point of Ix on B˜, then x has a minimal period sM.Proof.
Suppose, for the sake of contradiction, thatx exists a minimal period sM/ϖ. In view of the condition G4, we have ϖ≥ms.
Similarly,xi can be written in the form of(46)xk=∑i=1sM−ϖ/2ϕqisinωϖisk,∀x∈B˜.
Thus,(47)Ix=12y∗Ly−∑k=1sMGk,xk≥2sin2ωϖ2sy2−∑k=1sMGk,xk,where y=Δx1,Δx2,…,ΔxsM∗. It is obvious that(48)y2=∑k=1sMΔn−2xk+1−Δn−2xk2≥4sin2ωϖ2s∑k=1sMΔn−2xk2≥⋯≥4sin2ωϖ2sn−1x2.
Therefore, byG5,(49)Ix≥124sin2ωϖ2snx2−∑k=1sMg2k,01/2x−C2x2≥−124sin2ωϖ/2sn−C∑k=1sMg2k,0≥−s24sin2ωϖ/2sn−C∑k=1sMg2k,0=Θϖ≥Θms.
That is,Ix≥Θms which is a contradiction with the condition Ix<Θms, and the proof of Lemma 5 is now complete.


## 4. Proofs of the Main Theorems

In this section, the proofs of Theorems1–3 and Corollaries 1 and 2 are given by using the critical point theory.Proof.
of Theorem1. In view of Lemma 2, Ix is bounded from above in B.
Set(50)m¯=supx∈BIx.
Thus, there is a sequencexi on B such that(51)m¯=limi⟶∞Ixi.
In addition, by the proof of Lemma2, for any x∈B,(52)Ix≤124n−2ϱx2+sMτ≤sMτ.
Therefore,limx⟶+∞Ix=−∞. This means that xi is bounded. Consequently, xi has a convergent subsequence. We define it as xik. Denote(53)x¯=limk⟶+∞xik.
On account of the continuity ofIx in x, there must be a point x¯∈B, Ix¯=m¯. Obviously, x¯∈B is a critical point of Ix.
For anyx∈B,x≤δ1, from the condition G1,(54)Ix=12y∗Ly−∑k=1sMGk,xk≥12y∗Ly−ρ∑k=1sMxk2≥2sin2πsMy2−ρx2,where y=Δx1,Δx2,…,ΔxsM∗. It is easy to see that(55)y2=∑k=1sMΔn−2xk+1−Δn−2xk2≥4sin2πsM∑k=1sMΔn−2xk2≥⋯≥4sin2πsMn−1x2.
Therefore,(56)Ix≥4sin2π/sMn2−ρx2.
Takec=4sin2π/sMn/2−ρδ12. Thus, for any x∈∂Bδ10∩B,(57)Ix≥c.
Consequently,(58)m¯=supx∈BIx≥c>0.
Furthermore, there are positive constantsc>0 and δ1>0 such that for any x∈∂Bδ10∩B,(59)Ix≥c.
For anyx∈A, note that ∑k=1sMΔnxk−1,Δnxk−1=0, we have(60)Ix=12∑k=1sMΔnxk−1,Δnxk−1−∑k=1sMGk,xk=−∑k=1sMGk,xk≤0.
Thus,x¯∉A and the critical point x¯ of Ix corresponding to the critical value m¯ is a nontrivial periodic solution of (1) with period sM.
Chooseμ∈∂B10∩B. Denote(61)x=tμ+ν,∀ν∈B,t∈ℝ.
We have(62)Ix=12∑k=1sMΔnxk−1,Δnxk−1−∑k=1sMGk,xk≤t22∑k=1sMΔnμk−1,Δnμk−1−∑k=1sMGk,tμk+νk≤t22z∗Lz−∑k=1sMϱtμk+νk2−τ≤2t2z2−ϱ∑k=1sMtμk+νk2+sMτ=2t2z2−ϱt2−ϱν2+sMτ,where z=Δμ1,Δμ2,…,ΔμsM∗. Since(63)z2=∑k=1sMΔn−2μk+1−Δn−2μk2≤4∑k=1sMΔn−2μk2≤⋯≤4n−1,then(64)Ix≤4n2−ϱt2−ϱν2+sMτ≤−ϱν2+sMτ.
Accordingly, there exists a positive numberς>δ1 such that(65)Ix≤0,∀x∈∂Ω,where Ω=B¯ς0∩B1⊕tμ|0<t<ς. Applying the linking theorem, Ix has a critical value c0≥c>0, where(66)c0=infd∈Υsupx∈ΩIdx,and Υ=d∈CΩ¯,B|d|∂Ω=id.
Similar to the Proof of Theorem 1 in [22], we can prove that (1) has at least two sM-periodic nontrivial solutions and so we omit it. By Lemma 5, it suffices to prove that(67)Ix<Θms,∀x∈B˜.
FromG3,(68)Gk,x=gk,0x+12×∂2Gk,φx∂x2x2≥gk,0x+D2x2,∀x≤κ.
Thus,(69)Ix=12∑k=1sMΔ2xk,Δ2xk−∑n=1sMGk,xk≤12∑k=1sMΔ2xk,Δ2xk−D2∑k=1sMxk2−∑k=1sMgk,0xk,∀x≤κ.
Choose(70)xk=κsinωks.
It comes fromg−k,0=gk,0 and gk+M,0=gk,0 that(71)gk,0=∑i=1M−1/2pisin2iπMk=∑i=1M−1/2pisin2iπsMsk,where pi is a constant. In addition,(72)∑k=1sMgk,0xk=∑i=1M−1/2κai∑k=1sMsin2iπsMsk⋅sin2πsMk=0.
Consequently,(73)Ix≤24sin2ω2sn−Dx2.
It is easy to see that(74)x=κsπω1/2.
Thus,(75)Ix=24sin2ω/2sn−Dκ2sπω<Θms.
The Proof of Theorem1 is completed.Proof.
of Corollary1. Since s is a prime integer and s>0, it is easy to see that ms=s. Therefore,(76)∑k=1sMg2k,0<4πκ24sin2ω/2n−CD−4sin2ω/2snω.
In virtue of Theorem1, the conclusion of Corollary 1 is obtained. The proof of Corollary 1 is fulfilled.Proof of Theorem2.
In fact, it is evident that conditionG6 implies G1 and condition G7 implies G2. As a result of Theorem 1, Theorem 2 holds. The result of Theorem 2 is achieved.Remark 1.
Similar to the proof of Theorem1, we can also prove that Theorem 3 is right. For simplicity, we omit its proof. Thanks to Theorem 3, the conclusion of Corollary 2 is obviously accomplished.Remark 2.
A real example is given by using the main results of this paper.
Forn=2ands=4, assume that(77)Δ4xk−2=16xk2sin2kπ5xk,k∈ℤ.
We have(78)ω=2π5,ms=2,M=5,gt,x=16x2sin2tπ5x.
It is easy to verify that all the assumptions of Theorem1 are satisfied. Consequently, (77) possesses at least two nontrivial periodic solutions with minimal period (20).



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