We study a higher order difference equation. By Lyapunov-Schmidt reduction methods and computations of critical groups, we prove that the equation has four M-periodic solutions.

1. Introduction

Considering the following higher order difference equation
(1)∑i=0kai(xn-i+xn+i)+f(n,xn)=0,n∈Z,
where k∈N, N and Z are the sets of all positive integers and integers, respectively, f∈C1(R×R,R), R is the set of all real numbers, and there exists a positive integer M such that, for any (t,z)∈(R×R), f(t+M,Z)=f(t,Z), F(t,z)=∫0zf(t,s)ds.

Throughout this paper, for a,b∈Z, we define Z(a):={a,a+1,…}, Z(a,b):={a,a+1,…,b} when a≤b.

When k=1, a0=-1, a1=1, (1) can be reduced to the following second order difference equation:
(2)Δ2xn-1+f(n,xn)=0,n∈Z.
Equation (2) can be seen as an analogue discrete form of the following second order differential equation:
(3)d2xdt2+f(t,x)=0.

In recent years, much attention has been given to second order Hamiltonian systems and elliptic boundary value problems by a number of authors; see [1–3] and references therein. On one hand, there have been many approaches to study periodic solutions of differential equations or difference equations, such as critical point theory (which includes the minimax theory, the Kaplan-Yorke method, and Morse theory), fixed point theory, and coincidence theory; see, for example, [4–20].

Among these approaches, Morse theory is an important tool to deal with such problems. However, there are, at present, only a few papers dealing with higher order difference equation except [21–23]. On the other hand, under some assumptions, the functional f may not satisfy the Palasis-Smale condition. Thus, we cannot apply the Morse theory to f directly. To go around this difficulty, Tang and Wu [24] and Liu [25] obtain many interesting results of elliptic boundary value problems by combining Morse theory with Lyapunov-Schmidt reduction method or minimax principle. Inspired by this, we study the existence of periodic solutions of a higher order difference equation (1) by combining computations of critical groups with Lyapunov-Schmidt reduction method, and an existence theorem on multiple periodic solutions for such an equation is obtained.

For a given integer M>0, let
(4)λj=-2∑s=0kascos2sπMj,j=1,…,M.

We denote p1=M/2 when M is even, or p1=(M+1)/2 when M is odd. Because of λM-j=λj, j∈Z(1,M), then, λj,j∈Z(1,M) has p1 different values. Therefore, we can write these numbers in such a way:
(5)λ1<λ2<⋯<λp1.
Assume λmin=min{λj,λj≠0,j=1,…,p1}, λmax=max{λj,λj≠0,j=1,…,p1}.

Combing Morse theory with Lyapunov-Schmidt reduction method, we have the following results.

Theorem 1.

Suppose that M≥2k+1, a0+∑s=1k|as|<0, and f(t,z)=f(z); we assume that

f(z)∈C1(R,R), f(0)=0,f′(0)<λmin<f∞=λm≤λmax, m∈N(1,p1), where f∞=lim|z|→∞f(z)/z;

there exists a constant γ≥λ1 such that f′(z)≤γ<λm+1;

for any t∈Z,
(6)F(z)-12λm|z|2⟶+∞,as|z|⟶∞.

Then (1) possesses at least four nontrivial M-periodic solutions.

This paper is divided into four parts. Section 2 presents variational structure. In Section 3, we present some propositions. The proof of Theorem 1 is given in Section 4.

2. Preliminaries

To apply Morse theory to study the existence of periodic solutions of (1), we will construct suitable variational structure.

Let S be the set of sequences x={xn}n=-∞+∞, where xn∈R. For any x,y∈S and a,b∈R, ax+by is defined by
(7)ax+bx:={axn+bxn}.
Then S is a vector space.

For any given positive integer M, EM is defined as a subspace of S by
(8)EM={x={xn}∈S∣xn+M=xn,n∈Z}.

EM can be equipped with inner product 〈·,·〉EM and norm ∥·∥EM as follows:
(9)〈x,y〉M=∑j=1Mxj·yj,∀x,y∈EM,∥x∥EM=(∑j=1Mxj2)1/2,∀x∈EM,
where |·| denotes the Euclidean Norm in RM, and xn·yn denotes the usual scalar product in R.

Define a linear map L:EM→RM by
(10)Lx=(x1,…,xM)T.

It is easy to see that the map L defined in (10) is a linear homeomorphism with ∥x∥EM=|Lx| and (EM,〈…,…〉)EM is a finite dimensional Hilbert space, which can be identified with RM.

For (1), we consider the functional I defined on EM by
(11)I(x)=-12∑n=1M∑i=0kai(xn-i+xn+i)xn-∑n=1MF(n,xn),∀x∈EM,
where xn+M=xn, ∀x∈EM, F(t,z)=∫0zf(t,s)ds.

Since EM is linearly homeomorphic to RM, by the continuity of f(t,z), I can be viewed as continuously differentiable functional defined on a finite dimensional Hilbert space. That is, I∈C1(EM,R). If we define x0:=xM, then
(12)∂I(x)∂xn=-[∑i=0kai(xn-i+xn+i)+f(n,xn)],
where n∈Z(1,M). Therefore, x∈EM is a critical point of I; that is, I′(x)=0 if and only if
(13)∑i=0kai(xn-i+xn+i)+f(n,xn)=0,n∈Z(1,M).

On the other hand, {xn}∈EM is M-periodic in n, and f(t,z) is M-periodic in t; hence, x∈EM is a critical point of I if and only if ∑i=0kai(xn-i+xn+i)+f(n,xn)=0 for any n∈Z, and x={xn} is a M-periodic solution of (1). Thus, we reduce the problem of finding M-periodic solutions of (1) to that of seeking critical points of the functional I in EM.

Apparently, I(x)∈C2(EM,R). Consider
(14)(I′(x),v)=-12∑n=1M∑i=1kai[(xn-i+xn+i)vn+(vn-i+vn+i)xn]-∑n=1Mf(n,xn)vn,(I′′(x)v,w)=-12∑n=1M∑i=1kai[(wn-i+wn+i)vn+(vn-i+vn+i)wn]-∑n=1Mf′(n,xn)vnwn,
for all x,v,w∈EM. For convenience, we write x∈EM as x=(x1,x2,…,xM)T.

In view of xn+M=xn, ∀x=(x1,x2,…,xM)T∈EM, n∈Z, when M≥2k+1, I can be rewritten as
(15)I(x)=12xTAx-∑n=1MF(n,xn),

Let the eigenvalues of A be λ1′,λ1′,…,λM′, and let A be a circulant matrix [18] denoted by
(17)A≝Circ{-2a0,-a1,-a2,…,-ak,0,…,0,-ak,-ak-1,…,-a2,-a1}.
By [18], the eigenvalues of A are
(18)λj′=-2a0-∑s=1kas{expi2jπM}s-∑s=1kas{expi2jπM}M-s=-2∑s=0kascos(2jsπM),
where j=1,…,M.

According to (18), for any positive integer M with M≥2k+1, we know that.

If a0+∑s=1k|as|<0, then λj′>0 (j=1,2,…,M). That is, the matrix A is positive definite.

Comparing (18) with (4), we know that λj′=λj (j=1,…,M), then, the matrix A has p1 different eigenvalues denoted in such a way:
(19)λ1<λ2<⋯<λp1.

3. Main Propositions

In order to prove our main results, we will give several propositions and notations as follows.

Definition 2 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let X be a Banach space, let J∈C1(X,R), and let Hq(A,B) be the qth singular relative homology group of the topological pair (A,B) with coefficients in an Abelian group G. βq=rankHq(A,B) is called the q-dimension Betti number. Let u be an isolated critical point of J with J(u)=c, c∈R, and let U be a neighborhood of u0 in which J has no critical points except u0. Then the group
(20)Cq(J,u0):=Hq(Jc⋂U,Jc⋂U∖{u0}),q=0,1,2,…
is called the qth critical group of J at u, here Jc=J-1(-∞,c]. Assume that J satisfies PS condition; J has no critical value less than α∈R; then the qth critical group at infinity of J is defined as
(21)Cq(J,∞):=Hq(X,Ja),q=0,1,2,….

If J′′(u0)=0, then the Morse index of J at u0 is defined as the dimension of the maximal subspace of X on which the quadratic form (J′′(u0)v,v) is negative definite. Define Kc={u∈X:J′(u)=0,J(u)=c}. We need the following condition.

(A) Suppose that a<b are two regular values of J; J has at most finitely many critical points on J-1[a,b] and the rank of the critical group for every critical point is finite.

Definition 3 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Assume that J satisfies condition (A); c1<c2<⋯<cm are all critical values of J in [a,b] and Kci={z1i,z2i,…,znii}, i=1,2,…,m. Choose 0<ϵ<min{c1-a,c2-c1,…,cm-cm-1,b-cm}. Define
(22)Mq=Mq(a,b)=∑i=1mrankHq(Jci+ϵ,Jci-ϵ)=∑i=1m∑j=1nirankCq(J,zji),q=0,1,…

Then Mq is called the qth Morse-type number of J about the interval [a,b].

Here the critical groups of J at an isolated critical point u describe the local behavior of J near u, while the critical groups of J at infinity describe the global property of J. The Morse inequality gives the relation between them.

Proposition 4 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Suppose that J∈C1(X,R) satisfies the PS condition and has only isolated critical points, and the critical values of f are bounded below. Then we have
(23)∑q=0∞Mqtq=∑q=0∞βqtq+(1+t)Q(t),
where Mq=∑J′(u)=0rankCq(J,u), βq=rankCq(J,∞); Q is a formal series with nonnegative integer coefficients.

Now we recall the Lyapunov-Schmidt reduction method.

Proposition 5 (see [<xref ref-type="bibr" rid="B5">5</xref>]).

Let X be a separable Hilbert space with inner product 〈u,v〉 and norm ∥u∥ and let X-and X+ be closed subspaces of X such that X=X-⊕X+. Let J∈C1(X,R). If there is a real number β>0 such that, for all v∈X-, w1,w2∈X+, there holds
(24)〈∇f(v+w1)-∇J(v+w2),w1-w2〉≥β∥w1-w2∥2,
then we have the following:

there exists a continuous function ψ:X-→X+ such that(25)J(v+ψ(v))=minw∈X+J(v+w),

and ψ(v) is the unique member of X+ such that(26)〈∇J(v+ψ(v)),w〉=0,∀w∈X+;

the functional φ∈C1(X-,R) defined by φ(v)=J(v+ψ(v)) and(27)〈∇φ(v),v1〉=〈∇J(v+ψ(v)),v〉,∀v,v1∈X-;

an element v∈X- is a critical point of φ if and only if v+ψ(v) is a critical point of J.

Proposition 6 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

Assume that the assumptions of Proposition 5 hold, then at any isolated critical point v of φ we have
(28)Cq(φ,v)≅Cq(f,ψ(v)),q=0,1,2,…..

Proposition 7 (see [<xref ref-type="bibr" rid="B25">25</xref>]).

Assume that the assumptions of Proposition 5 hold, if there exists a compact mapping T:X→X such that, for any u∈X, we have ∇J(u)=u-T(u), then we have φ:
(29)ind(∇φ,v)=ind(∇J,v+ψ(v))
at any isolated critical point v of φ.

4. Proof of Theorem

Consider the following C1 functional:
(30)I(x)=12xTAx-∑n=1MF(n,xn).

As we know, the PS condition is an important part of critical point theory. However, under our assumptions (f1)–(f3), the functional I may not satisfy PS condition. Thus, we cannot apply the Morse theory directly. But the truncated functional I± does satisfy the PS condition. So we can obtain two critical points of I via mountain pass lemma; then we can obtain other critical points by combing Morse theory with Lyapunov-Schmidt reduction method.

At first, we consider the truncated problem
(31)∑i=1kai(xn-i+xn+i)+f+(xn)=0,n∈Z,k∈N,
where
(32)f+(z)={f(n,z),z≥0,0,z<0,(31)′∑i=1kai(xn-i+xn+i)+f-(xn)=0,n∈Z,k∈N,
where
(33)f-(z)={f(n,z),z≤0,0,z>0.

Then the functional I+:Z×R→R corresponding to (31) can be written as
(34)I+(x)=12xTAx-∑n=1MF+(xn),
where F+(n,z)=∫0zf+(n,s)ds. Apparently, I+∈C1.

The functional I-:Z×R→R corresponding to (31)′ can be written as
(35)I-(x)=12xTAx-∑n=1MF-(xn),
where F-(n,z)=∫0zf-(n,s)ds. Apparently, I-∈C1.

We only consider the case of I+; the case of I- is similar and omitted.

By (f1), we know that
(36)limz→-∞f+(z)z=0,limz→+∞f+(z)z=f∞.

Then there exist real number ϵ>0 (small enough) and Cϵ>0 such that
(37)f+(z)=f∞z+Cϵ,ifz⟶+∞.

Lemma 8.

Under the conditions of Theorem 1, the functional I+(x) satisfies the PS condition.

Proof.

Let {xq}∈EM be such a sequence; that is, there exists a positive constant M1 such that |I+(xq)|≤M1, ∀q∈N, and that |(I+′(xq),v)|→0 as q→+∞, ∀v∈EM.

That is, {xq}∈EM is a bounded sequence in the finite dimensional space EM. Consequently, it has a convergent subsequence. Thus, we obtain Lemma 8.

Let z+=max(z,0), z-=max(-z,0), and z=z+-z-.

Lemma 9.

If x∈EM is a local minimizer of I+, then x must be a local minimizer of I.

Proof.

Let x>0 be a local minimizer of I+; then for any sequence {xq}⊂EM, xq→x(q→∞), for big enough q, we have I(xq)≥I(x).

In fact,
(39)I(xq)-I(x)=I(xq)-I+(x)≥I(xq)-I+(xq)=∑n=1M[F+(xnq)-F(xnq)]=-∑n∈Z(1,M),xnq<0F(xnq).

Because xq→x, xnq=(xnq)+-(xnq)-, and xn=(xn)+-(xn)-, so (xnq)+→(xn)+=xn, -(xnq)-→0-.

For any n∈Z(1,M), if (xnq)-=0, then I(xq)=I(x).

If -(xnq)-→0-, by (f1), f(0)=0, and 0<f′(z)<γ, then f(z)<0 for z→0-. Therefore, -∑n∈Z(1,M),xnq<0F(xnq)>0; that is, I(xq)>I(x).

The proof of Lemma 9 is complete.

It is easy to see that the zero function 0 is a local minimizer of I+, and I+(sϕ1)→-∞ as s→+∞, where ϕ1 is a first eigenfunction corresponding to the first nonzero eigenvalue of A. Thus, by the mountain pass lemma we obtain a critical point x+ of I+. However, it is true that x+ is a critical point of I if x+ is a critical point of I+; then we deduce that x+ is a critical point of I with
(40)Cq(I,x+)≅δq,1G,x+>0inEM.
Similarly, we obtain another critical point x- of I and
(41)Cq(I,x-)≅δq,1G,x-<0inEM.

Next we will prove that I has two more nonzero critical points. We decompose EM=X-⊕X+ according to f∞=λm. We set
(42)X-=⨁i=1mKer(A-λiI),X+=⨁i=m+1MKer(A-λiI),EM=X-⨁X+.

Since f′(z)≤γ<λm+1, for any v∈X- and w1,w2∈X+, we have
(43)〈∇I(n,v+w1)-∇I(n,v+w2),w1-w2〉≥β∥w1-w2∥2,
where β=1-γλm+1-1. Then, by Proposition 5, there exist a continuous map ψ:X-→X+ and a C1-functional φ:X-→R such that
(44)φ(v)=I(v+ψ(v))=minw∈X+.

We need to show that φ has at least five critical points. Hence, we assume that φ has no critical value less than some α∈R.

Lemma 10.

Suppose that f∈C1(R,R) satisfies (f1)–(f3), then the functional φ is anticoercive.

Proof.

According to (f3), there exists R>0 such that
(45)12λmz2-F(z)≤0,|z|≥R.
Then, for any z∈R, we have
(46)12λmz2-F(z)≤T=max|z|≤R|12λmz2-F(z)|.
Assume that {vt}t=1∞ is a sequence in X- such that ∥vt∥→∞. Let ξt=vt/∥vt∥, then ∥ξt∥=1. Because of dimX-<∞, there exist some ξ∈X- such that, up to subsequence ∥ξt-ξ∥→0, ∥ξ∥=1.

In particular, ξ≠0, measΘ=meas{n∈Z[1,M]:ξn≠0}>0. For n∈Θ, |vnt|→∞. Hence, by (f3),
(47)∑n∈Θ(12λm∥vt∥2-F(vnt))⟶-∞,ast⟶∞.
By the above discussion, we have(48)φ(vt)≤I(vt)=-12∑n=1M∑i=1kai(vn-it+vn+it)vnt-∑n=1MF(vnt)≤12λm∥vt∥2-∑n=1MF(vnt)=∑n∈Θ[12λm∥vt∥2-F(vnt)]+∑n∈[1,M]∖Θ[12λm∥vt∥2-F(vnt)]≤∑n∈Θ[12λm∥vt∥2-F(vnt)]+MT⟶-∞.
This concludes the proof.

Because φ is anticoercive, we choose a<b<α and ρ>r>0 such that
(49)Aρ⊂φa⊂Ar⊂φb,
where Aρ={v∈X-:∥v∥≥ρ}. Since φ has no critical value in [a,b], H*(φb,φa)=0.

Thus, we have the following commutative diagram with exact rows:(50)where all the homomorphisms except ∂* are induced by inclusions. The exactness of rows implies that i*, k* are isomorphisms. Hence l*:Hq(X-,φa)→Hq(X-,Ar) is also an isomorphism, and we get
(51)Cq(φ,∞)=Hq(X-,φa)≅Hq(X-,Ar)=δq,mG.

Because the anticoercive functional φ is defined on the m-dimensional X-, it has a critical point v, with
(52)Cq(φ,v)≅δq,mG.

Let 0, v+, v- be the projection of 0, x+, x- in X-, respectively. Then they are all critical points of φ. By (11), (14), and Proposition 6, and 0 is a local minimizer of I, we have
(53)Cq(φ,v±)≅Cq(I,x±)≅δq,1Q,Cq(φ,0)≅Cq(I,0)≅δq,0Q.
If 0, v+, v-, v are the only critical points of φ, then by Proposition 4 with t=-1,
(54)(-1)0+2×(-1)1+(-1)m=(-1)m.
This is impossible. Thus φ has at least five critical points. So I also has five critical points, four of which are nonzero. Therefore, (1) has at least four nontrivial solutions. This completes the proof of Theorem 1.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research was supported by the Research Foundation of Education Bureau of Hunan Province, China (12C0632).

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