R E S E A R C H
Open Access
Periodic solutions of second order difference
systems with even potential
Huafeng Xiao
**Correspondence:
huafeng@gzhu.edu.cn School of Mathematics and Information Science, Guangzhou University, Guangzhou, China Key Laboratory of Mathematics and Interdisciplinary Science of Guangdong, Guangzhou, China
Abstract
In this article, we study the multiplicity and minimality of periodic solutions to difference systems, which are globally superquadratic or subquadratic. A new technique is given to detect the minimal period of periodic solutions to autonomous systems. Some weaker conditions than a globally subquadratic condition are obtained to guarantee the existence of periodic solutions with prescribed minimal period to autonomous system.
Keywords: difference systems; periodic solution; minimal period; superquadratic; subquadratic
1 Introduction
On one hand, difference equations have been widely used to describe real-life situations in computer science, economics, neural network, ecology, cybernetics,etc.On the other hand, they are also natural consequences of the discretization of differential equations. So it is worthwhile to explore this topic. Many tools are used to study the existence of all kinds of solutions to discrete systems. A powerful tool is critical point theory, which firstly was introduced by Guo and Yu in to study the existence of periodic solutions to a difference system (cf.[–]). Since then, the study of discrete dynamic systems has got considerable development. We refer to boundary value problems (cf.[, ]), periodic solutions (cf.[, ]), homoclinic orbits (cf.[, ]), and heteroclinic orbits (cf.[, ]).
As is well known, the minimal periodic problem is an important but difficult problem. As far as the author knows, the study of solutions with a prescribed minimal period began in . In that year, by estimating the energy of a variational functional, Yuet al.(cf. []) studied the existence of subharmonic solutions with a prescribed minimal period to a discrete forced pendulum equation. More recently, by making use of the Clark dual method, Bin (cf.[]), Long (cf.[]) and Longet al.(cf.[]) studied the existence and multiplicity of periodic solutions with a prescribed minimal period to difference systems. Because of the lack of methods, results in this field are scarce.
It is well known that the Nehari manifold has been introduced by Nehari in (cf.[, ]) and developed by Szulkin and Weth in (cf.[]). It has been used widely to study the existence of ground state solutions to ordinary differential systems, partial differential systems, and difference systems (cf.[–]). A ground state solution is a solution which possesses the minimal energy of all solutions. Since such a minimality can be used to prove
the minimal periods of solutions, it has been used to study the existence of period solutions with prescribed minimal period to ordinary differential equations (cf.[–]).
Motivated by the above references, in this paper, one attempts to make use of a Nehari manifold to study the multiplicity and minimality of periodic solutions to difference sys-tems. When the systems are globally superquadratic or subquadratic, by restricting our discussion to the Nehari manifold, firstly, we study the existence of multiple periodic so-lutions to nonautonomous systems; secondly, we study the existence of periodic soso-lutions with a prescribed minimal period to autonomous systems. Also, some subquadratic con-ditions, which are weaker than the globally subquadratic condition, are obtained to guar-antee the existence of periodic solutions with prescribed minimal period to autonomous system.
For convenience, we denote byN,Z,Rthe sets of all natural numbers, integers, and real numbers, respectively. Fora,b∈Zwitha≤b, defineZ[a,b] ={a,a+ , . . . ,b}. Form∈N, denote byRmthe Euclidean space with the usual inner product (·,·) and norm| · |.
Consider the nonautonomous difference system
x
n–+f(n,xn) = , n∈Z, ()
wherexn∈Rm,is a difference operator defined byxn=xn+–xnandxn=(xn).
Assume that:
(F) there exists an even functionF∈C(R×Rm,R)such thatf(t,z)is the gradient of
F(t,z)with respect toz,i.e.,
F(–t, –z) =F(t,z), f(t,z) =∇zF(t,z), ∀(t,z)∈R×Rm;
(F) there exists a positive integerT> such that
F(t+T,z) =F(t,z), ∀(t,z)∈R×Rm;
(F) there exists aα> such that
<αf(t,z),z≤f(t,z)z,z, ∀z∈Rm\ {},
wheref(t,·)denotes the Hermite matrix ofF(t,·); (F) there exists aβ∈(, )such that
<f(t,z)z,z≤βf(t,z),z, ∀z∈Rm\ {}.
Remark IfFsatisfies (F), (F), and (F), without loss of generality, we can assume that F(t, ) = for allt∈R. Otherwise, there exists a twice continuously differentiable function g(t) such thatF(t, ) =g(t). LetF(t,z) =F(t,z) –g(t). ThenF also satisfies (F), (F), and (F) withFreplaced byF. Similarly, ifFsatisfies (F), (F), and (F), we can assume that F(t, ) = for allt∈R.
Theorem Suppose that F satisfies(F), (F),and(F).Then()has at least m[(T– )/] – distinct pairs of different T -periodic solutions.
Theorem Suppose that F satisfies(F), (F),and(F).Then()has at least m[(T– )/] – distinct pairs of different T -periodic solutions.
Consider the autonomous difference system
x
n–+f(xn) = , n∈Z. ()
Assume thatFsatisfies
(F) there exists an even functionF∈C(Rm,R)such thatf(z)is the gradient ofF(z),
i.e.,
F(–z) =F(z), f(z) =∇zF(z), ∀z∈Rm;
(F) there exists aα> such that
<αf(z),z≤f(z)z,z, ∀z∈Rm\ {},
wheref(·)denotes the Hermite matrix ofF(·); (F) there exists aβ∈(, )such that
<f(z)z,z≤βf(z),z, ∀z∈Rm\ {}.
Remark IfFsatisfies (F) and (F) (respectively, (F) and (F)), without loss of gener-ality, we can assume thatF() = .
For an autonomous system, we have the following two results.
Theorem Suppose that F satisfies(F)and(F).Then,for any integer P> , ()possesses at least a periodic solution with minimal period P.
Theorem Suppose that F satisfies(F)and(F).Then,for any integer P> , ()possesses at least a periodic solution with minimal period P.
Now, one weakens the conditions (F) and (F). Assume thatFsatisfies the following conditions:
(F) there exists an even functionF∈C(Rm,R)such thatf(z)is the gradient ofF(z),
i.e.,
F(–z) =F(z), f(z) =∇zF(z), ∀z∈Rm;
(F) there exist constantsG> , <γ < such that for allx∈Rmand|x| ≥G,
(F) there exist constants <G< ,M> and <γ< such that for allx∈Rmand |x|<G,
F(z)≥M|z|γ.
Remark Assume thatFsatisfies (F) and (F). Then there exist some positive constants MandMsuch that, for allx∈Rm,
F(x)≤M|x|γ+M.
Theorem Suppose that F satisfies(F), (F), and(F). Then, for any integer P> , ()possesses at least a periodic solution with minimal period P.
The rest of this paper is divided into two parts. In Section , we study the multiplicity of periodic solutions to a nonautonomous system. In Section , firstly, we study the existence of periodic solutions with a prescribed minimal period to an autonomous system, which is globally superquadratic or subquadratic; secondly, some conditions, which are weaker than the globally subquadratic condition, are given to guarantee the existence of periodic solutions with prescribed minimal period to autonomous system.
2 Nonautonomous difference system 2.1 Preparations
By a similar argument to [], we can build the spaceET
ET=
x= (. . . ,x–n, . . . ,x–,x,x, . . . ,xn, . . .)|xn+T=xn,xn∈Rm,∀n∈Z
.
ETis a Hilbert space equipping with the following norm and inner product
x :=
It is easy to check thatETis linearly homeomorphic toRmT, which can also be identified
withRmT.
on it are equivalent. Hence, there exists aC,s> such that
C,s
x s≤ x ≤C,s x s, ∀x∈ET.
The variational functional corresponding to () defined onETis
where x= (xτ
,xτ, . . . ,xτT)τ, A=P–CP, C =diag(B,B, . . . ,B)mT×mT. Here τ denotes the
transposition of a vector andB,Pare matrices defined in [].
SinceETis linearly homeomorphic toRmT, it follows from (F) thatJcan be viewed as a
twice continuously differentiable functional defined on a finite-dimensional Hilbert space. Thus, for anyx,y,z∈ET, one has
It is easy to check that the critical points ofJareT-periodic solutions of (). Similarly to reference [], the eigenvalues ofBcan be given by
Now, we define a subspace ofETas follows:
ET={x∈ET|xnis odd inn}.
Obviously,ET=W. Thusdim(ET) =m(T– )/. For anyx∈ET, no matterTis even or
odd,xnhas a Fourier expansion
xn=
T–
k=
aksin
knπ
T .
Consider the following eigenvalue problem:
Ax=λx, x∈ET. ()
Problem () has(T– )/eigenvalues, each of them of multiplicitym. The eigenvalues of () areλk= sin(kπ/T), wherek= , , . . . ,(T– )/. Obviously, the smallest eigenvalue
isλand the largest eigenvalue isλ(T–)/, which is denoted byλmax. Now we give a useful lemma. Since the proof is standard, we omit it.
Lemma If x is a critical point of J restricted onET,then x is also a critical point of J
on ET.
At the end of this subsection, two useful lemmas are given.
Suppose thatHbe a real Banach space andMis a closed symmetricC-submanifold of Hwith /∈M. Suppose thatφ∈C(M,R).
Lemma [] Suppose thatφis even and bounded below.Define
cj:= inf A∈j
sup
x∈A
φ(x), j= , , . . . ,
wherej:={A⊂M:A= –A,A⊂H\ {},A is compact andγ(A)≥j}.Ifk=∅for some
k≥and ifφsatisfies the(PS)c for all c=cj,j= , , . . . ,k,thenφhas at least k distinct
pairs of critical points.
Lemma [] Ifφ is bounded below and satisfies the(PS)condition,then c:=infMφ is
attained and is a critical value ofφ.
2.2 Superquadratic case
In this subsection, we consider the existence of multiple periodic solutions of (), whereF satisfies (F), (F), and (F). Arguing similarly to [], we can prove the following lemma.
Lemma If F satisfies(F), (F),and(F),then < ( +α)F(t,x)≤(f(t,x),x)for all x∈ Rm\ {}.Also,
F(t,x)≤M|x|α+, when|x| ≤, F(t,x)≥M|x|α+ when|x| ≥,
Leth(x) =J(x),x. Define a Nehari manifoldMonETas follows:
M=x∈ET\ {} |h(x) =
.
By a similar argument to reference [], one can prove the following lemma.
Lemma Mis C-manifold with dimension m(T– )/– .If xis a critical point of J
restricted onM,then xis also a critical point of J restricted onET.
By Lemmas and , to search for periodic solutions of (), we need find critical points ofJrestricted onM.
Lemma Fixing x∈ET\ {},there exists a unique tx> such that txx∈M.
implies that there exists atx> such that
ϕx(tx) = . ()
Claim: There exists a uniquetx> satisfying ().
Suppose, to the opposite, that there exist <t<tsatisfying (). By a direct
Fori= , , sinceϕx(ti) = , thentix,Ax=
Lemma J satisfies the(PS)condition onM.
Proof Assume that{xk} ⊂Mis a (PS) sequence forJ. Then there exists aM
SinceFis continuous, there exists aM> such that
M|x|+α–F(t,x)≤M, ∀t∈[,T],x∈Rmwith|x| ≤. ETis a finite-dimensional space, there exists a convergent subsequence of{xk}.
Define a new map
g:S→M,
x→txx.
It follows from Lemma thatgis a bijection whose inverseg–is given byg–(x) =x/ x .
Proof For anyx∈M, sinceJ(x) =supt∈(,+∞)J(tx) =supt∈(,+∞)J(tx/ x ), it follows that
inf
x∈MJ(x) =xinf∈Mt∈(,+∞)sup J(tx) =xinf∈S sup
t∈(,+∞)
J(tx).
To prove thatC> , one only need to show thatinfx∈Ssupt∈(,+∞)J(tx) > .
Arguing similarly to Lemma , there exists at> , which is independent ofx, such that
ϕx(t)≥λt/, ∀ <t<t,∀x∈S.
Settingt=t/, one gets
J
t
x
=ϕx
t
≥λt
> , ∀x∈S
.
Thus
C=inf
x∈St∈(,∞)sup J(tx)≥ λt
> .
Proof of Theorem Because of (F),Mis a closed symmetric manifold and /∈M. It follows from Lemma thatMis aCmanifold with dimensionm(T– )/– . By
Lem-mas and ,Jis bounded from below and satisfies the (PS) condition. It is easy to check thatJ is even. Then Lemma implies thatJhas at leastm(T – )/– distinct pairs of critical points. Thus () possesses at leastm(T – )/– distinct pairs of periodic
solutions.
2.3 Subquadratic case
In this subsection, we consider the multiplicity of periodic solutions of (), whereFsatisfies (F), (F), and (F). In order to prove Theorem , we consider the functionalJ= –Jand the
Nehari manifoldMis defined asM:={x∈ET\ {} | J(x),x= }. Since the technique
of the proof of Theorem is just the same as that of Theorem , where J andMare replaced byJandM, we omit it here.
3 Autonomous difference equations 3.1 Variational framework
In this section, we consider the existence of periodic solutions with any prescribed mini-mal period of (), that is, for any given positive integerP, we search for periodic solutions of () with minimal periodP.
The argument of this subsection is similar to that in Section .. Let
EP=
x={. . . ,x–n, . . . ,x–,x,x, . . . ,xn, . . .} |xn+P=xn,xn∈Rm,n∈Z
,
and equipEPwith the inner product·,·Pas follows:
x,yP= P
n=
ThenEPis a Hilbert space, which is homeomorphic toRmP. Denote by · Pthe norm
introduced by·,·P.
Define theLsnorm onE Pby
x Ls=
P
n= |x|s
s .
ThenLsis equivalent to · Pand there existsC,s> such that
C,s
x Ls≤ x P≤C,s x Ls, ∀x∈EP.
The variational functional corresponding to () defined onEPis
I(x) =
P
n=
|xn|
–F(x
n)
=
Dx,xP–
P
n=
F(xn),
where #Dis a matrix of ordermp. ThenIis twice continuously differentiable. The critical points ofIareP-periodic solutions of ().
Define a subspace ofEPas follows:
EP={xn∈EP|xnis odd inn}.
Then the eigenvalue problem
Dx=λx, x∈EP,
has(P– )/solutions, each of them of multiplicitym. The smallest eigenvalue isλ=
sin(π/P) and the largest eigenvalue isλ(P–)/, which is denoted byλmax.
Lemma If x is a critical point of I restricted onEP,then x is also a critical point of I on
the whole space EP.
At the end of this subsection, we define a Nehari manifoldMonEPas follows:
M=x∈EP\ {} |
I(x),xP= .
3.2 Superquadratic case
In this subsection, we assume thatFsatisfies (F) and (F). The main target of this subsec-tion is to prove Theorem . A similar argument to Secsubsec-tion ., one can check the following facts:
(i) < ( +α)F(x)≤(f(x),x)for allx∈Rm\ {}. Also,
F(x)≤M|x|α+, when|x| ≤, F(x)≥M|x|α+ when|x| ≥,
whereM=max|x|=F(x)andM=min|x|=F(x);
(iii) critical points ofIrestricted onMare also critical points ofIrestricted onEP;
(iv) for anyx∈EP\ {}, there exists a uniquetxsuch thattxx∈Mand
I(txx) =maxt∈(,∞)I(tx);
(v) Irestricted onMsatisfies the (PS) condition;
(vi) Irestricted onMis bounded from below andC=infx∈MI(x) > .
Proof of Theorem It follows from Lemma thatC is a critical value. Denote byxthe
critical point ofIcorresponding toC. Thenxis aP-periodic solution of (). It is easy to
check thatxis a nonconstant periodic solution. Claim:xhasPas its minimal period.
Suppose, to the opposite, that there exists a positive integerk≥ such thatxhasP/kas its minimal period. Definey={yl|l∈Z[,P]}as follows:
In this subsection, we assume thatFsatisfies (F) and (F). A similar argument to Sec-tion ., we can prove the following facts:
(I) < (f(x),x)≤( +β)F(x)for allx∈Rm\ {}. Also,
F(x)≥M|x|β+, when|x| ≤, F(x)≤M|x|β+, when|x| ≥,
whereM,Mare defined in Section .; (II) Mis aCmanifold;
(III) critical points ofIrestricted onMare critical points ofIrestricted onEP;
(IV) for anyx∈EP\ {}, there exists a uniquetxsuch thattxx∈Mand
I(txx) =mint∈(,∞)I(tx);
Lemma Assume that F satisfies(F)and(F).Then I restricted onEPis bounded from
Because of Lemma , I restricted on M is bounded from below. Denote by C= infx∈MI(x). A similar argument to Lemma , we can prove thatC< .
Proof of Theorem It follows from Lemma thatCis a critical value. Denote byxthe
critical point ofIcorresponding toC. Thenxis a nonconstantP-periodic solution of ().
By a similar discussion to the proof of Theorem , we can prove thatxhasPas its minimal
period.
3.4 Subquadratic case (II)
In this subsection, we assume thatFsatisfies (F), (F), and (F). By a similar argument to Lemma , we have the following lemma.
Lemma Assume that F satisfies(F), (F),and(F).Then I restricted onEPis bounded
Lemma I restricted onEPsatisfies the(PS)condition.
SinceETis a finite-dimensional space, there exists a convergent subsequence of{xk}.
Proof of Theorem Let{xk} ⊂E
Pbe a minimal sequence ofI, that is,I(xk)→Cask→ ∞.
By the Ekeland variational principle,I(xk)→ ask→ ∞. SinceIsatisfies the (PS)
con-dition,Cis a critical point ofIrestricted onEP. Denote byxthe critical point ofI
corre-sponding to critical valueC. Thenxis a periodic solution of (). It is easy to check that
xis a nonconstantP-periodic solution. By a similar discussion to the proof of Theorem , we can prove thatxhasPas its minimal period.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This project is supported by NSFC (No. 11126063).
Received: 24 March 2014 Accepted: 11 July 2014 Published:04 Aug 2014
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