# Homoclinic Orbits for a Class of Noncoercive Discrete Hamiltonian Systems

Authors: Long Yuhua
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

A class of first-order noncoercive discrete Hamiltonian systems are considered. Based on a generalized mountain pass theorem, some existence results of homoclinic orbits are obtained when the discrete Hamiltonian system is not periodical and need not satisfy the global Ambrosetti-Rabinowitz condition.


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## Body

## 1. Introduction

LetN, Z, and R denote the set of all natural numbers, integers, and real numbers, respectively. Throughout this paper, without special statement, |·| denotes the usual norm in RN with N∈N, u·v denotes the inner product of u∈RN and v∈RN.Consider the noncoercive discrete Hamiltonian systems(1.1)JΔx(t)-M(t)Sx(t)+v*×v*H′(t,v×v(Sx(t)))=0,t∈Z,
where v:RN→Rm(1≤m≤N) is a nontrivial linear operator, v* is its adjoint, v×v is the tensorial product of v, v:(v×v)(p,q)=(v(p),v(q))(1.2)M(t)=(0v*L(t)vv*L(t)v0),
with L(t) is an (N×N) symmetric matrix valued function and H:R×RN, (t,y)↦H(t,y) is a continuous function, differentiable with respect to the second variable with continuous derivative H'(t,y)=(∂H/∂y)(t,y). S is the shift operator defined as Sx(t)=(x1(t+1)x2(t)) and x(t)=(x1(t)x2(t)), where x1,x2∈RN. Δxi(t)=xi(t+1)-xi(t), i=1,2, is the forward difference operator. J is the standard symplectic matrix J=(0-ININ0), where IN is the identity matrix on RN.As usual, assuming that a solutionx(t)=0 an equilibrium for (1.1), we say that a solution x(t) is homoclinic to 0 if x(t) satisfies x(t)≠0, and the asymptotic condition x(t)→0 as |t|→+∞. Such solutions have been found in various models of continuous dynamical systems and frequently have tremendous effects on the dynamics of such nonlinear systems. So the homoclinic orbits have been extensively studied since the time of Poincaré, see [1–7] and references therein.In recent years, there has been much research activity concerning the theory of difference equations. To a large extent, this due to the realization that difference equations are important in applications. New applications that involve difference equations continue to arise with frequency in the modelling of computer science, economics, neural network, ecology, cybernetics, and so forth, we can refer to [8–13] for detail. Many scholars have investigated discrete Hamiltonian systems independently main for two reasons. The first one is that the behaviour of discrete Hamiltonian systems is sometimes sharply different from the behaviour of the corresponding continuous systems. The second one is that there is a fundamental relationship between solutions of continuous systems and the corresponding discrete systems by employing discrete variable methods (see [8] for detail).The general form of (1.1) is
(1.3)JΔx(t)-H′(t,x(t))=0,t∈Z,
which was studied by many scholars in various fields. By making use of minimax theory and geometrical index theory, [14] gave results on subharmonic solutions with prescribed minimal periods. When (1.3) are superquadratic systems, Guo and Yu [15] obtained some existence and multiplicity results by Z2 index theory and linking theorem. In [16], when H is subquadratic at infinity, the authors gave some existence results of periodic solutions. As to homoclinic orbits for discrete systems, [17–19] studied the second order discrete systems by critical point theory recently. While for the first order discrete systems, such as (1.1) or (1.3), to the authors' best knowledge, it seems there exists no similar results.Moreover, we may regard (1.1) as being a discrete analogue of Hamiltonian systems
(1.4)Jx˙(t)-M(t)x+H′(t,x(t))=0,t∈R.
Equation (1.1) is the best approximation of (1.4) when one lets the step size not be equal to 1 but the variable's step size go to zero, so solutions of (1.1) can give some desirable numerical features for (1.4). (1.4) is one form of classical Hamiltonian systems appearing in the study of various fields and many well-known results were given.In view of above reasons, the goal of this paper is to study the existence of homoclinic orbits for the first order discrete Hamiltonian system (1.1) when H satisfies superquadratic conditions and need not satisfy the global Ambrosetti-Rabinowitz (AR) condition:(AR): there exist two constantsμ>2 and r>0 such that for all t∈Z and x∈R2N, |x|≥r(1.5)0<μH(t,x)≤H′(t,x(t))·x.Letl(t) denotes the smallest eigenvalue of v*L(t)v, that is,
(1.6)l(t)=infξ∈RN,|ξ|=1v*L(t)v(ξ)·ξ,∀t∈Z.
For later use, we need the following assumptions:(
L
1
) there exists 1<γ<2 such that l(t)|t|γ-2→+∞ as |t|→+∞;(
H
1
)
H
(
t
,
y
)
/
|
y
|
2
→
+
∞ as |y|→+∞, t∈Z;(
H
2
)
|
H
'
(
t
,
y
)
|
/
|
y
|
→
0 as |y|→+0 and t∈Z;(
H
3
) there exist a>0 and α>1 such that
(1.7)|H′(t,y)|≤a(|y|α+1),∀t∈Z,∀y∈R2N;(
H
4
) there exist β>α, b>0 and r>0 such that
(1.8)H′(t,y)·y-2H(t,y)≥b|y|β,∀t∈Z,∀|y|≥r;(
H
5
) for all t∈Z and all y∈R2N(1.9)H′(t,y)y≥2H(t,y)≥0.Remark 1.1.
By assumption(H1) and (H2), we know that H(t,y) satisfies the superquadratic condition at both infinity and 0 respect to the second variable y.The rest of the paper is organized as follows. In Section2, we shall establish the variational structure for (1.1) and turn the problem of looking for homoclinic orbits for (1.1) to the problem for seeking critical points of the corresponding functional. In order to apply the generalized mountain pass theorem, we give some preliminary results in Section 3. In Section 4, we shall state our main result and complete the proof of our result.


## 2. Variational Structure

SetS={x={x(t)}∣x(t)=(x1(t)x2(t))∈R2N,xj(t)∈RN,j=1,2,t∈Z} is a space which is composed of the following vectors,
(2.1)x={x(t)}t∈Z={…,xT(-t),…,xT(-1),xT(0),xT(1),…,xT(t),…}.
Define the subspace X of S as
(2.2)X={x∈S∣∑t∈Z‍[(JΔSx(t-1),x(t))+(M(t)Sx(t),x(t))]<+∞}.
Denote u=v×v, u*=v*×v*, define another subspace E of X as follows:
(2.3)E={x∈X∣x(t)∈(Keru)⊥}.
The space E is a Hilbert space with the inner product
(2.4)〈x,y〉=∑t∈Z‍[(JΔSx(t-1),y(t))+(M(t)Sx(t),y(t))],∀x,y∈E,
and the norm introduced from the inner product as follows:
(2.5)∥x∥2=〈x,x〉=∑t∈Z‍[(JΔSx(t-1),x(t))+(M(t)Sx(t),x(t))],∀x∈E.Define a functionalF(x) on E as follows:
(2.6)F(x)=-12∑t∈Z‍[(JΔSx(t-1),x(t))+(M(t)Sx(t),x(t))]+∑t∈Z‍G(t,u(Sx(t-1))),
according to the definition of ∥x∥, F(x) can be written in another form as follows:
(2.7)F(x)=-12∥x∥2+∑t∈Z‍G(t,u(Sx(t-1))).
The functional F(x) is a well-defined C1 on E, and next we prove that the problem of looking for homoclinic orbits for (1.1) can be turned to the problem for seeking critical points of the corresponding functional F(x) (see (2.6) or (2.7)).Let(2.8)F1(x)=-12∑t∈Z‍(JΔSx(t-1),x(t))=-12∑t∈Z‍((0-ININ0)(Δx1(t)Δx2(t-1)),(x1(t)x2(t)))=-12∑t∈Z‍[(x2(t-1),x1(t))-(x2(t),x1(t))+(x1(t+1),x2(t))-(x1(t),x2(t))],
while
(2.9)∑t∈Z‍(x2(t-1),x1(t))-(x2(t),x1(t))=∑t∈Z‍(x2(t-1),x1(t))-∑t∈Z‍(x2(t),x1(t))=∑t∈Z‍(x2(t),x1(t+1))-(∑t∈Z(x2(t),x1(t)))=∑t∈Z‍(x2(t),Δx1(t)),
then
(2.10)F1(x)=-∑t∈Z‍(Δx1(t),x2(t)),
it follows that
(2.11)F(x)=-∑t∈Z‍(Δx1(t),x2(t))-12∑t∈Z‍(M(t)Sx(t),x(t))+∑t∈Z‍G(t,u(Sx(t-1))).
Write Fxi(t)′=∂F(x)/∂xi(t), i=1,2, for any given t∈Z, there holds
(2.12)Fx1(t)′=x2(t-1)-x2(t)-v*Lvx2(t-1)+u*Gx1(t)(t-1,u(Sx(t-1))),Fx2(t)′=x1(t+1)-x1(t)-v*Lvx1(t+1)+u*Gx2(t)(t,u(Sx(t))).
Then we can draw a conclusion that F′(x)=0 is true if and only if
(2.13)Fx1(t)′=0,Fx2(t)′=0,
so
(2.14)x2(t-1)-x2(t)-v*Lvx2(t-1)+u*Gx1(t)(t-1,u(Sx(t-1)))=0,x1(t+1)-x1(t)-v*Lvx1(t+1)+u*Gx2(t)(t,u(Sx(t)))=0,
which can be reformed as
(2.15)(0-ININ0)(Δx1(t)Δx2(t))-(0v*L(t)vv*L(t)v0)(x1(t+1)x2(t))+u*(Gx1(t)(t-1,u(Sx(t)))Gx2(t)(t,u(Sx(t))))=0,
that is
(2.16)JΔx(t)-M(t)Sx(t)+u*H′(t,u(Sx(t)))=0,
which is just (1.1). Therefore, we obtain the following lemma.Lemma 2.1.
x
=
{
x
(
t
)
}
≠
0 is a homoclinic orbit of (1.1) if and only if x is a critical point of functional F(x) in E.


## 3. Preliminary Results

In order to apply the critical point theory to look for critical points for (2.6), we give some lemmas which will be of fundamental importance in proving our main result.LetE be a real Hilbert space with the norm ∥·∥. Suppose that E has an orthogonal decomposition E=E1⊕E2 with both E1 and E2 being infinite dimensional. Suppose (νn) (resp., (ωn)) is an orthogonal basis for E1 (resp. E2), and set En1=span{ν1,ν2,…,νn}, En1=span{ω1,ω2,…,ωn}, En=En1⊕En2, and fn=f∣En, the restriction of f on En. We say that f satisfies the (PS)* condition if any sequence (x(n)) in E, (x(n))∈En such that fn(x(n))≤C a constant, and fn′(x(n))→0 possesses a convergent subsequence.We state a basic theorem introduced in [20] by Rabinowitz which is used to obtain the critical points of the functional F(x).Lemma 3.1 (Generalized mountain pass lemma).
Letf∈C1(E,R) satisfy
(
f
1
) the (PS)*condition;
(
f
2
) there are ρ,δ>0 such that
(3.1)f(x)≥δ,
for all x∈Sρ={x∈E2∣∥x∥=ρ};
(
f
3
) there are r>ρ, M>0, e∈E12 with ∥e∥=1 such that
(3.2)f∣∂Q≤0,f∣Q≤M,
where Q={(Br⋂‍E1)}⊕{se0≤s≤r}.
Thenf has a critical point x with f(x)≥δ.Next we consider the eigenvalue problem.(3.3)JΔSx(t-1)+M(t)Sx(t)=λx(t).
Equation (3.3) can be reformed as follows:
(3.4)x1(t+1)=(I+v*Lv)-1x1(t)+λ(I+v*Lv)-1x2(t),x2(t+1)=λ(v*Lv-I)-1(I+v*Lv)-1x1(t)+[λ2(v*Lv-I)-1(I+v*Lv)-1-I]x2(t),
Denote
(3.5)A(λ)=((I+v*Lv)-1λ(I+v*Lv)-1λ(v*Lv-I)-1(I+v*Lv)-1λ2(v*Lv-I)-1(I+v*Lv)-1-I),
then (3.3) can be expressed by the following:
(3.6)x(t+1)=A(λ)x(t).
Therefore a standard argument shows that σ(A), the spectrum of A, consists of eigenvalues numbered by, (counted in their multiplicities) the following:
(3.7)⋯λ-2≤λ-1≤0≤λ1≤λ2⋯,
with λk→±∞ as k→±∞, and denote the corresponding system of eigenfunctions of A by (ek).LetE0= KerA, E+=span{e1,…,en} and E-=(E0⊕E+)⊥E, where S⊥E stands for the orthogonal complementary subspace of S in E. Then
(3.8)E=E-⊕E0⊕E+,
so the functional (2.7) can be rewritten as follows
(3.9)F(x)=-12(∥x+∥2-∥x-∥2)+∑t∈Z‍G(t,u(Sx(t-1))),
for all x=x-+x0+x+∈E-+E0+E+.Setl2={x={x(t)}∈S∣∑t∈Z‍|x(t)|2<+∞} and l∞={x={x(t)}∈S∣|x(t)|<+∞, for all t∈Z} and their norms are defined by the following:
(3.10)∥x∥l22=∑t∈Z‍|x(t)|2=(x,x)l2,∥x∥l∞=supt∈Z|x(t)|,
respectively. For any given 1≤r<+∞, define lr={x={x(t)}∈S∣∑t∈Z‍|x(t)|r<+∞} with the norm
(3.11)∥x∥lrr=(∑t∈Z‍|x(t)|r)=(x,x)lr.Define a selfadjoint operatorA on E by the following:
(3.12)(Ax,x)=∑t∈Z‍(JΔSx(t-1)+M(t)Sx(t),x(t))=∥x∥2,(Ax,Jx)l2=∑t∈Z‍(M(t)Sx(t),Jx(t)),|A| is the absolute value. Give another norm the domain of A by the following:
(3.13)∥x∥1=∥(I+|A|)x∥l2,
it is easy to get, for all x∈E,
(3.14)(Ax,Jx)l2≤∥x∥12(x,x)l2≤∥x∥12.Now we state a fundamental proposition, which will be used in the later.Proposition 3.2.
LetL satisfy (L1). Then for all 1≤p∈(2/(3-γ),+∞) there exists a constant λp>0 such that
(3.15)∥x∥lp≤λp∥x∥,∀x∈E.Proof.
We complete the proof of Proposition3.2 by 3 steps.
Step  1. When (L1) holds and p=2, we prove that
(3.16)∥x∥l2≤λ2∥x∥,∀x∈E,
Note that, by(L1), l(t)→+∞ as |t|→+∞, that is, l(t) is bounded from below and so there is a a~>0 such that
(3.17)v*L(t)v+a~IN≥0,t∈Z,β(R)=inf|t|≥Rl(t)→∞,as|t|→∞.
For R>0, choose a subsequence x(k)(t)∈E, one has
(3.18)∑|t|≥R‍|x(k)|2≤∑|t|≥R‍Jx(k)(t)·(v*L(t)v+a~J)x(k)(t)l(t)≤1β(R)∑t∈Z‍Jx(k)(t)·(v*L(t)v+a~J)x(k)(t)=1β(R)∑t∈Z‍(Jx(k)(t)·v*L(t)v·x(k)(t)+a~Jx(k)(t)·Jx(k)(t))=1β(R)(Ax(k),Jx(k))l2+1β(R)a~(Jx(k),Jx(k))l2≤1β(R)∥x(k)∥12.
For any given ϵ>0, by (3.18), one can take R0 so large that
(3.19)∑|t|≥R0‍|x(k)|2≤ϵ24.
Without loss of generality, we can assume that x(k)⇀0 in E. Define EI={x∈SI∣∑t∈Z‍[(JΔSx(t-1),x(t))+(M(t)Sx(t),x(t))]<+∞}, SI={x={x(t)}∣x(t)∈E,t∈I} and I={t∣|t|≤R0}. So x(k) is bounded in EI, which implies that x(k) is bounded in lI2. This together with the uniqueness of the weak limit in lI2, we have x(k)→0 in EI, so there exists a k0 such that
(3.20)∑t∈I‍|x(k)|2≤ϵ24,∀k≥k0.
Combing (3.19) and (3.20), we have x(k)→0 in l2. It follows that (3.16) is true.
Step  2. For all p>2, there exists a constant λp>0 such that (3.15) holds.
For anyp>2 and x∈E, by the Hölder inequality, we have
(3.21)∑t∈Z‍|x|p=∑t∈Z‍|x|·|x|p-1≤(∑t∈Z‍|x|2)1/2·(∑t∈Z‍|x|2(p-1))(p-1)/2(p-1)≤C∥x∥p-1∥x∥l2,
which together with (3.16) yields (3.15).
Step  3. Since (L1) implies l(t)→+∞ as |t|→∞, by Step 1 and 2, it remains to consider the case for 1≤p∈(2/(3-γ),2).
Let(3.22)β~(R)=inf|t|>Rl(t)|t|γ-2.
By (L1), β~(R)→∞ as R→∞.
Writeα=(2-γ)/(2-p), then αp>1. Set for R>0 and x∈E, ER1(x)={t∣|t|≥R and |t|α·|x(t)|>1} and ER2(x)={t∣|t|≥R and |t|α·|x(t)|≤1}. Then
(3.23)∑t∈ER1(x)‍|x|p=∑t∈ER1(x)‍(|t|α|x(t)|)p·|t|-αp≤∑t∈ER1(x)‍(|t|α|x(t)|)2·|t|-αp=∑t∈ER1(x)‍|t|2α|x(t)|2|t|-αp=∑t∈ER1(x)‍|t|2α-αp|x(t)|2=∑t∈ER1(x)‍|t|2-γ|x(t)|2,(3.24)∑t∈Z‍(M(t)Sx(t),Jx(t))=(Ax,Jx)l2≤∥|A|1/2x∥l2·∥|A|1/2Jx∥l2≤C∥x∥2∑t∈ER1(x)‍|x|2≤∑t∈ER1(x)‍Jx·(M+a~J)xl(t)≤1β~(R)|t|γ-2((M+a~J)x,Jx)l2,
From (3.24), we get
(3.25)∑t∈ER1(x)‍|x|p≤∑t∈ER1(x)‍|x|2|t|2-γ≤1β~(R)((M+a~J)x,Jx)l2≤1β~(R)C∥x∥2
and so
(3.26)∑|t|≥R‍|x|p=∑t∈ER1(x)‍|x|p+∑t∈ER2(x)‍|x|p=∑t∈ER1(x)‍(|t|α|x(t)|)p|t|-αp+∑t∈ER2(x)‍(|t|α|x(t)|)p|t|-αp≤1β~(R)C∥x∥2+∑t∈ER2(x)‍(|t|α|x(t)|)p|t|-αp≤1β~(R)C∥x∥2+∑t∈ER2(x)‍|t|-αp.
Since αp>1 and ∑t∈ER2(x)‍|t|-αp=∑|t|≥R‍|t|-αp then there exists a constant s such that
(3.27)∑|t|≥R‍|t|-αp=s,
which together with (3.26) yields
(3.28)∑|t|≥R‍|x|p≤1β~(R)C∥x∥2+s.
Give ϵ>0, by (3.28), choose R0>0 so large that
(3.29)∑|t|≥R0‍|x|p<ϵp2.
DenoteI={t∈Z∣|t|≤R0}, EI={x∣x(t)∈E,t∈I}. Any given subsequence (x(k))∈E, we can suppose x(k)⇀0 on E, now x(k) is bounded in lIp. This together with the uniqueness of the weak limit in lIp on I. For any x∈EI, we have
(3.30)∥x∥lIpp<ϵp2.
Combining (3.29) and (3.30), it follows
(3.31)∑t∈Z‍|x|p<ϵ,
that is, ∥x∥lp<ϵ, there has a constant λp>0 such that
(3.32)∥x∥lp≤λp∥x∥,
while 1≤p∈(2/(3-γ),2).


## 4. Main Results and Proofs

In the previous section, we turned the homoclinic orbits problem of (1.1) to the corresponding critical point problem of the functional (2.6) or (2.7). Next, we state our main results and complete their proofs by Lemma 3.1.Our main result is as follows.Theorem 4.1.
Suppose thatH satisfies (L1) and (H1)–(H5). Then the discrete Hamiltonian system (1.1) has a nontrivial homoclinic orbit.Remark 4.2.
Observe that ifx(t) is a homoclinic solution of (1.1), then y(t)=x(-t) is a homoclinic solution of  the following:
(4.1)JΔy(t)+M(-t)Sy(t)-v*×v*H′(-t,v×v(Sx(t)))=0.
Moreover, -H(-t,y) satisfies (H1)‒(H5) whenever H(t,y) satisfies (H1′), (H2), (H3), (H4′) and (H5′), where (H1′)
H(t,y)/|y|2→-∞ as |y|→+∞, t∈Z;(H4′)
there existβ>α, b>0 and r>0 such that
(4.2)H′(t,y)·y-2H(t,y)≤-b|y|β,∀t∈Z,∀|y|≥r;(H5′)
for allt∈Z and all y∈R2N(4.3)H′(t,y)y≤2H(t,y)≤0.So in the following, we will give another theorem and can omit its proof.Theorem 4.3.
The conclusion of Theorem4.1 holds when replacing (H1), (H4), and (H5) with (H1′), (H4′), and (H5′).With the aid of previous sections, we will verify thatF(x) satisfies the assumptions of Lemma 3.1. We will proceed by successive lemmas.Lemma 4.4.
IfH satisfies assumptions of Theorem 4.1, then there are constants ρ>0, δ>0 such that
(4.4)F|B≥δ,
where B={x∈E2∣∥x∥=ρ}.Proof.
For anyx∈E, it is easy to see that there exist two constants 0<m0<M0 such that
(4.5)m0|x|≤|u(x)|≤M0|x|.
By (H1), (H3), and (4.5), for all ϵ>0, there exist a constant Cϵ>0 such that
(4.6)|H′(t,u(x))|≤1M0[2ϵ|x|+(α+1)Cϵ|x|α],∀x∈E.
Now by mean value Theorem, (4.5) and (4.6), for all x∈E and t∈Z, we have
(4.7)|H(t,u(x))|=∫01‍H′(t,su(x))·u(x)ds≤ϵ|x|2+Cϵ|x|α+1,
on the other hand,
(4.8)∥Sx∥l22=∑t∈Z‍|Sx(t)|2=∑t∈Z‍|x1(t+1)|2+|x2(t)|2=∥x∥l22,
similarly,
(4.9)∥Sx∥lα+1α+1=∥x∥lα+1α+1,
it follows that
(4.10)∑t∈Z‍|H(t,u(Sx(t)))|≤∑t∈Z‍[ϵ|Sx|2+Cϵ|Sx|α+1]=ϵ∑t∈Z‍|Sx|2+Cϵ∑t∈Z‍|Sx|α+1=ϵ∥x∥l22+Cϵ∥x∥lα+1α+1,
By Proposition 3.2 and (4.10), for any x∈E, it holds
(4.11)F(x)=12∥x∥2-∑t∈Z‍H(t,u(Sx(t)))≥12∥x∥2-ϵ∥x∥l22-Cϵ∥x∥lα+1α+1≥12∥x∥2-ϵλ22∥x∥2-Cϵλα+1α+1∥x∥α+1,
Choosing ϵ such that ϵλ22=1/4, we obtain, for any x∈E2,
(4.12)F(x)≥14∥x∥2-Cϵλα+1α+1∥x∥α+1.
Since α>1, then there are constants ρ>0, δ>0 such that
(4.13)F∣B≥δ,
which completes the proof of Lemma 4.4.Lemma 4.5.
Under assumptions of Theorem4.1, let e∈E12 with ∥e∥=1, there exist r1,r2>0 such that
(4.14)F(x)≤0,x∈∂Q,
where Q={se∣0≤s≤r1}⊕{x∈E1∣∥x∥≤r2}.Proof.
Lete∈E12 with ∥e∥=1 and F=E1⊕span{e}. For x=x-+x0+x+∈F-{0} and ϵ>0, denote
(4.15)Ωx={t∈Z∣|x(t)|≥ϵ∥x∥},
then there exists ϵ1>0 such that
(4.16)♯Ωx=♯{t∈Z∣|x(t)|≥ϵ1∥x∥}≥[1ϵ1]+1,
where ♯Ωx is the number of t in Ωx and [·] is the greatest integer function.
By(H1), for d=1/2ϵ12m02, there exists R1>0 such that
(4.17)H(t,y)≥d|y|2,∀|y|≥R1,t∈Z,
where m0 was defined by (4.5). Then it follows
(4.18)H(t,u(Sx(t)))≥d|u(Sx(t))|2≥12ϵ13m02|u(Sx(t))|2≥12ϵ13|Sx(t)|2≥12ϵ1∥Sx(t)∥2=12ϵ1∥x∥2,
for all x∈F-{0} with ∥x∥≥R1/m0ϵ1 and t∈Ωx. Hence, from (4.16) and (4.18), one has
(4.19)F(x)=12∥x+∥2-12∥x-∥2-∑t∈Z‍H(t,u(Sx(t)))≤12∥x+∥2-∑t∈Ωx‍H(t,u(Sx(t)))≤12∥x+∥2-12ϵ1∥Sx(t)∥2([1ϵ1]+1)≤12∥x+∥2-12∥x∥2≤0,
is true for all x∈F-{0} with ∥x∥≥R1/m0ϵ1. Let r1>0 and denote
(4.20)Q={se∣0≤s≤r1}⊕{x∈E1∥x∥≤r2}.
Then by (4.19), for all r1>max{ρ,R1/m0ϵ1}, we have
(4.21)F(x)≤0,∀x∈∂Q,
where ρ is defined by Lemma 4.5, this is just (4.14). We completed the proof of Lemma 4.5.In order to verify thatF(x) satisfies (f1) of Lemma 3.1, we need the following lemma.Lemma 4.6.
Write(4.22)h(x)=∑t∈Z‍H(t,u(Sx(t))),∀x∈E.
Then h(x)∈C1(E,R).Proof.
Letφ(z)=h(t,x+zy), 0≤z≤1, for all x,y∈E, since H∈C1(E,R)(4.23)φ′(0)=limz→0φ(z)-φ(0)z=limz→0h(t,x+zy)-h(t,x)z=limz→01z∑t∈Z‍[H(t,u(S(x+zy)(t)))-H(t,u(Sx(t)))]=limz→0∑t∈Z‍[∇H(t,u(S(xt+θtzyt)))·u(Syt)]=∑t∈Z‍[∇H(t,u(S(xt+θtzyt)))·u(Syt)],
where 0<θt<1, then h(x) is Gâteaux differential on E and
(4.24)h′(x)y=∑t∈Z∇H(t,u(Sx(t)))·u(Sy(t)),∀x,y∈E.
Letx(k)⇀x weakly in E, by Proposition 3.2, one can assume that x(k)→x strongly in lp for p∈[1,+∞). By (4.24), we have
(4.25)∥h′(x(k))-h′(x)∥=sup∥y∥=1|∑t∈Z‍(∇H(t,u(Sx(k)(t)))-∇H(t,u(Sx(t)))·u(Sy(t)))|.
By (4.5) and (4.6), there exists a constant C1>0 such that for any R>0, there holds
(4.26)|∑|t|≥R‍(∇H(t,u(Sx(k)(t)))-∇H(t,u(Sx(t)))·u(Sy(t)))|≤∑|t|≥R‍[|∇H(t,u(Sx(k)(t)))|+|∇H(t,u(Sx(t)))|]·|u(Sy(t))|≤∑|t|≥R‍C1[|x(k)|+|x|+|x(k)|α+|x|α]|y|≤C1(∑|t|≥R‍(|x(k)|+|x|)2)1/2(∑|t|≥R‍|y|2)1/2+C1(∑|t|≥R‍(|x(k)|+|x|)α+1)α/(α+1)(∑|t|≥R|y|α+1‍)1/(a+1)≤C1∥y∥l2(∑|t|≥R‍(|x(k)|+|x|)2)1/2+C1∥y∥lα+1(∑|t|≥R‍(|x(k)|+|x|)α+1)α/(α+1),
so by Proposition 3.2, there exists a constant C2>0 such that for any ∥y∥=1,
(4.27)|∑|t|≥R‍(∇H(t,u(Sx(k)(t)))-∇H(t,u(Sx(t)))·u(Sy(t)))|≤C2[(∑|t|≥R‍|x(k)|2+|x|2)1/2+(∑|t|≥R‍|x(k)|α+1+|x|α+1)α/(α+1)].
We deduce from (4.27) that for any ϵ>0, there has R>0 so large that
(4.28)|∑|t|≥R‍(∇H(t,u(Sx(k)(t)))-∇H(t,u(Sx(t)))·u(Sy(t)))|<ϵ2,
for all k∈N and y∈E with ∥y∥=1. On the hand, it is well known that since x(k)→x strongly in l2, then when x∈EIR(4.29)∥∇H(t,u(Sx(k)))-∇H(t,uS(x))∥l2→0,
as k→∞, where EIR={x∣x(t)∈E,t∈IR} and IR={t∣t∈Z,|t|<R}. Thus there is k0∈N such that
(4.30)|∑|t|≤R‍(∇H(t,u(Sx(k)(t)))-∇H(t,u(Sx(t)))·u(Sy(t)))|<ϵ2,
is true for all integer k≥k0 and all y∈E with ∥y∥=1. Combining (4.28) and (4.30), it is easy to see
(4.31)∥∇H(t,u(Sx(k)))-∇H(t,u(Sx))∥<ϵ,k≥k0.
It follows from the arbitrariness of ϵ that h(x)∈C1(E,R).
Finally, let us complete the proof of Theorem4.1 by verifying F(x) satisfies the Palais-Smale condition.Lemma 4.7.
With assumptions of Theorem4.1, F(x) satisfies the (PS)* condition.Proof.
Let(x(k)) be a (PS)* sequence, that is, x(k)∈En, for all k∈N, and F(x(k))≤C, F′(x(k))→0, as k→∞. We claim that (x(k)) is bounded. If not, passing to a subsequence if necessary, we may assume that ∥x(k)∥→∞ as k→∞.
DenoteI1={t∈Z∣|x(k)(t)|≥r/m0} and I1={t∈Z∣|x(k)(t)|<r/m0} for all k∈N. By (H4), (H5), (4.5), and Lemma 4.6, we have
(4.32)2F(x(k))-F′(x(k))·x(k)=∑t∈Z‍[H′(t,u(Sx(k)(t)))·u(Sx(k)(t))-2H(t,u(Sx(k)(t)))]≥∑{t∈Z|u(Sx(k)(t))|≥r}‍b|u(Sx(k)x(t))|β≥∑t∈I1‍m0β|x(k)|β,
which implies that
(4.33)1∥x(k)∥∑t∈I1‍|x(k)|β→0,ask→∞.
Making use of (4.5) and (4.6), we obtain that there is a constant C3 such that
(4.34)|H′(t,u(Sx(k)(t)))|≤C3(|x(k)|+|x(k)|α),∀x∈E.
Hence from (4.34), there holds
(4.35)Fn′(x(k))·x(k)+=12∥x(k)+∥2-∑t∈ZH(t,u(Sx(k)(t)))·u(Sx(k)+(t))≥12∥x(k)+∥2-∑t∈Z|H(t,u(Sx(k)(t)))|·|u(Sx(k)+(t))|≥12∥x(k)+∥2-∑t∈ZC3(|x(k)|+|x(k)|α)|u(Sx(k)+(t))|≥12∥x(k)+∥2-∑t∈Z‍C3M0|x(k)||x(k)+|-∑t∈Z‍C3M0|x(k)|α|x(k)+|.
By Hölder inequality and Proposition 3.2, we achieve
(4.36)∑t∈Z|x(k)+|α‍|x(k)+|=∑t∈I1‍|x(k)|α|x(k)+|+∑t∈I2‍|x(k)|α|x(k)+|≤(rm0)α∑t∈I1‍|x(k)+|+(∑t∈I2‍(|x(k)|α)β/α)α/β·(∑t∈I2‍|x(k)+|β/(β-α))(β-α)/β≤(rm0)α∑t∈Z‍|x(k)+|+(∑t∈I2‍|x(k)|β)α/β·(∑t∈Z‍|x(k)+|β/(β-α))(β-α)/β=(rm0)α∥x(k)+∥l1+(∑t∈I2‍|x(k)|β)α/β∥x(k)+∥lβ/(β-α)≤(rm0)αλ1∥x(k)+∥+(∑t∈I2‍|x(k)|β)α/βλβ/(β-α)∥x(k)+∥=[(rm0)αλ1+λβ/(β-α)(∑t∈I2‍|x(k)|β)α/β]∥x(k)+∥.
Similarly,
(4.37)|x(k)||x(k)+|≤[rm0λ1+λβ/(β-1)(∑t∈I2‍|x(k)|β)1/β]∥x(k)+∥.
Combing (4.35)–(4.37), yields
(4.38)Fn′(x(k))·x(k)+≥12∥x(k)+∥2-λ1C3M0[(rm0)α+rm0]∥x(k)+∥-C3M0λβ/(β-α)(∑t∈I2‍|x(k)|β)α/β∥x(k)+∥-C3M0λβ/(β-1)(∑t∈I2|x(k)|β)1/β∥x(k)+∥,
Since 1<α<β, we deduce from (4.33) and (4.38) that
(4.39)∥x(k)+∥∥x(k)∥→0,ask→∞,
similarly,
(4.40)∥x(k)-∥∥x(k)∥→0,ask→∞.
Now, let
(4.41)y(k)(t)={x(k)(t),if|x(k)(t)|≤rm00,if|x(k)(t)|>rm0,(4.42)z(k)(t)=x(k)(t)-y(k)(t),∀k∈N,t∈Z.
Then
(4.43)∑t∈Z|z(k)|β‍=∑t∈Z‍|x(k)-y(k)|β=∑t∈I1‍|x(k)-y(k)|β+∑t∈I2‍|x(k)-y(k)|β=∑t∈I1|x(k)|β‍≤∑t∈Z‍|x(k)|β,
that is
(4.44)∥z(k)∥lββ≤∥x(k)∥lββ.
For β>1, by (4.32), (4.41) and Proposition 3.2, there exists a constant C4>0 such that
(4.45)C4(1+∥x(k)∥)≥∥z(k)∥lββ,∀k∈N.
Since E0 is of finite dimension, using Hölder inequality and (4.45), for any x(k)0∈E0, we have
(4.46)∥x(k)0∥l22=(x(k)0,x(k))l2=(x(k)0,z(k)+y(k))l2=(x(k)0,z(k))l2+(x(k)0,y(k))l2≤∑t∈Z‍|x(k)0||z(k)|+∑t∈Z‍|x(k)0||y(k)|≤∑t∈Z‍|x(k)0|rm0+(∑t∈Z‍|x(k)0|β/(β-1))(β-1)/β(∑t∈Z‍|z(k)|β)1/β=rm0∥x0(k)∥l1+∥x(k)0∥lβ/(β-1)∥z(k)∥lβ≤C5∥x(k)0∥l2(1+∥z(k)∥lβ),
where C5>0 is a constant.
Hence by (4.45) and (4.46), there exist positive constants C6, C7 such that
(4.47)∥x(k)0∥≤C6C5(1+∥z(k)∥lβ)≤C7(1+∥z(k)∥1/β),
which implies
(4.48)∥x(k)0∥∥x(k)∥→0,ask→∞,
when β>1.
By (4.39), (4.40), and (4.48), it follows
(4.49)1=∥x(k)∥∥x(k)∥=∥x(k)+∥+∥x(k)-∥+∥x(k)0∥∥x(k)∥→0,ask→∞,
which is a contradiction. Therefore, (x(k)) must be bounded. That is, F(x) satisfies the (PS)* condition.By Lemma3.1, F(x) possesses a critical point x∈E such that F(x)≥δ>0 and (1.1) has a nontrivial homoclinic orbit.



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*Source: 720139-2012-10-03.xml*