# Infinitely Many Homoclinic Orbits for 2nth-Order Nonlinear Functional Difference Equations Involving the

Authors: Xiaofei He
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

By establishing a new proper variational framework and using the critical point
theory, we establish some new existence criteria to guarantee that the 2nth-order nonlinear difference equation containing both advance and retardation with p-Laplacian Δn(r(t−n)φp(Δnu(t−1)))+q(t)φp(u(t))=f(t,u(t+n),…,u(t),…,u(t−n)), n∈ℤ(3), t∈ℤ, has infinitely many homoclinic orbits, where φp(s) is p-Laplacian operator;φp(s)=|s|p−2s(1<p<∞)r, q, f are nonperiodic in t. Our conditions on the potential are rather relaxed, and some existing results in the literature are improved.


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## Body

## 1. Introduction

In this paper, we shall be concerned with the existence of homoclinic orbits of the nonlinear difference equation(1.1)Δn(r(t-n)φp(Δnu(t-1)))+q(t)φp(u(t))=f(t,u(t+n),…,u(t),…,u(t-n)),n∈Z(3),t∈Z,
where Δ is the forward difference operator defined by Δu(t)=u(t+1)-u(t),Δ2u(t)=Δ(Δu(t)). φp(s) is p-Laplacian operator φp(s)=|s|p-2s(1<p<∞). As usual, we say that a solution u(t) of (1.1) is homoclinic (to 0) if u(t)→0 as t→±∞. In addition, if u(t)≢0, then u(t) is called a nontrivial homoclinic solution. It is well known that homoclinic orbits play an important role in analyzing the chaos of dynamical systems; we refer the interested reader to [1–15].We may think of (1.1) as being a discrete analogue of the 2nth-order differential equation(1.2)[r(t)φp(x(n))](n)+q(t)φp(x)-f(t,x(t+n),…,x(t),…,x(t-n))=0,t∈R.Ifp=2, (1.1) reduces to the following equation:(1.3)Δn(r(t-n)Δnu(t-n))+q(t)u(t)=f(t,u(t+n),…,u(t),…,u(t-n)),n∈Z(3),t∈Z.
Recently, Chen and Tang [2] applied the critical point theory to prove the existence of homoclinic solutions for (1.3).In some recent papers [1, 2, 8, 9, 16–18], the authors studied the existence of periodic and homoclinic solutions of second-order nonlinear difference equation by using the critical point theory. These papers show that the critical point method is an effective approach to the study of periodic solutions of second-order difference equations. Compared to one-order or second-order difference equations, the study of higher-order equations has received considerably less attention (see, e.g., [2, 3, 19–21] and references contained therein). But to the best knowledge of the authors, results on existence of homoclinic solutions of (1.1) have not been found in the literature.Motivated by the recent papers [2, 5, 14, 22], the aim of this paper is to consider problem (1.1) in a more general sense. More exactly our results represent the extensions to equations with p-Laplacian. Throughout the paper, for a function G, we let Gi′(x1,x2,…,xi,…,xn) denote the partial derivative of G on the i variable.


## 2. Main Results

Theorem 2.1.
Assume thatq and  F satisfy the following assumptions: (r)
For everyt∈ℤ, r(t)>0;(q)
For everyt∈ℤ, q(t)>0, and  lim|t|→+∞q(t)=+∞;(F1)
There exists a functionalF(t,xn(t),…,x0(t)) which is continuously differentiable in the variable  from xn to x0 for every t∈ℤ and satisfy
(2.1)∑i=-n0F2+n+i′(t+i,xn+i,…,xi)=f(t,xn,xn-1,…,x0,x-1,…,x-n),|f(t,xn,xn-1,…,x0,x-1,…,x-n)|=o(∑i=-nn|xi|p-1),as∑i=-nn|xi|p-1⟶0,|F(t,xn,…,x0)|=o(∑i=-nn|xi|p),as∑i=-nn|xi|p⟶0
uniformly in t∈ℤ∖J, where J is defined in (F2);(F2)
F(t,xn,…,x0)=W(t,x0)-H(t,xn,…,x0), for every  t∈ℤ,  W,H are continuously differentiable in x0 and xn,…,x0, respectively. Moreover, there is a bounded set J⊂ℤ such that(2.2)H(t,xn,…,x0)≥0;(F3)
There is a constantμ>p such that(2.3)0<μW(t,x0)≤W2′(t,x0)x0,∀(t,x0)∈Z×(R∖{0});(F4)
H(t,0,…,0)≡0, and there is a constant  ϱ∈(p,μ) such that(2.4)∑i=-n0H2+n+i′(t,xn,…,x0)x-i≤ϱH(t,xn,…,x0);(F5)
There exists a constantb>0 such that(2.5)H(t,xn,…,x0)≤bγϱ,fort∈Z,γ>1,
where γ=(∑i=0n|xi|p)1/p.(F6)
F(t,-xn,…,-x0)=F(t,xn,…,x0), for all (t,xn,…,x0)∈ℤ×ℝn+1.Then (1.1) possesses an unbounded sequence of homoclinic solutions.Theorem 2.2.
Assume thatr, q, and  F satisfy (r), (q), (F1), (F3)–(F5) and the following assumption: (F2’)
F(t,xn,…,x0)=W(t,x0)-H(t,xn,…,x0), for every  t∈ℤ,  W,H are continuously differentiable in x0 and xn,…,x0, respectively, and(2.6)|F(t,xn,…,x0)|=o(γp)asγ⟶0,
where γ=(∑i=0n|xi|p)1/p uniformly in t∈ℤ,
Then (1.1) possesses an unbounded sequence of homoclinic solutions.Theorem 2.3.
Assume thatr,  q, and  F satisfy (r), (q), (F1) and satisfy the following assumptions: (F7)
For anyt∈ℤ,(2.7)F(t,xn,…,x0)≥F(t,x0)≥0;(F8)
For anyr>0, there exist  a=a(r),  b=b(r)>0, and  ν<p such that(2.8)(p+1a+b(∑i=0n|xi|p)ν/p)F(t,xn,…,x0)≤∑i=-n0F2+n+i′(t,xn,…,x0)x-i,∀t∈Z,(∑i=0n|xi|p)1/p≥r;(F9)
For anyt∈ℤ(2.9)lim⁡s→+∞[s-pmin⁡|x|=1F(t,sx)]=+∞.
Then there exists an unbounded sequence of homoclinic solutions for (1.1).


## 3. Preliminaries

To apply critical point theory to study the existence of homoclinic solutions of (1.1), we shall state some basic notations and lemmas, which will be used in the proofs of our main results.Let(3.1)S={{u(t)}t∈Z:u(t)∈R,t∈Z},E={u∈S:∑t∈Z[|r(t-1)Δnu(t-1)|p+q(t)|u(t)|p]<+∞}
and for u∈E, let(3.2)‖u‖={∑t∈Z[|r(t-1)Δnu(t-1)|p+q(t)|u(t)|p]}1/p.
Then E is a uniform convex Banach space with this norm.LetI:E→ℝ be defined by(3.3)I(u)=1p‖u‖p-∑t∈ZF(t,u(t+n),…,u(t)).
If (q) and (F1) hold, then I∈C1(E,ℝ) and one can easily check that(3.4)〈I′(u),v〉=∑t∈Z[r(t-1)|Δnu(t-1)|p-2Δnv(t-1)+q(t)|u(t)|p-2v(t)-f(t),u(t+n),…u(t),…u(t-n)v(t)],∀u,v∈E.
By using(3.5)Δnu(t-1)=∑k=0n(-1)k(nk)u(t+n-k-1),
we can compute the partial derivative as(3.6)∂I(u)∂u(t)=Δn(r(t-n)φp(Δnu(t-1)))+q(t)φp(u(t))-f(t,u(t+n),…,u(t),…,u(t-n)).
So, the critical points of I in E are the solutions of (1.1) with u(±∞)=0.Lemma 3.1 (see [12]).
LetE be a real Banach space and  I∈C1(E,ℝ) satisfy (PS)-condition with  I even. Suppose that  I satisfies the following conditions: (i)
I(0)=0;(ii)
There exist constantsρ,α>0 such that I|∂Bρ(0)≥α;(iii)
For each finite dimensional subspaceE′⊂E, there is r=r(E′)>0 such that I(u)≤0 for u∈E′∖Br(0), where Br(0) is an open ball in E of radius r centered at 0.Then I possesses an unbounded sequence of critical values.Lemma 3.2.
Foru∈E,
(3.7)β‖u‖∞p≤β‖u‖lpp≤‖u‖p,
where β=inft∈ℤq(t).Lemma 3.3.
Assume that (F3) holds. Then for every(t,x)∈ℤ×ℝ,  s-μW(t,sx) is nondecreasing on  (0,+∞).The proof of Lemmas3.2 and 3.3 is routine and so we omit it.


## 4. Proofs of Theorems

Proof of Theorem2.1.
It is clear thatI(0)=0. Our proof is devided into three steps.
Step 1 (PS Condition).
Assume that{uk}k∈ℕ⊂E is a sequence such that {I(uk)}k∈ℕ is bounded and I′(uk)→0 as k→+∞. Then there exists a constant c>0 such that
(4.1)|I(uk)|≤c,‖I′(uk)‖E*≤ϱcfork∈N.
From (2.2), (2.3), (2.4), (4.1), (F3), and (F4), we obtain
(4.2)pc+pc‖uk‖≥pI(uk)-pϱ〈I′(uk),uk〉=ϱ-pϱ‖uk‖p-p∑t∈Z[W(t,uk(t))-1ϱW2′(t,uk(t))uk(t)]+p∑t∈ZH(t,uk(t+n),…,uk(t))-pϱ∑t∈Z∑i=-n0H2+n+i′(t+i,uk(t+n+i),…,uk(t+i))uk(t)=ϱ-pϱ‖uk‖p-p∑n∈Z[W(t,uk(t))-1ϱW2′(t,uk(t))uk(t)]+p∑t∈ZH(t,uk(t+n),…,uk(t))-pϱ∑t∈Z∑i=-n0H2+n+i′(t,uk(t+n),…,uk(t))uk(t-i)≥ϱ-pϱ‖uk‖p,k∈N.
By (4.2), there exists a constant A>0 such that
(4.3)‖uk‖≤Afork∈N.
It can be assumed that uk⇀u0 in E. For any given number ɛ>0, by (F1), we can choose ζ>0 such that
(4.4)|f(t,u(t+n),…,u(t),…,u(t-n))|≤ɛξp-1fort∈Z∖J,(u(t+n),…,u(t),…,u(t-n))∈Rn+1,
where ξ=(∑i=-nn|u(t+i)|p)1/p≤ζ.
Sinceq(t)→∞,  we can also choose an integer Π>max{|k|:k∈J} such that
(4.5)q(t)≥(2n+1)Apζp,|t|≥Π.
By (4.2) and (4.4), we have
(4.6)|uk(t)|p=1q(t)q(t)(uk(t))p≤ζp(2n+1)Ap‖uk‖p≤ζp2n+1for|t|≥Π,k∈N.
Since uk⇀u0 in E, it is easy to verify that uk(t) converges to u0(t) pointwise for all t∈ℤ, that is
(4.7)lim⁡k→∞uk(t)=u0(t),∀t∈Z.
Hence, we have by (4.6) and (4.7)
(4.8)|u0(t)|p≤ζp2n+1for|t|≥Π.
It follows from (4.7) and the continuity of f(t,u(t+1),…,u(t),…,u(t-n)) on u(t+1),…,u(t),…,u(t-n) that there exists k0∈ℕ such that
(4.9)∑t=-ΠΠ|f(t,uk(t+n),…,uk(t),…,uk(t-n))-f(t,u0(t+n),…,u0(t),…,u0(t-n))|<ɛfork≥k0.
On the other hand, it follows from (F1), (2.4), (4.2), (4.4), (4.6), and (4.8) that
(4.10)∑|t|>Π|f(t,uk(t+n),…,uk(t),…,uk(t-n))-f(t,u0(t+n),…,u0(t),…,u0(t-n))|×|uk(t)-u0(t)|≤∑|t|>Π(12|f(t,uk(t+n),…,uk(t),…,uk(t-n))|+|f(t,u0(t+n),…,u0(t),…,u0(t-n))|12)×(|uk(t)|+|u0(t)|)≤ɛ∑|t|>Π[(∑i=-nn|uk(t+i)|p-1)+(∑i=-nn|u0(t+i)|p-1)](|uk(t)|+|u0(t)|)≤(2n+1)ɛ∑t∈Z(|uk(t)|p-1+|u0(t)|p-1)(|uk(t)|+|u0(t)|)≤(4n+2)ɛ∑t∈Z(|uk(t)|p+|u0(t)|p)≤(4n+2)ɛβ(Ap+‖u0‖p).
Since ɛ is arbitrary, combining (4.9) with (4.10), we get
(4.11)∑t∈Z|f(t,uk(t+n),…,uk(t),…,uk(t-n))-f(t,u0(t+n),…,u0(t),…,u0(t-n))||uk(t)-u0(t)|⟶0ask⟶∞.
Using the Hölder’s inequality
(4.12)ac+bd≤(ap+bp)1/p(cq+dq)1/q,
where a,b,c,d are nonnegative numbers and 1/p+1/q=1, p>1, it follows from (3.3) and (3.4) that
(4.13)〈I′(uk)-I′(u0),uk-u0〉=∑t∈Z|r(t-1)Δnuk(t-1)|p-2(Δnuk(t-1),Δnuk(t-1)-Δnu0(t-1))+∑t∈Zq(t)|uk(t)|p-2(uk(t),uk(t)-u0(t))-∑t∈Z|r(t-1)Δnu0(t-1)|p-2(Δnu0(t-1),Δnuk(t-1)-Δnu0(t-1))-∑t∈Zq(t)|u0(t)|p-2(u0(n),uk(t)-u0(t))-∑t∈Z(f(t,uk(t+n),…,uk(t),…,uk(t-n))-f(t,u0(t+n),…,u0(t),…,u0(t-n)),uk(n)-u0(n))=‖uk‖p+‖u0‖p-∑n∈Z|r(t-1)Δnuk(t-1)|p-2(Δnuk(t-1),Δnu0(t-1))-∑t∈Zq(t)|uk(t)|p-2(uk(t),u0(t))-∑t∈Z|r(t-1)Δnu0(t-1)|p-2(Δnu0(t-1),Δnuk(t-1))-∑t∈Zq(t)|u0(t)|p-2(u0(t),uk(t))-∑t∈Z(f(t,uk(t+n),…,uk(t),…,uk(t-n))-f(t,u0(t+n),…,u0(t),…,u0(t-n)),uk(n)-u0(n))≥‖uk‖p+‖u0‖p-(∑t∈Z|r(t-1)Δnu0(t-1)|p)1/p(∑t∈Z|r(t-1)Δnuk(t-1)|p)1/q-(∑t∈Zq(t)|u0(t)|p)1/p(∑t∈Zq(t)|uk(t)|p)1/q-(∑t∈Z|r(t-1)Δnuk(t-1)|p)1/p(∑t∈Z|r(t-1)Δnu0(t-1)|p)1/q-(∑t∈Zq(t)|uk(t)|p)1/p(∑t∈Zq(t)|u0(t)|p)1/q-∑t∈Z(f(t,uk(t+n),…,uk(t),…,uk(t-n))-f(t,u0(t+n),…,u0(t),…,u0(t-n)),uk(t)-u0(t))≥‖uk‖p+‖u0‖p-(∑t∈Z[|r(t-1)Δnu0(t-1)|p+q(t)|u0(t)|p])1/p⋅(∑t∈Z[|r(t-1)Δnuk(t-1)|p+q(t)|uk(t)|p])1/q-(∑t∈Z[|r(t-1)Δnuk(n-1)|p+q(t)|uk(t)|p])1/p⋅(∑t∈Z[|r(t-1)Δnu0(n-1)|p+q(t)|u0(t)|p])1/q-∑t∈Z(f(t,uk(t+n),…,uk(t),…,uk(t-n))-f(t,u0(t+n),…,u0(t),…,u0(t-n)),uk(t)-u0(t))=‖uk‖p+‖u0‖p-‖u0‖‖uk‖p-1-‖uk‖‖u0‖p-1-∑t∈Z(f(t,uk(t+n),…,uk(t),…,uk(t-n))-f(t,u0(t+n),…,u0(t),…,u0(t-n)),uk(t)-u0(t))=(‖uk‖p-1-‖u0‖p-1)(‖uk‖-‖u0‖)-∑t∈Z(f(t,uk(t+n),…,uk(t),…,uk(t-n))-f(t,u0(t+n),…,u0(t),…,u0(t-n)),uk(t)-u0(t)).
Since I′(uk)→0 as k→+∞ and uk⇀u0 in E, it follows from (4.11) and (4.13) that
(4.14)〈I′(uk)-I′(u0),uk-u0〉⟶0ask→∞,
which yields that ∥uk∥→∥u∥ as k→+∞. By the uniform convexity of E and the fact that uk⇀u0 in E, it follows from the Kadec-Klee property that uk→u0 in E. Hence, I satisfies (PS)-condition.
Step 2 (Condition (ii) of Lemma3.1).
By (F1), there existsη∈(0,1) such that
(4.15)|F(t,u(t+n),…,u(t))|≤2-pβ4p(n+1)∑i=0n|u(t+i)|pfort∈Z∖J,(∑i=0n|u(t+i)|p)1/p≤η.
Set
(4.16)M=sup⁡{W(t,u)∣t∈J,u∈R,|u|=1},
and δ=min{(β/(4pM+1))(μ-p),η}. If ∥u∥=β1/pδ:=ρ, then by Lemma 3.2, |u(t)|≤δ≤η<1 for t∈ℤ. By (4.16) and Lemma 3.2, we have
(4.17)∑t∈JW(t,u(t))≤∑t∈J,u(t)≠0W(t,u(t)|u(t)|)|u(t)|μ≤M∑t∈J|u(t)|μ≤Mδμ-p∑t∈J|u(t)|p≤Mδμ-pβ∑t∈Jq(t)|u(t)|p≤14p∑t∈Jq(t)|u(t)|p.
Set α=βδp/2p. Hence, from (3.3), (4.15), (4.17), (q), (F1), and (F2), we have
(4.18)I(u)=1p‖u‖p-∑t∈ZF(t,u(t+n),…,u(t))=1p‖u‖p-∑t∈Z∖JF(t,u(t+n),…,u(t))-∑t∈JF(t,u(t+n),…,u(t))≥1p‖u‖p-2-pβ4p(n+1)∑t∈Z∖J(∑i=0n|u(t+i)|p)-∑t∈JW(t,u(t))+∑t∈JH(t,u(t+n),…,u(t))≥1p‖u‖p-β4p∑t∈Z|u(t)|p-14p∑t∈Jq(t)|u(t)|p≥1p‖u‖p-14p∑t∈Zq(t)|u(t)|p-14p∑t∈Jq(t)|u(t)|p≥1p‖u‖p-14p‖u‖p-14p‖u‖p=12p‖u‖p=α.
equation (4.18)shows that ∥u∥=ρ implies that I(u)≥α, that is, I satisfies assumption (ii) of Lemma 3.1.
Step 3 (Condition (iii) of Lemma3.1).
LetE′ be a finite dimensional subspace of E. Assume that dimE′=m and u1,u2,…,um is a base of E′.
For anyu∈E′, there exist λi∈ℝ, i=1,2,…,m such that
(4.19)u(t)=∑i=1mλiui(t)fort∈Z.
Let
(4.20)‖u‖*=∑i=1m|λi|‖ui‖.
It is easy to verify that ∥·∥* defined by (4.20) is a norm of E′. Since all the norms of a finite dimensional normed space are equivalent, so there exists a constant d>0 such that
(4.21)‖u‖*≤d‖u‖∞foru∈E′.
Assume that
(4.22)‖ui‖=d,i=1,2,…,m.
Sinceui∈E, by Lemma 3.2, we can choose an integer Π1>0 such that
(4.23)|ui(t)|<ηm,i=1,2,…,m,|t|>Π1,
where η is given in (4.15). Set
(4.24)Θ={∑i=1mλiui(t):λi∈R,i=1,2,…,m;∑i=1m|λi|=1}={u∈E′:‖u‖*=d}.
Hence, for u∈Θ, let t0=t0(u)∈ℤ such that
(4.25)|u(t0)|=‖u‖∞.
Then, by (4.19)–(4.25), we have
(4.26)d=d∑i=1m|λi|=∑i=1m|λi|‖ui‖=‖u‖*≤d‖u‖∞=d|u(t0)|≤d∑i=1m|λi||ui(t0)|,u∈Θ.
This shows that |u(t0)|≥1 and there exists i0∈{1,2,…,m} such that |ui0(t0)|≥1/m, which, together with (4.23), implies that |t0|≤Π1.
Set(4.27)τ=min⁡{W(t,x):|t|≤Π1,|x|=1}.
Since W(t,x)>0 for all t∈ℤ and x∈ℝ∖{0}, and W(t,x) is continuous in x, so τ>0. It follows from (4.27) and Lemma 3.3 that
(4.28)∑n=-Π1Π1W(t,u(t))≥W(t0,u(t0))≥W(t0,u(t0)|u(t0)|)|u(t0)|μ≥[min⁡|x|=1W(t0,x)]|u(t0)|μ≥τforu∈Θ.
For any u∈E, it follows from (F5) that
(4.29)∑t=-Π1Π1H(t,u(t+n),…,u(t))=∑{t∈Z(-Π1,Π1),(∑i=0n|u(t+i)|p)1/p>1}H(t,u(t+n),…,u(t))+∑{t∈Z(-Π1,Π1),(∑i=0n|u(t+i)|p)1/p}≤1H(t,u(t+n),…,u(t))≤b∑{t∈Z(-Π1,Π1),(∑i=0n|u(t+i)|p)1/p>1}(∑i=0n|u(t+i)|p)ϱ/p+∑t=-Π1Π1max⁡(∑i=0n|u(t+i)|p)1/p≤1H(t,u(t+n),…,u(t))≤(n+1)ϱ/p+1b∑t∈Z|u(t)|ϱ+∑t=-Π1Π1max⁡(∑i=0n|u(t+i)|p)1/p≤1H(t,u(t+n),…,u(t))≤(n+1)ϱ/p+1β-ϱ/pb‖u‖ϱ+∑t=-Π1Π1max⁡(∑i=0n|u(t+i)|p)1/p≤1H(t,u(t+n),…,u(t))=M1‖u‖ϱ+M2,
where
(4.30)M1=(n+1)ϱ/p+1β-ϱ/pb,M2=∑t=-Π1Π1max⁡(∑i=0n|u(t+i)|p)1/p≤1H(t,u(t+n),…,u(t)).
From (3.3), (3.7), (4.28), and (4.29), we have for u∈Θ and σ>1(4.31)I(σu)=σpp‖u‖p-∑t∈ZF(t,σu(t+n),…,σu(t))=σpp‖u‖p-∑|t|>Π2F(t,σu(t+n),…,σu(t))-∑|t|≤Π2F(t,σu(t+n),…,σu(t))≤σpp‖u‖p+∑|t|>Π2∑i=0nβσp4p(n+1)|u(t+i)|p-∑|t|≤Π2F(t,σu(t+n),…,σu(t))≤σpp‖u‖p+σp4p‖u‖p-∑|t|≤Π2F(t,σu(t+n),…,σu(t))=σpp‖u‖p+σp4p‖u‖p-∑|t|≤Π2W(t,σu(t))+∑|t|≤Π2H(t,σu(t+n),…,σu(t))≤σpp‖u‖p+σp4p‖u‖p+σϱ(M1‖u‖ϱ+M2)-τσμ=(cσ)pp+cpσp4p+M1(cσ)ϱ+M2σϱ-τσμ.
Since μ>ϱ>p, we deduce that there is σ0=σ0(d,M1,M2,τ)=σ0(E′)>1 such that
(4.32)I(σu)<0foru∈Θ,σ≥σ0.
That is
(4.33)I(u)<0foru∈E′,‖u‖≥dσ0.
This shows that (iii) of Lemma 3.1 holds. By Lemma 3.1, I possesses an unbounded sequence {dk}k∈ℕ of critical values with dk=I(uk), where uk is such that I′(uk)=0 for k=1,2,…. If {∥uk∥}k∈ℕ is bounded, then there exists B>0 such that
(4.34)‖uk‖≤Bfork∈N.
By a similar fashion for the proof of (4.15), for the given η in (4.15), there exists Π3>0 such that
(4.35)|uk(t)|≤ηfor|t|≥Π3,k∈N.
Thus, from (F2), (3.7), (4.34), and (4.35), we have
(4.36)1p‖uk‖p=dk+∑t∈ZF(t,uk(t+n),…,uk(t))=dk+∑|t|>Π3F(t,uk(t+n),…,uk(t))+∑t=-Π3Π3F(t,uk(t+n),…,uk(t))≥dk-β4p(n+1)∑|t|>Π3∑i=0n|uk(t+i)|p-∑n=-Π3Π3H(t,uk(t+n),…,uk(t))≥dk-14p‖uk‖p-∑n=-Π3Π3max⁡|uk|≤B/β|H(t,uk(t+n),…,uk(t))|.
It follows that
(4.37)dk≤54p‖uk‖p+∑t=-Π3Π3max⁡|uk|≤B/β|H(t,uk(t+n),…,uk(t))|<+∞.
This contradicts the fact that {dk}k∈ℕ is unbounded, and so {∥uk∥}k∈ℕ is unbounded.Proof of Theorem2.2.
In the proof of Theorem2.1, the condition that H(t,u(t+n),…,u(t))≥0 for (t,u(t+n),…,u(t))∈J×ℝn+1, γ=(∑i=0n|u(t+i)|p)1/p≤1 in (F1) is only used in the proofs of assumption (ii) of Lemma 3.1. Therefore, we only prove that assumption (ii) of Lemma 3.1 still holds using (F2') instead of (F2). By (F2'), it follows that
(4.38)|F(t,u(t+n),…,u(t))|≤2-pβ2p(n+1)(∑i=0n|u(t+i)|p)fort∈Z,(∑i=0n|u(t+i)|p)1/p≤η.
If ∥u∥=β1/pη:=ρ, then by Lemma  3.2, |u(t)|≤η for t∈ℤ. Set α=βηp/2p. Hence, from (3.3), (4.38), and Lemma 3.2, we have
(4.39)I(u)=1p‖u‖p-∑t∈ZF(t,u(t+n),…,u(t))≥1p‖u‖p-2-pβ2p(n+1)(∑i=0n|u(t+i)|p)≥1p‖u‖p-12p‖u‖p=12p‖u‖p=α.
equation (4.39) shows that ∥u∥=ρ implies that I(u)≥α, that is, assumption (ii) of Lemma 3.1 holds. The proof of Theorem 2.2 is completed.Proof of Theorem2.3.
We first show thatI satisfies condition (C). Assume that {uk}k∈ℕ⊂E is a (C) sequence of I, that is, {I(uk)}k∈ℕ is bounded and (1+∥uk∥)∥I′(uk)∥→0 as k→+∞. Then it follows from (3.3) and (3.4) that
(4.40)C1≥pI(uk)-〈I′(uk),uk〉=∑t∈Z[∑i=-n0F2+n+i′(t,uk(t+n),…,uk(t))uk(t-i)-pF(t,uk(t+n),…,uk(t))].
It follows from (F8) that there exists η∈(0,1) such that (4.15) holds. By (F7) and (F8), we have
(4.41)∑i=-n0F2+n+i′(t,u(t+n),…,u(t))u(t-i)>pF(t,uk(t+n),…,u(t))≥0,
and for t∈ℤ, ∑i=0n|u(t+i)|p≥ηp, we have
(4.42)F(t,uk(t+n),…,uk(t))≤[a+b(∑i=0n|u(t+i)|p)ν/p]×[∑i=-n0F2+n+i′(t,uk(t+n),…,uk(t))uk(t-i)-pF(t,uk(t+n),…,uk(t))].
It follows from (3.2), (3.3), (4.38), (4.40), (4.41), and (4.42) that
(4.43)1p‖uk‖p=I(uk)+∑t∈ZF(t,uk(t+n),…,u(t))=I(uk)+∑t∈Z((∑i=0n|u(t+i)|p)1/p≤η)F(t,uk(t+n),…,u(t))+∑t∈Z((∑i=0n|u(t+i)|p)1/p>η)F(t,uk(t+n),…,u(t))≤I(uk)+2-pβ2p(n+1)∑t∈Z((∑i=0n|u(t+i)|p)1/p≤η)|uk(t+i)|p+∑t∈Z((∑i=0n|u(t+i)|p)1/p>η)[a+b(∑i=0n|u(t+i)|p)ν/p]×[∑i=-n0F2+n+i′(t,uk(t+n),…,uk(t))uk(t-i)-pF(t,uk(t+n),…,uk(t))]≤C2+12p‖uk‖2+∑t∈Z[a+b(∑i=0n|u(t+i)|p)ν/p]×[∑i=-n0F2+n+i′(t,uk(t+n),…,uk(t))uk(t-i)-pF(t,uk(t+n),…,uk(t))]≤C2+12p‖uk‖p+(a+b‖uk‖∞ν)∑t∈Z[(∇W(n,uk(n)),uk(n))-pW(n,uk(n))]≤C2+12p‖uk‖p+C1(a+b(n+1)‖uk‖∞ν)≤C2+12p‖uk‖p+C1{a+β-ν/pb(n+1)‖uk‖ν},k∈N.
Since ν<p, it follows from (4.43) that {∥uk∥}k∈ℕ is bounded. Similar to the proof of Theorem 2.1, we can prove that {uk} has a convergent subsequence in E. Hence, I satisfies condition (C).
It is obvious thatI is even and I(0)=0, and so assumption (i) of Lemma 3.1 holds. The proof of assumption (ii) of Lemma 3.1 is the same as in the proof of Theorem 2.2.
Finally, it remains to show thatI satisfies assumption (iii) of Lemma 3.1. Let E′ be a finite dimensional subspace of E. Since all norms of a finite dimensional normed space are equivalent, so there is a constant d>0 such that (4.21) holds. Assume that dimE′=m and u1,u2,…,um are the bases of E′ such that (4.22) holds. Let η, Π1, and Θ be the same as in the proof of Theorem 2.1. Then (4.23), (4.25), and (4.26) hold. For the Π1 given in the proof of Theorem 2.1. By (F9), there exists σ0=σ0(d,Π1)>1 such that
(4.44)s-pmin⁡|x|=1W(n,sx)≥(2d)pfors≥σ02,n∈Z(-Π1,Π1).
It follows from (F7), (F9), (3.3), and (4.44) that
(4.45)I(σu)=σpp‖u‖p-∑t∈ZF(t,σu(t+n),…,σu(t))≤σpp‖u‖p-F(t0,σu(t0))≤σpp‖u‖p-min⁡|x|=1F(t0,σ|u(t0)|x)≤(dσ)pp-(dσ)p|u(t0)|p≤(dσ)pp-(dσ)p=-(dσ)pq,u∈Θ,σ≥σ0.
We deduce that there is σ0=σ0(d,Π1)=σ0(E′)>1 such that
(4.46)I(σu)<0foru∈Θ,σ≥σ0.
That is
(4.47)I(u)<0foru∈E′,‖u‖≥dσ0.
This shows that condition (iii) of Lemma 3.1 holds. By Lemma 3.1, I possesses an unbounded sequence {dk}k∈ℕ of critical values with dk=I(uk), where uk is such that I′(uk)=0 for k=1,2,…. Since
(4.48)1p‖uk‖p=dk+∑n∈ZF(t,uk(t+n),…,uk(t))≥dk
and {dk}k∈ℕ is unbounded, it follows that {∥uk∥}k∈ℕ is unbounded. The proof is complete.


## 5. Examples

In this section, we give some examples to illustrate our results.Example 5.1.
In (1.1), let q(t)→+∞ as |t|→+∞ and
(5.1)F(t,u(t+n),…,u(t))=[|u(t)|4+|t|-(∑i=0nu2(t+i))(5+3|t|)/2(4+|t|)].
Let p=2, μ=4, ϱ=3, J={-3,-2,-1,0,1,2,3} and
(5.2)W(t,u(t))=|u(t)|4+|t|,H(t,u(t+n),…,u(t))=(∑i=0nu2(t+i))(5+3|t|)/2(4+|t|).
Then it is easy to verify that all conditions of Theorem 2.1 are satisfied. By Theorem 2.1, (1.1) has a nontrivial homoclinic solution.Example 5.2.
In (1.1), let r(t)>0, q(t)→+∞ as |t|→+∞ and
(5.3)F(t,u(t+n),…,u(t))=(∑k=1m1ak|u(t)|μk-∑j=1m2bj(∑i=0n|u(t+i)|p)ϱj/p),
where μ1>μ2>⋯>μm1>ϱ1>ϱ2>⋯>ϱm2>p, ak,bj>0, k=1,2,…,m1; j=1,2,…,m2. Let μ=μm1, ϱ=ϱ1, and
(5.4)W(t,u(t))=∑k=1m1ak|u(t)|μk,H(t,u(t+n),…,u(t))=∑j=1m2bj(∑i=0n|u(t+i)|p)ϱj/p.
Then it is easy to verify that all conditions of Theorem 2.2 are satisfied. By Theorem 2.2, (1.1) has a nontrivial homoclinic solution.Example 5.3.
In (1.1), let
(5.5)F(t,u(t+n),…,u(t))=q(t)∑i=0n|u(t+i)|pln⁡[1+(∑i=0n|u(t+i)|p)1/p],
where q:ℤ→(0,∞) such that q(t)→+∞ as |t|→+∞. Since
(5.6)∑i=-n0F2+n+i′(t,u(t+n),…,u(t))u(t-i)=q(t)[p∑i=0n|u(t+i)|pln⁡[1+(∑i=0n|u(t+i)|p)1/p]+(∑i=0n|u(t+i)|p)p+1/p1+(∑i=0n|u(t+i)|p)1/p]≥(p+11+(∑i=0n|u(t+i)|p)1/p)F(t,u(t+n),…,u(t))≥0,∀(t,u(t+n),…,u(t))∈Z×Rn+1.
This shows that (F8) holds with a=b=ν=1. It is easy to verify that assumptions (q), (F1), and (F7) of Theorem 2.3 are satisfied. By Theorem 2.3, (1.1) has an unbounded sequence of homoclinic solutions.



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*Source: 297618-2012-01-31.xml*