doi:10.1155/2010/470375
Research Article
Existence of Homoclinic Solutions for a Class of
Nonlinear Difference Equations
Peng Chen and X. H. Tang
School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China
Correspondence should be addressed to X. H. Tang,[email protected]
Received 5 May 2010; Accepted 2 August 2010
Academic Editor: Jianshe Yu
Copyrightq2010 P. Chen and X. H. Tang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By using the critical point theory, we establish some existence criteria to guarantee that the nonlinear difference equationΔpnΔxn−1δ−qnxnδ fn, xn has at least one homoclinic solution, wheren ∈ Z, xn ∈ R, andf : Z×R → Ris non periodic inn. Our conditions on the nonlinear termfn, xnare rather relaxed, and we generalize some existing results in the literature.
1. Introduction
Consider the nonlinear difference equation of the form
ΔpnΔun−1δ−qnxnδfn, un, n∈Z, 1.1
whereΔis the forward difference operator defined by Δun un 1−un,Δ2un
ΔΔun,δ > 0 is the ratio of odd positive integers,{pn}and{qn}are real sequences, {pn}/0.f :Z×R → R. As usual, we say that a solutionunof1.1is homoclinicto 0 ifun → 0 asn → ±∞. In addition, ifun/≡0, thenunis called a nontrivial homoclinic solution.
difference equation, such as stability, attractiveness, periodicity, oscillation, and boundary value problem. Recently, there are some new results on periodic solutions of nonlinear difference equations by using the critical point theory in the literature; see1–3.
In general, 1.1 may be regarded as a discrete analogue of a special case of the following second-order differential equation:
ptϕx ft, x 0, 1.2
which has arose in the study of fluid dynamics, combustion theory, gas diffusion through porous media, thermal self-ignition of a chemically active mixture of gases in a vessel, catalysis theory, chemically reacting systems, and adiabatic reactorsee, e.g.,4–6and their references. In the case ofϕx |x|δ−2x,1.2has been discussed extensively in the literature; we refer the reader to the monographs7–10.
It is well known that the existence of homoclinic solutions for Hamiltonian systems and their importance in the study of the behavior of dynamical systems have been already recognized from Poincar´e; homoclinic orbits play an important role in analyzing the chaos of dynamical system. In the past decade, this problem has been intensively studied using critical point theory and variational methods.
In some recent papers 1–3, 11–14, the authors studied the existence of periodic solutions, subharmonic solutions, and homoclinic solutions of some special forms of1.1by using the critical point theory. These papers show that the critical point method is an effective approach to the study of periodic solutions for difference equations. Along this direction, Ma and Guo13applied the critical point theory to prove the existence of homoclinic solutions of the following special form of1.1:
ΔpnΔun−1−qnun fn, un 0, 1.3
wheren∈Z,u∈R,p, q:Z → R, andf :Z×R → R.
Theorem Asee13. Assume thatp, q,andfsatisfy the following conditions:
ppn>0for all n∈Z;
qqn>0for alln∈Zandlim|n| → ∞qn ∞;
f1there is a constantμ >2such that
0< μ
x
0
fn, sds≤xfn, x, ∀n, x∈Z×R\ {0}; 1.4
f2 limx→0fn, x/x0uniformly with respect ton∈Z.
Then1.3possesses a nontrivial homoclinic solution.
AR For everyn∈Z,Wis continuously differentiable inx, and there is a constant μ >
2 such that
0< μWn, x≤∇Wn, x, x, ∀n, x∈Z× RN\ {0}. 1.5
However, it seems that results on the existence of homoclinic solutions of 1.1 by critical point method have not been considered in the literature. The main purpose of this paper is to develop a new approach to the above problem by using critical point theory.
Motivated by the above papers 13, 14, we will obtain some new criteria for guaranteeing that1.1has one nontrivial homoclinic solution without any periodicity and generalize Theorem A. Especially, Fn, xsatisfies a kind of new superquadratic condition which is different from the corresponding condition in the known literature.
In this paper, we always assume thatFn, x 0xfn, sds,F1n, x
x
0 f1n, sds,
F2n, x
x
0 f2n, sds. Our main results are the following theorems.
Theorem 1.1. Assume thatp, q,andfsatisfy the following conditions:
ppn>0for alln∈Z;
qqn>0for alln∈Zandlim|n| → ∞qn ∞;
F1Fn, x F1n, x−F2n, x, for everyn∈Z,F1andF2are continuously differentiable inx, and there is a bounded setJ ⊂Zsuch that
F2n, x≥0, ∀n, x∈J×R, |x| ≤1,
1
qnfn, xo |x|
δ asx−→0
1.6
uniformly inn∈Z\J;
F2there is a constantμ > δ 1such that
0< μF1n, x≤xf1n, x, ∀n, x∈Z×R\ {0}; 1.7
F3F2n,0≡0,and there is a constant∈δ 1, μsuch that
xf2n, x≤F2n, x, ∀n, x∈Z×R. 1.8
Theorem 1.2. Assume thatp,q,andFsatisfyp,q,F2,F3,and the following assumption:
F1’Fn, x F1n, x−F2n, x, for everyn∈Z,F1andF2are continuously differentiable inx,and
1
qnfn, xo |x|
δ as x−→0 1.9
uniformly inn∈Z. Then1.1possesses a nontrivial homoclinic solution.
Remark 1.3. Obviously, both conditionsF1andF1are weaker thanf1. Therefore, both Theorems 1.1and1.2generalize Theorem A by relaxing conditionsf1andf2.
When Fn, x is subquadratic at infinity, as far as the authors are aware, there is no research about the existence of homoclinic solutions of 1.1. Motivated by the paper
16, the intention of this paper is that, under the assumption thatFn, xis indefinite sign and subquadratic as|n| → ∞, we will establish some existence criteria to guarantee that
1.1has at least one homoclinic solution by using minimization theorem in critical point theory.
Now we present the basic hypothesis onp,q,andF in order to announce the results in this paper.
F4For every n ∈ Z, F is continuously differentiable inx, and there exist two constants 1< γ1< γ2< δ 1and two functionsa1, a2∈lδ 1/δ 1−γ1Z,0, ∞such that
|Fn, x| ≤a1n|x|γ1, ∀n, x∈Z×R, |x| ≤1,
|Fn, x| ≤a2n|x|γ2, ∀n, x∈Z×R, |x| ≥1.
1.10
F5There exist two functions b ∈ lδ 1/δ 1−γ1Z,0, ∞ and ϕ ∈ C0, ∞,0, ∞
such that
fn, x≤bnϕ|x|, ∀n, x∈Z×R 1.11
whereϕs Osγ1−1as|s| ≤c,cis a positive constant.
F6There existn0∈Zand two constantsη >0andγ3∈1, δ 1such that
Fn0, x≥η|x|γ3, ∀x∈R, |x| ≤1. 1.12
Theorem 1.4. Assume thatp,q,andFsatisfyp,q,F4,F5,andF6. Then1.1possesses at least one nontrivial homoclinic solution.
ByTheorem 1.4, we have the following corollary.
Corollary 1.5. Assume thatp,q, andFsatisfyp,q,and the following conditions:
F7Fn, x anVx, whereV ∈ C1R,Rand a ∈ lδ 1/δ 1−γ1Z,0, ∞,γ
1 ∈
1, δ 1is a constant such thatan0>0for somen0∈Z.
F8There exist constantsM, M>0,γ2∈γ1, δ 1,andγ3∈1, δ 1such that
M|x|γ3≤Vx≤M|x|γ1, ∀x∈R, |x| ≤1,
0< Vx≤M|x|γ2, ∀x∈R, |x| ≥1,
1.13
F9Vx O|x|γ1−1as|x| ≤c, cis a positive constant.
Then1.1possesses at least one nontrivial homoclinic solution.
2. Preliminaries
Let
S{{un}n∈Z:un∈R, n∈Z},
E
u∈S: n∈Z
pnΔun−1δ 1 qnunδ 1< ∞
,
2.1
and foru∈E, let
u
n∈Z
pnΔun−1δ 1 qnunδ 1< ∞
1/δ 1
, u∈E. 2.2
ThenEis a uniform convex Banach space with this norm. As usual, for 1≤p < ∞, let
lpZ,R
u∈S: n∈Z
|un|p< ∞
,
l∞Z,R
u∈S: sup
n∈Z|un|< ∞
,
and their norms are defined by
up
n∈Z
|un|p
1/p
, ∀u∈lpZ,R; u∞sup
n∈Z|un|, ∀u∈l
∞Z,R, 2.4
respectively.
For anyn1, n2 ∈ Z with n1 < n2, we let Zn1, n2 n1, n2∩Z, and for function
f:Z → Randa∈R, we set
Zfn≥an∈Z:fn≥a, Zfn≤an∈Z:fn≤a. 2.5
LetI:E → Rbe defined by
Iu 1 δ 1u
δ 1− n∈Z
Fn, un. 2.6
Ifp,q,andF1,F1,orF4holds, thenI ∈C1E,R,and one can easily check that
Iu, v
n∈Z
pnΔun−1δΔvn−1 qnunδvn−fn, unvn ∀u, v∈E.
2.7
Furthermore, the critical points ofIinEare classical solutions of1.1withu±∞ 0. We will obtain the critical points ofIby using the Mountain Pass Theorem. We recall it and a minimization theorem as follows.
Lemma 2.1see15,17. LetEbe a real Banach space andI ∈C1E,Rsatisfy (PS)-condition.
Suppose thatIsatisfies the following conditions:
iI0 0;
iithere exist constantsρ, α >0such thatI|∂Bρ0 ≥α;
iiithere existse∈E\Bρ0such thatIe≤0.
ThenIpossesses a critical valuec≥αgiven by
cinf g∈Γsmax∈0,1I
gs, 2.8
Lemma 2.2. Foru∈E
quδ∞ 1≤ uδ 1, 2.9
whereqinfn∈Zqn.
Proof. Sinceu∈E, it follows that lim|n| → ∞|un|0. Hence, there existsn∗∈Zsuch that
|un∗|max
n∈Z|un|. 2.10
So, we have
uδE 1≥
n∈Z
qnunδ 1≥q
n∈Z
|un|δ 1≥quδ∞ 1. 2.11
The proof is completed.
Lemma 2.3. Assume thatF2andF3hold. Then for everyn, x∈Z×R,
is−μF
1n, sxis nondecreasing on0, ∞;
iis−F
2n, sxis nonincreasing on0, ∞.
The proof ofLemma 2.3is routine and so we omit it.
Lemma 2.4see18. LetEbe a real Banach space andI ∈C1E,Rsatisfy the (PS)-condition. If
Iis bounded from below, thencinfEIis a critical value ofI.
3. Proofs of Theorems
Proof ofTheorem 1.1.In our case, it is clear thatI0 0. We show thatI satisfies thePS -condition. Assume that {uk}k∈N ⊂ E is a sequence such that {Iuk}k∈N is bounded and Iuk → 0 ask → ∞. Then there exists a constantc >0 such that
|Iuk| ≤c, IukE∗≤c fork∈N. 3.1
From2.6,2.7,3.1,F2, andF3, we obtain
δ 1c δ 1cuk
≥δ 1Iuk−
δ 1
Iuk, uk
−δ 1
uk
δ 1 δ 1 n∈Z
F2n, ukn− 1
uknf2n, ukn
−δ 1 n∈Z
F1n, ukn− 1
uknf1n, ukn
≥ −δ 1
uk
δ 1, k∈N.
It follows that there exists a constantA >0 such that
uk ≤A fork∈N. 3.3
Then,ukis bounded inE. Going if necessary to a subsequence, we can assume thatuk u0 inE. For any given numberε >0, byF1, we can chooseζ >0 such that
fn, x≤εqn|x|δ forn∈Z\J, x∈R, |x| ≤ζ. 3.4
Sinceqn → ∞, we can also choose an integerΠ>max{|k|:k∈J}such that
qn≥ A
δ 1
ζδ 1, |n| ≥Π. 3.5
By3.3and3.5, we have
|ukn|δ 1 1
qnqn|ukn|
δ 1≤ ζδ 1
Aδ 1uk
δ 1≤ζδ 1, for|n| ≥Π, k∈N. 3.6
Sinceuk u0inE, it is easy to verify thatuknconverges tou0npointwise for alln∈Z, that is,
lim
k→ ∞ukn u0n, ∀n∈Z. 3.7
Hence, we have by3.6and3.7
|u0n| ≤ζ, for|n| ≥Π. 3.8
It follows from3.7and the continuity offn, xonxthat there existsk0∈Nsuch that
Π
n−Π
On the other hand, it follows from3.3,3.4,3.6, and3.8that
It follows from2.7and the H ¨older’s inequality that
≥ ukδ 1 u0δ 1−
Equation3.18shows thatuρimplies thatIu≥α, that is,Isatisfies assumptioniiof
Lemma 2.1. Finally, it remains to show thatIsatisfies assumptioniiiofLemma 2.1. For any
Sinceμ > ≥ δ 1 andm > 0,3.23implies that there existsσ0 > 1 such thatσ0ω > ρ andIσ0ω < 0. Sete σ0ωn. Thene ∈ E,e σ0ω > ρ,andIe Iσ0ω < 0. By
Lemma 2.1,Ipossesses a critical valued≥αgiven by
dinf g∈Γsmax∈0,1I
gs, 3.24
where
Γ g∈C0,1, E:g0 0, g1 e. 3.25
Hence, there existsu∗∈Esuch that
Iu∗ d, Iu∗ 0. 3.26
Then function u∗ is a desired classical solution of 1.1. Since d > 0, u∗ is a nontrivial homoclinic solution. The proof is complete.
Proof ofTheorem 1.2. In the proof ofTheorem 1.1, the condition thatF2n, x ≥ 0 forn, x ∈
J×R,|x| ≤1 inF1, is only used in the the proofs of assumptioniiofLemma 2.1. Therefore, we only proves assumptioniiofLemma 2.1still hold that usingF1’instead ofF1. By
F1’, there existsη∈0,1such that
fn, x≤ 1 2qn|x|
δ forn, x∈Z×R, |x| ≤η. 3.27
SinceFn,0≡0, it follows that
|Fn, x|≤ 1
2δ 1qn|x|
δ 1 forn, x∈Z×R, |x| ≤η. 3.28
Ifuq1/δ 1η:ρ, then byLemma 2.2,|un| ≤ηforn∈Z. Setαqηδ 1/2δ 1. Hence, from2.6and3.28, we have
Iu 1 δ 1u
δ 1− n∈Z
Fn, un
≥ 1
δ 1u
δ 1− 1 2δ 1
n∈Z
qnunδ 1
≥ 1
δ 1u
δ 1− 1 2δ 1u
δ 1
1
2δ 1u δ 1
α.
3.29
Equation3.29shows thatuρimplies thatIu≥α, that is, assumptioniiofLemma 2.1
Since 1< γ1 < γ2 < δ 1,3.30implies thatIu → ∞asu → ∞. Consequently,I is bounded from below.
Next, we prove that I satisfies the PS-condition. Assume that {uk}k∈N ⊂ E is a sequence such that{Iuk}k∈Nis bounded andIuk → 0 ask → ∞. Then by2.6,2.9, and3.30, there exists a constantA >0 such that
uk∞≤q−1/δ 1uk ≤A, k∈N. 3.31
So passing to a subsequence if necessary, it can be assumed thatuk u0inE. It is easy to verify thatuknconverges tou0npointwise for alln∈Z, that is,
lim
k→ ∞ukn u0n, ∀n∈Z. 3.32
Hence, we have, by3.31and3.32,
u0∞≤A. 3.33
ByF5, there existsM2>0 such that
ϕ|x|≤M2|x|γ1−1, ∀x∈R, |x| ≤A. 3.34
For any given numberε >0, byF5, we can choose an integerΠ>0 such that
⎛ ⎝
|n|>Π
|bn|δ 1/δ 1−γ1
⎞ ⎠
δ 1−γ1/δ 1
< ε. 3.35
It follows from3.32and the continuity offn, xonxthat there existsk0∈Nsuch that
Π
n−Π
On the other hand, it follows from3.31,3.33,3.34,3.35, andF5that
Similar to the proof ofTheorem 1.1, it follows from3.12that
IsatisfiesPS-condition.
ByLemma 2.4,c infEIuis a critical value ofI, that is, there exists a critical point
Finally, we show thatu∗/0. Letu0n0 1 andu0n 0 forn /n0. Then byF4and
F6, we have
Isu0
sδ 1
δ 1u0
δ 1− n∈Z
Fn, su0
sδ 1
δ 1u0
δ 1−Fn
0, su0n0
≤ sδ 1
δ 1u0
δ 1−ηsγ3|u
0n0|γ3, 0< s <1.
3.40
Since 1 < γ3 < δ 1, it follows from3.40thatIsu0 < 0 fors > 0 small enough. Hence
Iu∗ c < 0, therefore u∗ is nontrivial critical point ofI, and sou∗ u∗nis a nontrivial homoclinic solution of1.1. The proof is complete.
Proof ofCorollary 1.5.Obviously,F7andF8imply thatF4holds, andF7andF9imply thatF5holds witha1n a2n bn |an|. In addition, byF7andF8, we have
Fn0, x an0Vx≥Man0|x|γ3, ∀x∈R, |x| ≤1. 3.41
This shows that F6 holds also. Hence, byTheorem 1.4, the conclusion ofCorollary 1.5 is true. The proof is complete.
4. Examples
In this section, we give some examples to illustrate our results.
Example 4.1. In1.1, letpn>0 and
Fn, x qna1|x|μ1 a2|x|μ2−2− |n||x|1−2− |n||x|2
, 4.1
whereq : Z → 0,∞ such thatqn → ∞ as|n| → ∞, μ1 > μ2 > 1 > 2 > δ 1,
a1, a2>0. Letμμ2, 1,J{−2,−1,0,1,2},and
F1n, x qn
a1|x|μ1 a2|x|μ2
, F2n, x qn
2− |n||x|1 2− |n||x|2. 4.2
Then it is easy to verify that all conditions ofTheorem 1.1are satisfied. ByTheorem 1.1,1.1
has at least a nontrivial homoclinic solution.
Example 4.2. In1.1, letpn>0,qn>0 for alln∈Zand lim|n| → ∞qn ∞, and let
Fn, x qn
⎛ ⎝m1
i1
ai|x|μi−
m2
j1
bj|x|j
⎞
whereμ1 > μ2 >· · · > μm1 > 1 > 2 >· · · > m2 > δ 1,ai, bj >0, i1,2, . . . , m1, andj
1,2, . . . , m2. Letμμm1, 1, and
F1n, x qn m1
i1
ai|x|μi, F
2n, x qn m2
j1
bj|x|j.
4.4
Then it is easy to verify that all conditions ofTheorem 1.2are satisfied. ByTheorem 1.2,1.1
has at least a nontrivial homoclinic solution.
Example 4.3. In1.1, letq:Z → 0,∞such thatqn → ∞as|n| → ∞and
Fn, x cosn
1 |n||x|
|4/3 sinn
1 |n||x|
3/2. 4.5
Then
fn, x 4 cosn
31 |n||x|
−2/3x 3 sinn 21 |n||x|
−1/2x,
|Fn, x| ≤ 2|x|
4/3
1 |n|, ∀n, x∈Z×R, |x| ≤1,
|Fn, x| ≤ 2|x|
3/2
1 |n|, ∀n, x∈Z×R, |x| ≥1,
fn, x≤8|x|
1/3 9|x|1/2
61 |n| , ∀n, x∈Z×R.
4.6
We can choosen0such that
cosn0>0, sinn0>0. 4.7
Let
η cosn0 sinn0
1 |n0|
. 4.8
Then
Fn0, x≥η|x|3/2, ∀x∈R, |x| ≤1. 4.9
These show that all conditions ofTheorem 1.4are satisfied, where
1< 4
3 γ1< γ2γ3 3
2 < δ 1, a1n a2n bn 2
1 |n|, ϕs
8s1/3 9s1/2
12 .
4.10
Acknowledgments
The authors would like to express their thanks to the referees for their helpful suggestions. This paper is partially supported by the NNSFno: 10771215of China and supported by the Outstanding Doctor degree thesis Implantation Foundation of Central South Universityno: 2010ybfz073.
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