Volume 2008, Article ID 247071,11pages doi:10.1155/2008/247071
Research Article
Existence Theorems of Periodic Solutions for
Second-Order Nonlinear Difference Equations
Xiaochun Cai1 and Jianshe Yu2
1College of Statistics, Hunan University, Changsha, Hunan 410079, China
2College of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, China
Correspondence should be addressed to Xiaochun Cai,[email protected]
Received 14 August 2007; Accepted 14 November 2007 Recommended by Patricia J. Y. Wong
The authors consider the second-order nonlinear difference equation of the typeΔpnΔxn−1δ
qnxδnfn, xn, n∈Z, using critical point theory, and they obtain some new results on the existence
of periodic solutions.
Copyrightq2008 X. Cai and J. Yu. 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.
1. Introduction
We denote byN, Z, Rthe set of all natural numbers, integers, and real numbers, respectively. Fora, b∈Z, defineZa {a, a1, . . .}, Za, b {a, a1, . . . , b}whena≤b.
Consider the nonlinear second-order difference equation
Δpn
Δxn−1
δ
qnxδnf
n, xn
, n∈Z, 1.1
where the forward difference operatorΔis defined by the equationΔxnxn1−xnand
Δ2x
n−1 ΔΔxn−1 Δxn−Δxn−1. 1.2
In 1.1, the given real sequences {pn}, {qn}satisfypnT pn > 0, qnT qn for anyn ∈ Z,
f :Z×R→Ris continuous in the second variable, andfnT, z fn, zfor a given positive integerTand for alln, z∈Z×R.−1δ −1, δ >0,andδis the ratio of odd positive integers. By a solution of1.1, we mean a real sequencex{xn}, n∈Z, satisfying1.1.
In1,2 , the qualitative behavior of linear difference equations of type
has been investigated. In3 , the nonlinear difference equation
ΔpnΔxn−1 qnxnfn, xn 1.4
has been considered. However, results on periodic solutions of nonlinear difference equations are very scarce in the literature, see 4, 5 . In particular, in6 , by critical point method, the existence of periodic and subharmonic solutions of equation
Δ2x n−1f
n, xn
0, n∈Z, 1.5
has been studied. Other interesting contributions can be found in some recent papers 7–11
and in references contained therein. It is interesting to study second-order nonlinear difference equations1.1because they are discrete analogues of differential equation
ptϕuft, u 0. 1.6
In addition, they do have physical applications in the study of nuclear physics, gas aerody-namics, infiltrating medium theory, and plasma physics as evidenced in12,13 .
The main purpose here is to develop a new approach to the above problem by using critical point method and to obtain some sufficient conditions for the existence of periodic solutions of1.1.
LetXbe a real Hilbert space,I ∈C1X,R, which implies thatIis continuously Fr´echet
differentiable functional defined onX.I is said to be satisfying Palais-Smale condition P-S conditionif any sequence{Iun}is bounded, andIun → 0 asn→ ∞possesses a
conver-gent subsequence inX. LetBρ be the open ball inX with radiusρand centered at 0, and let
∂Bρdenote its boundary.
Lemma 1.1 mountain pass lemma, see14 . LetX be a real Hilbert space, and assume thatI ∈ C1X,Rsatisfies the P-S condition and the following conditions:
I1there exist constantsρ > 0anda > 0such thatIx≥ afor allx ∈∂Bρ, whereBρ {x∈ X:
xX < ρ};
I2I0≤0and there existsx0∈Bρsuch thatIx0≤0.
Thencinfh∈Γsups∈0,1Ihsis a positive critical value ofI, where
Γ h∈C0,1 , X:h0 0, h1 x0
. 1.7
Lemma 1.2saddle point theorem, see14,15 . LetXbe a real Banach space,XX1 ⊕X2,where
X1/{0}and is finite dimensional. SupposeI∈C1X,Rsatisfies the P-S condition and I3there exist constantsσ,ρ >0such thatI|∂BρX1 ≤σ;
I4there ise∈Bp
X1and a constantω > σsuch thatI|eX2 ≥ω.
ThenIpossesses a critical valuec≥ωand
cinf
h∈Γu∈maxBρX1
Ihu, 1.8
whereΓ {h∈CBρ
2. Preliminaries
In this section, we are going to establish the corresponding variational framework for1.1. LetΩbe the set of sequences
xxn
n∈Z
. . . , x−n, . . . , x−1, x0, x1, . . . , xn, . . .
, 2.1
that is,
Ω xxn
:xn∈R, n∈Z
. 2.2
For anyx, y∈Ω, a, b∈R,axbyis defined by
axby:axnbyn
∞
n−∞. 2.3
ThenΩis a vector space. For given positive integerT, ET is defined as a subspace ofΩby
ET
xxn
∈Ω:xnT xn, n∈Z
. 2.4
Clearly,ET is isomorphic toRT, and can be equipped with inner product
x, y
T
i1
xiyi, ∀x, y ∈ET, 2.5
by which the norm·can be induced by
x:
T
i1
x2i
1/2
, ∀x∈ET. 2.6
It is obvious that ET with the inner product defined by 2.5 is a finite-dimensional Hilbert
space and linearly homeomorphic toRT. Define the functionalJonE
T as follows:
Jx 1
δ1
T
n1
pnΔxn−1δ1− 1
δ1
T
n1
qnxnδ1 T
n1
Fn, xn, ∀x∈ET, 2.7
where Ft, z 0z ft, sds. Clearly,J ∈ C1E
T,R, and for any x {xn}n∈Z ∈ ET, by using
x0xT, x1xT1, we can compute the partial derivative as
∂J
∂xn −ΔpnΔxn−1
δ −q
nxδnfn, xn, n∈Z1, T. 2.8
Thusx{xn}n∈Zis a critical point ofJ onET i.e., Jx 0if and only if
ΔpnΔxn−1δ qnxnδ fn, xn, n∈Z1, T. 2.9
By the periodicity of xn and fn, z in the first variable n, we have reduced the existence
of periodic solutions of 1.1 to that of critical points of J on ET. In other words, the
func-tionalJ is just the variational framework of 1.1. For convenience, we identifyx ∈ ET with
x x1, x2, . . . , xTT. Denote W {x1, x2, . . . , xTT ∈ ET : xi ≡ v, v ∈ R, i ∈ Z1, T} and W⊥ Y such that E
T Y⊕W. Denote other norm ·r on ET as follows see, e.g., 16 :
xr
T
i1|xi|r1/r, for all x ∈ ET and r > 1. Clearly, x2 x. Due to ·r1 and ·r2
being equivalent whenr1, r2 > 1,there exist constantsc1,c2, c3, andc4such that c2 ≥ c1 > 0,
c4≥c3>0, and
c1x ≤ xδ1≤c2x, 2.10
c3x ≤ xβ≤c4x, 2.11
3. Main results
In this section, we will prove our main results by using critical point theorem. First, we prove two lemmas which are useful in the proof of theorems.
Lemma 3.1. Assume that the following conditions are satisfied:
F1there exist constantsa1>0,a2>0, andβ > δ1such that
ThenA0>0. Also, by the above inequality, we have
In view of
T
n1
xl
n δ1≤Tβ−δ−1/β
T
n1
xl
n β
δ1/β
, 3.7
we have
xlβ
β ≥T
δ1−β/δ1xlβ
δ1. 3.8
Then we get
−M≤Jxl≤ A0 δ1x
lδ1
δ1−a1Tδ
1−β/δ1xlβ
δ1a2T. 3.9
Therefore, for anyl∈N,
a1Tδ1−β/δ1xlβδ1− A0
δ1x
lδ1
δ1≤Ma2T. 3.10
Sinceβ > δ1,the above inequality implies that{xl}is a bounded sequence inET.Thus{xl}
possesses convergent subsequences, and the proof is complete.
Theorem 3.2. Suppose thatF1and following conditions hold: F3for eachn∈Z,
lim
z→0
fn, z
zδ 0; 3.11
F4
qn<0, ∀n∈Z1, T. 3.12
Then there exist at least two nontrivialT-periodic solutions for1.1.
Proof. We will useLemma 1.1to proveTheorem 3.2. First, byLemma 3.1,Jsatisfies P-S condi-tion. Next, we will prove that conditionsI1andI2hold. In fact, byF3, there existsρ >0
such that for any|z|< ρandn∈Z1, T,
|Fn, z| ≤ − qmax
2δ1z
δ1
, 3.13
whereqmax maxn∈Z1,Tqn<0.Thus for anyx∈ET, x ≤ρfor alln∈Z1, T,we have
Jx≥ −qmax δ1
T
n1
xδn1 qmax
2δ1
T
n1
xδn1
− qmax
2δ1x
δ1 δ1
≥ − qmax
2δ1c
δ1
1 xδ
1
2 .
Takinga−cδ1
1 qmax/2δ1ρδ1,we have
Jx|∂Bρ ≥a >0, 3.15
and the assumptionI1is verified. Clearly,J0 0.For any givenw∈ET withw1 and a
constantα >0,
Jαw 1
δ1
T
n1
pnαwn−αwn−1δ1−qnαwnδ1 T
n1
Fn, αwn
≤ 1
δ1
T
n1
pn2αδ1−qnαδ1 −a1 T
n1
|αwn|βa2T
≤ 1
δ1
T
n1
2δ1pn−qn αδ1αδ1−a1T2−β/2αβa2T
−→ − ∞, α−→ ∞.
3.16
Thus we can easily choose a sufficiently largeαsuch thatα > ρand forxαw∈ET,Jx<0.
Therefore, byLemma 1.1, there exists at least one critical valuec≥a >0.We suppose thatxis a critical point corresponding toc, that is,Jx c,andJx 0.By a similar argument to the proof ofLemma 3.1, for anyx∈ET,there existsx∈ET such thatJx cmax. Clearly,x /0.If
x /x, and the proof is complete; otherwise,xxandccmax.ByLemma 1.1,
cinf
h∈Γ
sup
s∈0,1
Jhs, 3.17
whereΓ {h∈C0,1 , ET|h0 0, h1 x}.Then for anyh∈Γ, cmax maxs∈0,1Jhs.
By the continuity of Jhs in s, J0 ≤ 0 and Jx < 0 show that there exists some s0 ∈ 0,1 such that Jhs0 cmax. If we choose h1, h2 ∈ Γ such that the intersection {h1s |
s ∈ 0,1}{h2s | s ∈ 0,1}is empty, then there exists1, s2 ∈ 0,1such thatJh1s1
Jh2s2 cmax.Thus we obtain two different critical pointsx1 h1s1,x2 h2s2ofJ in
ET. In this case, in fact, we may obtain at least two nontrivial critical points which correspond
to the critical valuecmax.The proof ofTheorem 3.2is complete. Whenfn, xn≡hn, we have
the following results.
Theorem 3.3. Assume that the following conditions hold:
G1
qn <0, ∀n∈Z1, T; 3.18
G2
1
cδ11
T
n1
h2n
δ1/2T
n1
−qn<
pminλ2δ1/2−qmax
T
n1
hn δ1
wherepmin minn∈Z1,Tpn, qmax maxn∈Z1,Tqn, c1is a constant in2.10, andλ2is the minimal positive eigenvalue of the matrix
A
possesses at least oneT-periodic solution.
First, we proved the following lemma.
Lemma 3.4. Assume thatG1holds, then the functional
Jx 1
satisfies P-S condition onET.
Proof. For any sequence{xl} ⊂ E
possesses a convergent subsequence, and the proof ofLemma 3.4is complete. Now we prove
as one finds by minimizing with respect tox.That is
This implies that the assumption of saddle point theorem is satisfied. Thus there exists at least one critical point ofJ onET, and the proof is complete. When qn > 0,we have the following
result.
Theorem 3.5. Assume that the following conditions are satisfied: G32δ1pnpn1 < qn, qn >0for alln∈Z1, T;
Then3.21possesses at least oneT-periodic solution.
Before provingTheorem 3.5, first, we prove the following result.
That is,
possesses convergent subsequences, and the proof is complete.
Proof ofTheorem 3.5. For anyw z, z, . . . , zT ∈W,we have
satisfies the assumption of saddle point theorem, that is, there exists at least one critical point ofJonET.This completes the proof ofTheorem 3.5.
Acknowledgment
References
1 C. D. Ahlbrandt and A. C. Peterson,Discrete Hamiltonian Systems. Difference Equations, Continued Frac-tions, and Riccati EquaFrac-tions, vol. 16 ofKluwer Texts in the Mathematical Sciences, Kluwer Academic, Dor-drecht, The Netherlands, 1996.
2 S. S. Cheng, H. J. Li, and W. T. Patula, “Bounded and zero convergent solutions of second-order dif-ference equations,”Journal of Mathematical Analysis and Applications, vol. 141, no. 2, pp. 463–483, 1989. 3 T. Peil and A. Peterson, “Criteria forC-disfocality of a selfadjoint vector difference equation,”Journal
of Mathematical Analysis and Applications, vol. 179, no. 2, pp. 512–524, 1993.
4 F. Dannan, S. Elaydi, and P. Liu, “Periodic solutions of difference equations,”Journal of Difference Equa-tions and ApplicaEqua-tions, vol. 6, no. 2, pp. 203–232, 2000.
5 S. Elaydi and S. Zhang, “Stability and periodicity of difference equations with finite delay,”Funkcialaj Ekvacioj, vol. 37, no. 3, pp. 401–413, 1994.
6 Z. M. Guo and J. S. Yu, “The existence of periodic and subharmonic solutions for second-order super-linear difference equations,”Science in China Series A, vol. 3, pp. 226–235, 2003.
7 R. P. Agarwal,Difference Equations and Inequalities: Theory, Methods, and Applications, vol. 228 of Mono-graphs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.
8 R. P. Agarwal and P. J. Y. Wong,Advanced Topics in Difference Equations, vol. 404 ofMathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1997.
9 M. Cecchi, Z. Doˇsl´a, and M. Marini, “Positive decreasing solutions of quasi-linear difference equa-tions,”Computers & Mathematics with Applications, vol. 42, no. 10-11, pp. 1401–1410, 2001.
10 P. J. Y. Wong and R. P. Agarwal, “Oscillations and nonoscillations of half-linear difference equations generated by deviating arguments,”Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 11–26, 1998, Advances in difference equations, II.
11 P. J. Y. Wong and R. P. Agarwal, “Oscillation and monotone solutions of second order quasilinear difference equations,”Funkcialaj Ekvacioj, vol. 39, no. 3, pp. 491–517, 1996.
12 M. Cecchi, M. Marini, and G. Villari, “On the monotonicity property for a certain class of second order differential equations,”Journal of Differential Equations, vol. 82, no. 1, pp. 15–27, 1989.
13 M. Marini, “On nonoscillatory solutions of a second-order nonlinear differential equation,”Bollettino della Unione Matematica Italiana, vol. 3, no. 1, pp. 189–202, 1984.
14 P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, vol. 65 ofCBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1986.
15 J. Mawhin and M. Willem,Critical Point Theory and Hamiltonian Systems, vol. 74 ofApplied Mathematical Sciences, Springer, New York, NY, USA, 1989.
