doi:10.1155/2009/360871
Research Article
Nonlinear Discrete Periodic Boundary Value
Problems at Resonance
Ruyun Ma
1and Huili Ma
21Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
2College of Economics and Management, Northwest Normal University, Lanzhou 730070, China Correspondence should be addressed to Ruyun Ma,ruyun [email protected]
Received 25 June 2009; Revised 4 October 2009; Accepted 6 December 2009
Recommended by Kanishka Perera
LetT ∈Nbe an integer withT >2, and letT:{1, . . . , T}. We study the existence of solutions of nonlinear discrete problemsΔ2ut−1 λ
katut gt, ut ht, t∈T,u0 uT, u1
uT1,wherea, h : T → Rwitha > 0, λkis thekth eigenvalue of the corresponding linear eigenvalue problem.
Copyrightq2009 R. Ma and H. Ma. 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
Initialed by Lazer and Leach1 , much work has been devoted to the study of existence result for nonlinear periodic boundary value problem
yx m2yx gx, yxex, x∈0,2π,
y0 y2π, y0 y2π, 1.1
However, relatively little is known about the discrete analog of1.1of the form
Δ2ut−1 λ
katut gt, ut ht, t∈T,
u0 uT, u1 uT1, 1.2
whereT:{1, . . . , T},a, h :T → Rwitha >0,gt, s :T×R → Ris continuous ins. The likely reason is that the spectrum theory of the corresponding linear problem
Δ2ut−1 λ
katut 0, t∈T,
u0 uT, u1 uT1 1.3
was not established until 14 . In 14 , Wang and Shi showed that the linear eigenvalue problem1.3has exactlyT real eigenvalues
μ0< μ1≤μ2<· · ·< μT−2≤μT−1, whenT is odd,
μ0< μ1≤μ2<· · · ≤μT−2< μT−1, whenT is even.
1.4
Suppose that these above eigenvalues haveN1 different valuesλk,k 0,1, . . . , N. Then
1.4can be rewritten as
λ0< λ1<· · ·< λN. 1.5
For each λk, we denote its eigenspace byMk. If dimMk 1, then we assume that Mk :
span{ψk}in whichψkis the eigenfunction ofλk. If dimMk 2, then we assume thatMk :
span{ψk, ϕk}in whichψkandϕkare two linearly independent eigenfunctions ofλk.
It is the purpose of this paper to prove the existence results for problem1.2when there occurs resonance at the eigenvalueλkand the nonlinear functiongmay “touching” the
eigenvalueλk1. To have the wit, we have what follows.
Theorem 1.1. Leta, h:T → Rwitha >0,gt, s:T×R → Ris continuous ins, and for some
r∗<0< R∗,
gt, x≥At, ∀x≥R∗,
gt, x≤Bt, ∀x≤r∗, 1.6
whereA, B:T → Rare two given functions. Suppose for some1≤k≤N−1,
dimMk12. 1.7
whereΓ:T → Ris a given function satisfying
0≤Γt≤λk1−λk, t∈T, 1.9
and for at leastT/2 2points in1, T ,
Γt< λk1−λk, 1.10
wherer denotes the integer part of the real numberr. Then1.2has at least one solution provided
T
t1
htvt<
vt>0
gtvt
vt<0
g−tvt, 1.11
wherev∈Mk,v /0, and
gt lim inf
u→∞ gt, u, g−t lim supu→ −∞ gt, u. 1.12
In12 , Iannacci and Nkashama proved the analogue ofTheorem 1.1for continuous-time nonlinear periodic boundary value problems1.1. Our paper is motivated by Iannacci and Nkashama 12 . However, as we will see below, there are big differences between the continuous case and the discrete case. The main tool we use is the Leray-Schauder continuation theoremsee Mawhin15, Theorem IV.5 .
Finally, we note that whenat ≡ 1 in1.2, the existence of odd solutions or even solutions was investigated by R. Ma and H. Ma16 under some parity conditions on the nonlinearities. The existence of solutions of second-order discrete problem at resonance was studied by Rodriguez in17 , in which the nonlinearity is required to be bounded. For other results on discrete boundary value problems, see Kelley and Peterson 18 , Agarwal and O’Regan19 , Rachunkova and Tisdell20 , Yu and Guo21 , Atici and Cabada22 , Bai and Xu23 . However, these papers do not address the problem under “asymptotic nonuniform resonance” conditions.
2. Preliminaries
Let
T{0,1, . . . , T1}. 2.1
Let
ThenDis a Hilbert space under the inner product
u, vT
t1
atutvt, 2.3
and the corresponding norm is
u: u, u T
t1
atutut 1/2
. 2.4
Thus,
ψk, ϕk0 if dimMk2,
ψj, ψk0, forj, k∈ {0,1, . . . , N}, j /k,
ϕj, ϕk0, forj, k∈ {0,1, . . . , N}, j /k.
2.5
In the rest of the paper, we always assume that
ψk1, fork∈ {0,1, . . . , N}, ϕk1 if dimMk2.
2.6
Define a linear operatorL:D → Dby
Lut −Δ2ut−1, t∈T, Lu0: Lu1, LuT1: LuT.
2.7
Lemma 2.1see16 . Letu, w∈D. Then
T
k1
wkΔ2uk−1 −T
k1
ΔukΔwk. 2.8
Similar to12, Lemma 3 , we can prove the following.
Lemma 2.2see12 . Suppose that
ithere existA, B:T → Rand real numbersr <0< R, such that
gt, x≥At, ∀x≥R,
iithere existα, β:T → 0,∞and a constantB0>0such that
gt, u≤αt|u|βt, t∈T, |u| ≥B0. 2.10
Then for each real numberκ >0, there is a decomposition
gt, x qκt, x eκt, x 2.11
ofgsatisfying
0≤xqκt, x, t∈T, x∈R, 2.12
qκt, u≤αt|u|βt κ, t∈T, |u| ≥max{1, B0}, 2.13
and there exists a functionσκ:T → 0,∞depending onr, R,andgsuch that
|eκt, x| ≤σκt, t∈T, x∈R. 2.14
3. Existence of Periodic Solutions
In this section, we need to give some lemmas first, which have vital importance to prove Theorem 1.1.
For convenience, we set
ϕk:0, as dimMk1. 3.1
Thus, for anyu∈D, we have the following Fourier expansion:
ut a0
N
i1
aiψit biϕit, t∈T. 3.2
Let us write
ut ut u0t ut, u⊥t ut−u0t, 3.3
where
ut a0
k−1
i1
aiϕit biψit,
u0t a
kϕkt bkψkt,
ut N
ik1
aiϕit biψit.
Set
In fact, we have fromLemma 2.1that
− N
ik1
N
jk1
bibjλk1
T
t1
atψitψjt
N
jk1
a2
j
λj−λk1
N
jk1
b2
j
λj−λk1
N
jk1
a2
jbj2
λj−λk1
≥0.
3.13
Obviously, ΛΓu 0 implies that ak2 · · · aN bk2 · · ·bN 0, and accordingly
ut A0ψk1t B0ϕk1tfor someA0, B0∈R.
Next we prove thatΛΓu 0 impliesu0. Suppose to the contrary thatu /0. We note thatuhas at mostT/2 1 zeros inT. Otherwise,umust have two consecutive zeros inT, and subsequently,u≡0 in0, T1 by1.3. This is a contradiction.
Using3.6and the fact thatuhas at mostT/2 1 zeros inT, it follows that
ΛΓu T
t1
λk1at−λkat−Γtatut 2
t∈T,ut/0
atλk1−λk−Γt ut 2
>0,
3.14
which contradictsΛΓu 0. Hence,u0.
We claim that there is a constantδ2δ2Γ>0 such that
ΛΓu≥δ2u2. 3.15
Assume that the claim is not true. Then we can find a sequence{un} ⊂Dandu∈D,
such that, by passing to a subsequence if necessary,
0≤ΛΓun≤ 1n, un1, 3.16
Proof. Using the computations in the proof ofLemma 3.1and3.22, we obtain
Proof ofTheorem 1.1. The proof is motivated by Iannacci and Nkashama12 .
whereq1ande1satisfy2.12and2.14withκ1. Moreover, by2.13
Therefore,1.2is equivalent to
Δ2ut−1 λ
katut γt, utut ft, ut ht,
To prove that1.2has at least one solution inD, it suffices, according to the Leray-Schauder continuation method15 , to show that all of the possible solutions of the family of equations
Δ2ut−1 λ
katut 1−ητatut ηγt, utut ηft, ut ηht, t∈T,
u0 uT, u1 uT1
3.36
in whichη ∈0,1 ,τ ∈0, λk1−λkwithτ < δ/4,τ fixedare bounded by a constantK0 which is independent ofηandu.
Notice that, by3.32, we have
0≤1−ητat ηγt, u≤
Γt δ
2
at, t∈T, u∈R. 3.37
It is clear that forη0,3.36has only the trivial solution. Now ifu∈Dis a solution of3.36for someη∈0,1, usingLemma 3.2and Cauchy’s inequality, we obtain
0
T
t1
ut u0t−utΔ2ut−1 λ
katut 1−ητat ηγt, utut
T
t1
ut u0t−utηft, ut−ηht
≥ δ
2
u⊥2−ζuuu0νh,
3.38
where
ζ
√
T
mint∈T !
at 2
. 3.39
So we conclude that
0≥ δ
2
u⊥2−βu⊥u0, 3.40
for some constantβ > 0, depending only ona, νandhbut not onuorη. Takingαβδ−1, we get
We claim that there existsρ > 0, independent ofuand η, such that for all possible solutions of3.36
u< ρ. 3.42
Suppose on the contrary that the claim is false. Then there exists{ηn, un} ⊂0,1×D
withun ≥nand for alln∈N,
Δ2u
nt−1 λkatunt 1−ηnτatunt ηngt, unt ηnht,
un0 unT, un1 unT1.
3.43
From3.41, it can be shown that
u0
n−→ ∞, un⊥u0n
−1
−→0, 3.44
and accordingly,u⊥nu0
n−1is bounded inD.
Settingvn un/un, we have
Δ2v
nt−1 λkatvnt τatvnt
ηn
ht un
ηnτatvnt−ηn
gt, u
nt
un
, t∈T,
vn0 vnT, vn1 vnT1.
3.45
Define an operatorA:D → Dby
Awt: Δ2wt−1 λkatwt τatwt, t∈T,
Aw0: AwT, Aw1: AwT1. 3.46
ThenA−1 : D → D is completely continuous sinceDis finite dimensional. Now,3.45is equivalent to
vnt A−1
"
ηn
h· un
ηnτa·vn·−ηn
g·, u
n·
un
#
t, t∈T. 3.47
By3.26, it follows that{g·, un·/un}is bounded. Using 3.47, we may assume that
taking a subsequence and relabeling if necessaryvn → vinD, · ,||v|| 1 andv0
vT,v1 vT1.
On the other hand, using3.41, we deduce immediately that
v⊥
Therefore,
vt akϕkt bkψkt, t∈T. 3.49
Rewritevnv0nvn⊥, and let, taking a subsequence and relabeling if necessary,
v0
n−→v∗, inD. 3.50
Set
I{t∈T:v∗t>0}, I−{t∈T:v∗t<0}. 3.51
Sinceut/≡0 inT,I/∅orI−/∅.
We claim that
lim
n→ ∞unt ∞, ∀t∈I, 3.52
lim
n→ ∞unt −∞, ∀t∈I−. 3.53
We may assume thatI/∅, and only deal with the caset∈ I. The other case can be treated by similar method.
It follows from3.50that
v0
n−v∗∞:maxvn0t−v∗t|t∈T
−→0, n−→ ∞, 3.54
which implies that for allnsufficiently large,
v0
nt≥ 12v∗t>0, ∀t∈I. 3.55
On the other hand, we have from 3.44,3.55, and the fact ||un|| ≥ ||u0n|| that there exists
N >0 such that forn > Nandt∈I,
unt u0nt u⊥nt un
v0
nt u
⊥ nt
un
≥ 1
2unv 0
nt. 3.56
This together with3.55implies that forn≥N,
unt≥ 1
4unv
∗t, t∈T
. 3.57
Now let us come back to3.43. Multiplying both sides of3.43byv0
However, this contradicts1.11.
Example 3.3. By16 , the eigenvalues and eigenfunctions of
Δ2yt−1 λyt 0,
y0 y7, y1 y8 3.60
can be listed as follows:
λ00, ϕ01,
Let us consider the nonlinear discrete periodic boundary value problem
then
Γ1 λ2−λ1; Γ
j< λ2−λ1, forj2, . . . ,7. 3.65
Now, it is easy to verify thatgsatisfies all conditions ofTheorem 1.1. Consequently, for any 7-periodic functionh:Z → R,3.62has at least one solution.
Acknowledgments
This work was supported by the NSFC no. 10671158, the NSF of Gansu Province no. 3ZS051-A25-016, NWNU-KJCXGC-03-17, NWNU-KJCXGC-03-18, the Spring-Sun program no. Z2004-1-62033, SRFDPno. 20060736001, and the SRF for ROCS, SEM2006311 .
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