Volume 2010, Article ID 839474,19pages doi:10.1155/2010/839474
Research Article
Existence of Solutions of a Discrete Fourth-Order Boundary Value Problem
Ruyun Ma, Chenghua Gao, and Yongkui Chang
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Ruyun Ma,ruyun [email protected] Received 4 December 2009; Revised 30 January 2010; Accepted 25 March 2010 Academic Editor: Leonid Shaikhet
Copyrightq2010 Ruyun Ma et al. 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.
Leta, bbe two integers withb−a≥5 and letT2{a2, a3, . . . , b−2}. We show the existence of solutions for nonlinear fourth-order discrete boundary value problemΔ4ut−2 ft, ut, Δ2ut−1,t∈T2,ua1 ub−1 Δ2ua Δ2ub−2 0 under a nonresonance condition involving two-parameter linear eigenvalue problem. We also study the existence and multiplicity of solutions of nonlinear perturbation of a resonant linear problem.
1. Introduction
The deformations of an elastic beam whose both ends are simply supported are described by a fourth-order two-point boundary value problem
ygxyex, 0< x <1,
y0 y1 y0 y1 0 1.1
See studies by Aftabizadeh 1 and Gupta in 2. The existence of solutions of nonlinear boundary value problems of fourth-order differential equations has been studied by many authors; see 1–12 and the references therein. For example, Aftabizadeh 1 proved an existence theorem for nonlinear boundary value problems
yf
x, y, y
, 0< x <1,
y0 y0, y0 y1, y1 y0, y1 y1, 1.2
under several conditions thatf is a bounded function. Yang3obtained existence results of 1.2under the following assumption.
AThere are constantsa, b, c≥0 witha/π4b/π2<1 such that f
x, y, u≤ayb|u|c. 1.3
Del Pino and Man´asevich4extended Yang’s result and proved the following.
Theorem A. Assume that the pairα, βsatisfies α
kπ4 β
kπ2/1 1.4 for allk∈Nand that there are positive constantsa, b, andcsuch that
amax
k∈N
1
k4π4−α−βk2π2
bmax
k∈N
λk
k4π4−α−βk2π2
<1, ft, u, v−
αu−βv≤a|u|b|v|c
1.5
for allx∈0,1,u, v∈R, then1.2possesses at least one solution.
Of course, the natural question is whether or not the similar existence can be established for the corresponding discrete analog of1.2of the form
Δ4ut−2 f
t, ut,Δ2ut−1
, t∈T2, 1.6
ua1 r1, ub−1 r2, Δ2ua r3, Δ2ub−2 r4, 1.7 whereT2{a2, a3, . . . , b−2},ri∈Rfori∈ {1,2,3,4}.
The purpose of this paper is to show that the answer is yes. To this end, we state and prove a spectrum result of two-parameter linear eigenvalue problem
Δ4ut−2 βΔ2ut−1−αut 0, t∈T2, 1.8
ua1 ub−1 Δ2ua Δ2ub−2 0. 1.9
This result is a slightly generalized version of Shi and Wang13, Theorem 2.1. InSection 3, we use Leray-Schauder principle to study the existence of solutions of1.6,1.7under some nonresonant conditions involving the spectrum of 1.8,1.9.Section 4is considered with some perturbations of resonant linear problems. We established some a priori bounds and used these together with bifurcation arguments to prove the existence and multiplicity of solutions.
Finally, we note that the existence of solutions of second-order discrete boundary value problems has also received much attention; see studies by Agarwal and Wong in14,
Henderson in15, and the references therein. However, relatively little is known about the existence of solutions of fourth-order discrete boundary value problems. The likely reason may be that the structure of spectrum of the corresponding linear eigenvalue problem is not very clear. To our best knowledge, only He and Yu16as well as Zhang et al.17dealt with the discrete problem of the form
Δ4ut−2 ft, ut, t∈T2, 1.10
ua ub Δ2ua Δ2ub−2 0. 1.11
As we will see inSection 2,1.9has more advantage than1.11in the study of the spectrum of two-parameter linear eigenvalue problems.
2. Spectrum of Two-Parameter Linear Eigenvalue Problem
Leta, bbe two integers withb−a≥5. Recall
T2{a2, a3, . . . , b−2}. 2.1 Let
T0{a, a1, . . . , b}, T1{a1, a2, . . . , b−1},
Λ {1,2, . . . , b−a−3}. 2.2
LetXbe the Banach space
X u|u:T0−→R, ua1 ub−1 Δ2ua Δ2ub−2 0
2.3
under the norm
uX :maxu
j|j∈T2
. 2.4
LetY be the Banach space
Y
y|y:T1−→R, ya1 yb−1 0
2.5
equipped with the norm
yY :maxy
j|j∈T2
. 2.6
LetZbe the Banach space
Z{z|z:T2−→R} 2.7
equipped with the norm
zZ:maxz
j|j∈T2
. 2.8
Remark 2.1. For anyz∈Zwith
z{za2, za3, . . . , zb−2}, 2.9
it determines a unique elementy∈Y by
y{0, za2, za3, . . . , zb−2,0} 2.10
and a unique elementx∈Xby
x{−za2,0, za2, za3, . . . , zb−2,0,−zb−2}. 2.11
Hence, the Banach spacesX,Y, andZare homomorphic with each other. Denote the natural homomorphism fromZtoXbyj.
Now, we define a linear operatorL:X → Zby
Lut: Δ4ut−2 βΔ2ut−1−αut, t∈T2. 2.12
Fork∈Λ, letλkbe thekth-eigenvalue of the second-order linear eigenvalue problem Δ2ut−1 λut 0, t∈T2,
ua1 ub−1 0. 2.13
It is well known thatλkis simple, and the corresponding eigenfunction ψkt:sinkπt−a−1
b−a−2 , t∈T1, k∈Λ. 2.14 See the study by Kelly and Peterson in18.
The following result is considered with the spectrum of two-parameter eigenvalue problem:
Δ4ut−2 βΔ2ut−1−αut 0, t∈T2, 2.15 ua1 ub−1 Δ2ua Δ2ub−2 0. 2.16
It is a slightly generalized version of Shi and Wang13, Theorem 2.1.
Proposition 2.2. α, βis an eigenvalue pair of 2.15,2.16if and only if
α λ2k β
λk 1 2.17
for somek∈Λ.
Proof. Letr1, r2∈Csuch that
r1r2 β, r1r2−α. 2.18
Define two second-order difference operatorsL1, L2:Y → Yby L1y
t: Δ2yt−1 r1yt, t∈T2, L1y
a1 0, L1y
b−1 0, L2y
t: Δ2yt−1 r2yt, t∈T2, L2y
a1 0, L2y
b−1 0.
2.19
Then, fory∈Y andt∈T1, L2◦L1yt L2
Δ2yt−1 r1yt
Δ2
Δ2yt−2 r1yt−1 r2
Δ2yt−1 r1yt
Δ4yt−2 r1r2Δ2yt−1 r1r2yt Δ4yt−2 βΔ2yt−1−αyt
L|Yyt.
2.20
We claim that if2.15,2.16possess a nontrivial solutiony, then eitherr1 λkorr2 λkfor somek ∈Λ. In either case, sinkπt−a−1/b−a−2,t∈T1, is a nontrivial solution of 2.15,2.16.
In fact, ifr2/λkfor allk∈Λ, then2.20implies that L1y
t 0, t∈T1. 2.21
This is
Δ2yt−1 r1yt 0, t∈T2,
ya1 yb−1 0. 2.22
Thus,r1λkfor somek∈Λ, and
yt sinkπt−a−1
b−a−2 , t∈T1. 2.23
Ifr2λkfor somek∈Λ, then2.20implies that
L1yt γψkt, t∈T1, 2.24
for someγ ∈R\ {0}. This is
Δ2yt−1 r1yt γψkt, t∈T2, 2.25
ya1 yb−1 0. 2.26
Sinceγ /0, it follows that
r1/λk, k∈Λ. 2.27
This implies that2.25,2.26have a unique solution
yt: L1−1γψkt, t∈T1. 2.28
We show that
r1λkγ, yt ψkt. 2.29
In fact, from2.25we have b−2 ta2
γψ2kt b−2
ta2
Δ2yt−1 r1yt ψkt
b−2
ta2
Δ2ψkt−1 r1ψkt yt
b−2
ta2
r1−λkψktyt
b−2
ta2
r1−λkψk2t,
2.30
which implies thatγr1−λk, and, subsequently,yt ψkt.
Therefore, the claim is true.
Now, 2.17 follows by substituting this solution into 2.15, 2.16. Reciprocally, if 2.17holds, then, clearly, sinkπt−a−1/b−a−2,t ∈ T1is a nontrivial solution of 2.15,2.16.
Remark 2.3. From the proof ofProposition 2.2, we see that if2.15subjects to1.9, then we can factorL|Y as follows:
L|Y L2◦L1. 2.31
However, this cannot be done if2.15subjects to1.11. So,1.9has more advantage than 1.11in the study of the spectrum of two-parameter linear eigenvalue problems.
Next, forj ∈N, let us set Lj
α, β
| α λ2j β
λj 1
. 2.32
In view of the Proposition 2.2, we call Lj an eigenline of 2.15, 2.16. We note that an eigenvalue pairα, βcan belong to at most two eigenlines. Ifα, βbelongs to just oneLj, then the corresponding eigenspace is spanned by sinkπt−a−1/b−a−2. Ifα, βbelongs toLj∩Lk j /k, then the corresponding eigenspace is spanned by sinjπt−a−1/b−a−2 and sinkπt−a−1/b−a−2.
Suppose that the pairα, βis not an eigenvalue pair of2.15,2.16, that is, α
λ2k β
λk/1 2.33
for allk∈Λ, and thath∈Z:
ht b−a−3
k1
hksinkπt−a−1
b−a−2 , t∈T2. 2.34
From the Fredholm Alternative, it follows that the boundary value problem Δ4ut−2 βΔ2ut−1−αut ht, t∈T2,
ua1 ub−1 Δ2ua Δ2ub−2 0
2.35
has a unique solution for each h ∈ Z. Moreover, this solution admits a Fourier series expansion of the form
ut b−a−3
k1
hksinkπt−a−1/b−a−2
λ2k−α−βλk , t∈T2,
ua1:0, ub−1:0, ua:−ua2, ub:−ub−2.
2.36
Also, we have
Δ2ut−1 −b−a−3
k1
λkhksinkπt −a−1/b−a−2
λ2k−α−βλk , t∈T2, Δ2ua:0, Δ2ub−2:0.
2.37
From2.36and2.37, we can easily see that the operatorsA:Z → Z,B :Z → Z defined by
Aht ut, Bht Δ2ut−1, t∈T2, 2.38
are compact linear operators. In 2.38,u is the solution of2.35,2.16corresponding to h∈Z. The norms ofAandBare, respectively, given by
AZ→Zmax
k∈Λ
1 λ2k−α−βλk
, BZ→Zmax
k∈Λ
λk
λ2k−α−βλk
. 2.39
Finally, as an immediate consequence ofProposition 2.2, we have the following.
Proposition 2.4. Letγandδbe two constants withγ, δ∈0,∞×0,∞andγδ >0. Then the generalized eigenvalues of problem
Δ4ut−2 μ
δΔ2ut−1−γut
0, t∈T2, ua1 ub−1 Δ2ua Δ2ub−2 0
2.40
are given by
μ1 γ, δ
< μ2 γ, δ
<· · ·< μb−a−3 γ, δ
, 2.41
where
μk γ, δ
λ2k
γδλk, k∈Λ. 2.42
The generalized eigenfunction corresponding toμkγ, δis
ψkt sinkπt−a−1
b−a−2 , t∈T1. 2.43
3. Existence Results for Nonresonant Problems
Theorem 3.1. Assume that the pairα, βsatisfies
α λ2k β
λk/1 3.1
for allk∈Λand that there are positive constantsa∗, b∗, andc∗such that
a∗max
k∈Λ
1 λ2k−α−βλk
b∗max
k∈Λ
λk
λ2k−α−βλk
<1, 3.2
ft, u, v−
αu−βv≤a∗|u|b∗|v|c∗ 3.3
for allt∈T2,u, v∈R, then1.6,1.7possess at least one solution.
Remark 3.2. It is not difficult to see that3.1,3.2imply that a∗
λ2k−α−βλk b∗λk
λ2k−α−βλk <1 3.4 fork ∈ Λ. It turns out that 3.4is equivalent to the fact that the squareα−a∗, αa∗× β−b∗, βb∗does not intersect any of the eigenlinesLjof2.15,2.16. From this point of view,3.1,3.2can be thought of as a two-parameter nonresonance condition relative to the eigenlinesLj.
Proof ofTheorem 3.1. It is easy to check that the problem
Δ4ut−2 0, t∈T2,
ua1 r1, ub−1 r2, Δ2ua r3, Δ2ub−2 r4
3.5
has a unique solutionlt. Set
yt:ut−lt, t∈T0. 3.6
Then1.6,1.7can be rewritten as Δ4yt−2 f
t, yt lt,Δ2
yt−1 lt−1
, t∈T2, ya1 yb−1 Δ2ya Δ2yb−2 0.
3.7
Since f
t, yt lt,Δ2
yt−1 lt−1
− α
yt lt
−β Δ2
yt−1 lt−1
≤a∗yb∗Δ2yt−1c∗∗
3.8
with
c∗∗a∗max
t∈T0
|lt|b∗max
t∈T1
Δ2lt−1c∗, 3.9
it follows that3.2and3.3still hold except thatc∗is replaced byc∗∗. So, we may suppose thatr1r2 r3r40 in1.7.
Let us defineT :Z×Z → Z×Zby Tu, v
A
f·, u, v−
αu−βv , B
f·, u, v−
αu−βv
, 3.10
whereAandBare the operators defined in2.38. The growth condition3.3together with the compactnessAandBimplies thatTis a completely continuous operator. ByRemark 2.1, the problem
Δ4ut−2 f
t, ut,Δ2ut−1
, t∈T2, ua1 ub−1 Δ2ua Δ2ub−2 0
3.11
is equivalent to the fixed point problem inZ×Z:
u, v Tu, v. 3.12
We will study this fixed point problem by means of the well-known Leray-Schauder principle 18. To do this, we show that there is a uniform bound independent of λ ∈ 0,1 for the solutions of the equation
u, v λTu, v. 3.13
Thus, letu, vbe a solution of3.13. From the definition ofTand3.3, we obtain the result that
uZ≤ AZ→Z{auZbvZc}, 3.14 vZ≤ BZ→Z{auZbvZc}. 3.15
Combining3.14and3.15and using3.2and2.38, we obtain the existence of a constant MMa∗, b∗, c∗,AZ→Z,BZ→Zsuch that
uZvZ≤M. 3.16
By the Leray-Schauder principle19, we conclude the existence of at least one solution of 3.12, and the theorem follows.
4. Existence and Multiplicity Results for Perturbations of Resonant Linear Problems
In this section, we consider the perturbations of resonant linear problems of the form Δ4ut−2
μkμ
δΔ2ut−1−γut
gt, ut ht, t∈T2, 4.1μ ua1 ub−1 Δ2ua Δ2ub−2 0, 4.1
whereγ, δ∈0,∞×0,∞withγδ >0,μkμkγ, δ, andgandhsatisfy the following.
H1 Sublinear growth conditiong :T2×R → Ris continuous, and there existα∈0,1, C1, C2∈0,∞such that
gt, s≤C1|s|αC2, s∈R, t∈T2 4.2
H2There existsβ >0 such that
sgt, s>0, fort∈T2, |s|> β 4.3
H3h:T2 → Rsatisfies
b−2 ta2
htψkt 0. 4.4
We will establish some a priori bounds and use these together with Leray-Schauder continuation and bifurcation arguments to reduce results which say that there are multiple solutions of4.1μ,4.1forμon one side of zero and guarantee the existence of at least one solution forμ0 andμon the other side of zero. To wit, we have the following.
Theorem 4.1. Let (H1), (H2), and (H3) hold. Then there existμ− < 0 < μ such that4.1μ,4.1 have
1at least one solution ifμ∈0, μ, 2at least three solutions ifμ∈μ−,0.
We have the following “dual” theorem ifH2is replaced by the assumption H2that there existsβ >0 such that
sgt, s<0, fort∈T2, |s|> β. 4.5
Theorem 4.2. Let (H1), (H2’), and (H3) hold. Then there existμ− <0 < μsuch that4.1μ,4.1 have
1at least one solution ifμ∈μ−,0, 2at least three solutions ifμ∈0, μ.
DefineL:X → Zby Ly
t Δ4ut−2 μk
δΔ2ut−1−γut
, t∈T2. 4.6
DefineF :Z → Zby
Fut gt, ut, t∈T2. 4.7
It is easy to check thatF:Z → Zis continuous. Obviously4.1μ,4.1are equivalent to Luμ
δΔ2ut−1−γut
Fu h. 4.7μ
Define an operatorP :X → Xby
Pxt ψktb−2
sa2
xsψks, t∈T0, 4.8
where
ψka:−ψka2, ψkb:−ψkb−2. 4.9
It is easy to show the following.
Lemma 4.3. Pis a projection and ImP KerL.
Define an operatorE:Z → Zby Ey
t yt−ψktb−2
sa2
ysψks, t∈T2. 4.10
Obviously, we have the following.
Lemma 4.4. Eis a projection and ImE ImL.
It is clear that
XXP ⊕XI−P, ZZI−E⊕ZE, 4.11
whereIrepresents the identity operator andXP, XI−P, ZI−E, andZEare the images ofP, I−P, I−E, andE, respectively.
It is obvious that the restriction ofLtoXI−Pis a bijection fromXI−PontoZE, the image ofL. We defineM:ZE → XI−P by
M: L|XI−P−1. 4.12
Since kerL span{ψk}, we see that eachx∈Xcan be uniquely decomposed into
xρψkv 4.13
for someρ∈R, andv∈XI−P. Forz∈Z, we also have the decomposition
zτψkh, 4.14
withτ ∈Randh∈ZE.
Lemma 4.5. Equations4.1μ,4.1are equivalent to the system
Lvμ
δΔ2vt−1−γvt EF
ρψkv
h, 4.9μ
μb−2
sa2
δΔ2ψks−1−γψks
ψks b−2
sa2
ψksf
s, ρψks vs
. 4.15
Lemma 4.6. Let (H1) and (H2) hold. Then there existsR0 such that any solutionyof4.1μ,4.1 satisfies
yX< R0 4.16
as long as
0≤μ≤δ: 1
2MJXI−P→XI−P, 4.17
whereJ:X → Zis defined by
Jxt:δΔ2xs−1−γxs, t∈T2. 4.18
Proof. ObviouslyLμJ|XI−P :XI−P → ZEis invertible for|μ| ≤δ. Moreover, by 4.17,
LμJ
|−1XI−P
ZE→XI−P
L|XI−P
IμMJ−1
ZE→XI−P
IμMJ−1X
I−P→XI−PMZE→XI−P ≤2MZE→XI−P. 4.19
Letyρψkvbe any solution of4.1μ,4.1. Then we have that, ifρ /0,
vX
LμJ−1X
I−PE h−F
ρψkv
X
≤
LμJ−1X
I−P
Z
E→XI−PEZ→ZE
hZC1ρψkZvZαC2
≤2MZE→XI−PEZ→ZE
hZC1ρψkZvZαC2
≤2MZE→XI−PEZ→ZE
hZC1ρψkXvXαC2 2MZE→XI−PEZ→ZE
hZC1ρψkXα
1 vX
ρψkX α
C2
≤2MZE→XI−PEZ→ZE
hZC1ρψkXα
1 αvX
ρψkX
C2
2MZE→XI−PEZ→ZE
×
⎡
⎣hZC1ρψkXα
⎛
⎝1 α
ρψkX1−α vX ρψkXα
⎞
⎠C2
⎤
⎦,
4.20
and hence
vX
ρψkXα ≤ C3
ρψkXα C4 αC4
ρψkX1−α vX
ρψkXα, 4.21
where
C3MZE→XI−PEZ→ZEhZC2, C42C1MZE→XI−PEZ→ZE. 4.22
If
ρ≥ 2αC41/1−α
ψkX :C, 4.23
then we have
vX
ρψkXα ≤ 2C3
CψkXα 2C4:C∗. 4.24
If we assume that the conclusion of the lemma is false, we obtain a sequence{ηn} with 0≤ ηn ≤ δandηn → 0, and a sequence of corresponding solutions{yn ρnψkvn} of4.1ηn,4.1such thatynX → ∞. From4.24, we conclude that it is necessary that
|ρn| → ∞. We may assume that
ρn−→∞, ρn ≥C ∀n∈N 4.25
since the other case can be treated by the same way. Thus4.24yields that
vnX:≤Cρnα 4.26 withC:C∗ψkαX.
Now from4.15, we get that
ρnηn
b−2 sa2
δΔ2ψks−1−γψks
ψks b−2
sa2
ψksf
s, ρnψks vns
. 4.27
By4.17and4.27, it follows that b−2 sa2
ψksf
s, ρnψks vns
≤0. 4.28
Let
A
t|t∈ {a2, . . . , b−2}, ψkt>0 , A−
t|t∈ {a2, . . . , b−2}, ψkt<0
. 4.29
It is easy to see that
A∪A−/∅, minψkt|t∈A∪A−
>0. 4.30
Combining4.30and4.26, we conclude that there exists a positive constantΓsuch that, for n∈N,
vnX ≤Γρnminψkt|t∈A∪A−α
, 4.31
which implies that
ρnlim→∞min
ρnψkt vnt|t∈A ∞,
ρnlim→∞min
ρnψkt vnt|t∈A−
−∞. 4.32
Applying4.32,4.30, andH2, we conclude that
b−2
sa2
ψksf
s, ρnψks vns
s∈A
ψksf
s, ρnψks vns
s∈A−
ψksf
s, ρnψks vns
>0,
4.33
which contracts4.28.
Using the similar arguments, we may establish the following lemma.
Lemma 4.7. Let (H1) and (H2’) hold. Then there existsR0such that any solutionyof4.1μ,4.1 satisfies
yX< R0 4.34
as long as
−δ≤μ≤0, 4.35
whereδis given in4.17.
Lemma 4.8. Let (H1) and (H2) hold. Then there existsR1 : R1 ≥R0such that, for 0 ≤μ≤δand R≥R1, one has
deg j◦
LμJ−F−h
, BR,0
deg j◦
LδJ
, BR,0
±1, 4.36
wherej : Z → X is the natural homomorphism,BR {u ∈ XuX < R}, and “deg” denotes Leray-Schauder degree whenμ /0 and coincidence degree whenμ 0 (see the study by Gaines and Mawhin in [20]). Therefore4.7μhas a solution inBRforμ∈0,δ.
Proof. ByLemma 4.6and the definition ofL, the degree deg
j◦
LμJ−F−h
, BR,0
4.37
is well defined forμ∈0,δ and is a constant with respect toμ.
Now ifμ, y∈0,1×Xis a solution of Lyδ
δΔ2ys−1−γys
−μ F
y h
0, 4.38
then we have yXμ
LδJ −1 hF
yX ≤LδJ −1Z
E→XI−P
hZC1yαXC2
. 4.39
Hence there existsR0 > 0 such thatyX < R0. Thus if R1 max{R0, R0}, then we have, wheneverR > R1, that
deg j◦
LδJ −F−h
, BR,0 deg
j◦
LδJ
, BR,0
±1, 4.40
which completes the proof.
By a similar manner we may establish the following.
Lemma 4.9. Let (H1) and (H2) hold. Then there existsR1: R1≥R0such that, for 0≥μ≥ −δand R≥R1, one has
deg j◦
LμJ−F−h
, BR,0
deg j◦
L −δJ
, BR,0
±1. 4.41
Therefore4.7μhas a solution inBRforμ∈−δ, 0.
Lemma 4.10. Let (H1) and (H2) hold. Then there existsδ1>0 such that, for−δ1< μ <0, one has deg
j◦
LμJ−F−h
, BR1,0 deg
j◦Lδ1J, BR1,0
±1. 4.42
Proof. Let
τ0 inf
x∈∂BR1∩Xj◦Lx−Fx−hX. 4.43
Then it is not difficult to check thatτ0 >0. Hence if we takeδ1so small thatδ1R1 < τ0, then forμ∈−δ1, δ1,
deg j◦
LμJ−F−h
, BR1,0 deg
j◦L −F−h, BR1,0
±1. 4.44
Lemma 4.11. Let (H1) and (H2) hold. Then there existsδ1>0 such that, for 0≤μ≤δ1, one has deg
j◦
LμJ−F−h
, BR,0 deg
j◦L −δ1J, BR,0
±1. 4.45
Proof ofTheorem 4.1. By the study of Massab `o and Pejsachowicz in21, Theorem 1.1,4.7μ has a continuumC∗ {μ, yμ}of solutions withyμX < R1andμ∈−δ1,δ. On the other hand, since F isL-completely continuous and satisfies H1and since μ 0 is a simple eigenvalue, it follows from the study by Rabinowitz in 22, Theorem 1.6that μ 0 is a bifurcation point from infinity for4.7μ. Moreover, there exist two continua
C±∞ μ, yμ
⊂R×X 4.46
of solutions of4.7μ, bifurcating from infinity atμ0, that is, there exists0∈0,1/R1, such that for all: 0< ≤0there exist two continuaC andC− with
C ⊂ C∞, C− ⊂ C−∞, C± ⊂!
μ, yμ
:yμX ≥ 1
, μ<
"
:U0,∞, 4.47
andC± connects0,∞to∂U0,∞. Notice thatμ, y ∈ C implies thatya2 > 0, and μ, y∈ C− implies thatya2<0. So,
C∩ C− ∅. 4.48
Now,Lemma 4.6implies that C± ⊆!
μ, yμ
|yμ
X≥ 1
, − < μ <0
"
. 4.49
This completes the proof.
Proof ofTheorem 4.2. Using similar arguments, we may get the desired results.
Acknowledgments
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFCno. 10671158, NWNU-KJCXGC-03-47, the Spring-sun programno. Z2004-1-62033, SRFDPno. 20060736001, and the SRF for ROCS, SEM2006 311.
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