Multiplicity Results Using Bifurcation Techniques for a Class of Fourth-Order -Point Boundary Value Problems

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Volume 2009, Article ID 970135,20pages doi:10.1155/2009/970135

Research Article

Multiplicity Results Using Bifurcation

Techniques for a Class of Fourth-Order

m

-Point

Boundary Value Problems

Yansheng Liu

1

_{and Donal O’Regan}

2

1_{Department of Mathematics, Shandong Normal University, Jinan 250014, China}

2_{Department of Mathematics, National University of Ireland, Galway, Ireland}

Correspondence should be addressed to Yansheng Liu,[email protected]

Received 13 March 2009; Accepted 12 April 2009 Recommended by Juan J. Nieto

By using bifurcation techniques, this paper investigates the existence of nodal solutions for a class of fourth-orderm-point boundary value problems. Our results improve those in the literature. Copyrightq2009 Y. Liu and D. O’Regan. 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

Consider the following fourth orderm-point boundary value problemBVP, for short

u4t fut, ut, t∈0,1

u0 0, u1

m−2

i1

αiu

ηi

u0 0, u1

m−2

i1

αiu

ηi

,

1.1

wheref : R×R → Ris a given sign-changing continuous function,m ≥3,ηi ∈0,1,and

αi>0 fori1, . . . , m−2 with

m−2

i1

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Multi-point boundary value problems for ordinary diﬀerential equations arise in diﬀerent areas of applied mathematics and physics. The existence of solutions of the second order multi-point boundary value problems has been studied by many authors and the methods used are the nonlinear alternative of Leray-Schauder, coincidence degree theory, fixed point theorems in cones and global bifurcation techniquessee1–9, and the references therein. In5, Ma investigated the existence and multiplicity of nodal solutions for

ut fut 0, t∈0,1;

u0 0, u1

m−2

i1

αiu

ηi

1.3

when

ηi ∈Qi1,2, . . . , m−2 with 0< η1< η2 <· · ·< ηm−2<1, 1.4

andαi >0 fori1, . . . , m−2 satisfying1.2. He obtained some results on the spectrum of

the linear operator corresponding to1.1. It should be pointed out that the main tool used in

5is results on bifurcation coming from the trivial solutions and we note no use was made of global results on bifurcation from infinity.

Recently 10 Wei and Pang studied the existence and multiplicity of nontrivial solutions for the fourth orderm-point boundary value problems:

u4t fut, ut, t∈0,1

u0 0, u1

m−2

i1

αiu

ηi

u0 0, u1

m−2

i1

αiu

ηi

,

1.5

wheref : R×R → Ris a given sign-changing continuous function,m ≥3,ηi ∈0,1, and

αi>0 fori1, . . . , m−2 satisfies1.2.

Motivated by5,10, in this paper we consider the existence and multiplicity of nodal solutions for BVP1.1. The method used here is Rabinowitz’s global bifurcation theorem. To the best of our best knowledge, only10seems to have considered the existence of nontrivial or positive solutions of the nonlinear multi-point boundary value problems for fourth order diﬀerential equations. As in5,10we suppose1.2is satisfied throughout.

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value ofL. The set of real characteristic values ofLwill be denoted byσL. The multiplicity ofμ∈σLis

dim ∞

j1

NI−μLj, 1.6

where NAdenotes the null space of A. Suppose that H : R×E → Eis compact and

Hλ, u ouatu0 uniformly on boundedλintervals. Then

uλLuHλ, u 1.7

possesses the line of trivial solutionsΘ {λ,0 |λ ∈R}. It is well known that ifμ ∈ R, a necessary condition forμ,0to be a bifurcation point of1.7with respect toΘis thatμ ∈ σL. Ifμis a simple characteristic value ofL, letvdenote the eigenvector ofLcorresponding toμnormalized sov 1. ByΣwe denote the closure of the set of nontrivial solutions of

1.7. A component ofΣis a maximal closed connected subset. It was shown inRabinowitz

11, Theorems 1.3, 1.25, 1.27, the following.

Lemma 1.1. Ifμ∈σLis simple, thenΣcontains a componentCμwhich can be decomposed into

two subcontinuaCμ,Cμ−such that for some neighborhoodBofμ,0,

λ, u∈Cμ

C−μ

∩B, λ, u/μ,0 1.8

impliesλ, u λ, αvwwhereα >0α <0and|λ−μ|o1,wo|α|atα0. Moreover, each ofCμ,Cμ−either

imeets infinity inΣ, or

iimeetsμ, 0where_{μ /}μ∈σL, or

iiicontains a pair of pointsλ, u,λ,−u,_{u /}0.

The following are global results for1.7on bifurcation from infinitysee, Rabinowitz

9, Theorem 1.6 and Corollary 1.8.

Lemma 1.2. SupposeLis compact and linear,Hλ, uis continuous onR×E,Hλ, u ouat

u ∞uniformly on boundedλintervals, andu2Hλ, u/u2is compact. Ifμ∈σLis of odd multiplicity, thenΣpossesses an unbounded componentDμwhich meetsμ,∞. Moreover ifΛ⊂R

is an interval such thatΛ∩σL {μ}and℘is a neighborhood ofμ,∞whose projection onRlies inΛand whose projection onEis bounded away from0, then either

iDμ\℘is bounded inR×Ein which caseDμ\℘meetsΘ {λ,0|λ∈R}or

iiDμ\℘is unbounded.

If (ii) occurs andDμ \℘has a bounded projection onR, thenDμ\℘meets μ, ∞where

μ /μ∈σL.

Lemma 1.3. Suppose the assumptions ofLemma 1.2 hold. Ifμ ∈ σLis simple, then Dμ can be

decomposed into two subcontinuaDμ,D−μ and there exists a neighborhoodI⊂℘ofμ,∞such that

λ, u ∈ DμD−μ∩I and λ, u/ μ,∞implies λ, u λ, αvwwhere α > 0α < 0and

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2. Preliminaries

LetX C0,1with the normu maxt∈0,1|ut|,Y {u ∈ C10,1 : u0 0, u1

m−2

i1 αiuηi}with the normu1 max{u,u},Z {u ∈ C20,1 : u0 0, u1

m−2

i1 αiuηi} with the norm u2 max{u,u,u

}. Then X, Y, and Z are Banach spaces.

For anyC1 _function _u_{, if}_u_t

0 0, thent0 is a simple zero ofuif ut0/0. For any

integerk∈Nand anyν∈ {±}, as in6, define setsT_kν⊂Zconsisting of the set of functions

u∈Zsatisfying the following conditions:

iu0 0, νu0>0 andu1/0;

iiuhas only simple zeros in0,1, and has exactlyk−1 such zeros;

iiiuhas a zero strictly between each two consecutive zeros ofu.

NoteT_k−−T_kand letTkT_k−∪T_k. It is easy to see that the setsT_k−andT_kare disjoint

and open inZ. Moreover, ifu∈Tν

k, thenuhas at leastk−2 zeros in0,1, and at mostk−1

zeros in0,1.

LetER×Yunder the product topology. As in12, we add the points{λ,∞:λ∈R}

to the spaceE. LetΦ_kR×T_k,Φ_k−R×T_k−, andΦkR×Tk.

We first convert BVP1.1into another form. Supposeutis a solution of BVP1.1. Letvt −ut. Notice that

ut vt 0, t∈I;

u0 0, u1

m−2

i1

αiu

ηi

.

2.1

Thusutcan be written as

ut Lvt, 2.2

where the operatorLis defined by

Lvt:

1

0

Ht, svsds, ∀v∈Y, 2.3

where

Ht, s Gt, s

m−2

i1 αiG

ηi, s

1− m_i₁−2αiηi

,

Gt, s

⎧ ⎨ ⎩

1−t, 0≤s≤t≤1;

1−s, 0≤t≤s≤1.

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Therefore we obtain the following equivalent form of1.1

vt fLvt,−vt 0, t∈0,1;

v0 0, v1

m−2

i1

αiv

ηi

.

2.5

For the rest of this paper we always suppose that the initial value problem

vt fLvt,−vt 0, t∈0,1;

vt0 vt0 0

2.6

has the unique trivial solution v ≡ 0 on 0,1 for any t0 ∈ 0,1; in fact some suitable

conditions such as a Lipschitz assumption orf∈C1_{guarantee this.}

Define two operators onYby

Avt: LFvt, Fvt:fLvt,−vt, t∈I, v∈Y. 2.7

Then it is easy to see the following lemma holds.

Lemma 2.1. The linear operatorLand operatorAare both completely continuous fromYtoYand

Lv1≤Mv ≤Mv1, ∀v∈Y, 2.8

whereMmax{1,1/81 _im₁−2αi/1− mi1−2αiηi}.

Moreover,u∈ C4₀_,₁_{is a solution of BVP}_1.1_{if and only if}_v ₋_u _{is a solution of the} operator equationvAv.

Let the functionΓsbe defined by

Γs coss−

m−2

i1

αicosηis, s∈R. 2.9

Then we have the following lemma.

Lemma 2.2. iFor eachk≥1,Γshas exactly one zerosk∈Ik: k−1π, kπ, so

s1< s2<· · ·< sk−→∞ k−→∞; 2.10

iithe characteristic value ofLis exactly given byμks2_k, k1,2, . . ., and the eigenfunction

φkcorresponding toμkisφkt cosskt;

iiithe algebraic multiplicity of each characteristic valueμkofLis1;

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Proof. From5and by a similar analysis as in the proof of6, Lemma 3.3we obtainiand

ii.

Now we assertiiiholds. Suppose, on the contrary, there existsy∈Y such thatI− μkLyμ_k−1φk. Theny∈Zand

−y−s2

kycosskt. 2.11

Fromy0 0 we know the general solution of this diﬀerential equation is

yCcosskt−

1 2skt

sinskt. 2.12

Fromiandiiof this lemma,Ccossktsatisfies the boundary condition. Thus

cossk m−2

i1

αicosηisk, sinsk m−2

i1

αiηisinηisk. 2.13

Then, by1.2,

1

_m₋₂

i1

αicosηisk

2

_m₋₂

i1

αiηisinηisk

2

≤m−2

i,j1

αiαjcosηiskcosηjsksinηisksinηjsk

≤ _m₋₂

i1

αi

2

<1,

2.14

a contradiction. Thus the algebraic multiplicity of each characteristic valueμkofLis 1.

Finally, fromsk∈k−1π, kπands1∈0, π/2, it is easy to see thativholds.

Lemma 2.3. Ford d1, d2∈R×R\ {0,0}, define a linear operator

Ldvt

d1L2d2L

vt, ∀t∈I, v∈Y, 2.15

whereLis defined as in2.3. Then the generalized eigenvalues ofLdare simple and are given by

0< λ1Ld< λ2Ld<· · ·< λkLd−→∞ k−→∞, 2.16

where

λkLd

μ2

k

d1d2μk.

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The generalized eigenfunction corresponding toλkLdis

φkt cosskt, 2.18

whereμk, sk, φkare as inLemma 2.2.

Proof. Suppose there existλand_{v /}0 such thatvλLdv. Setut Lvt. Then from2.2–

2.7and2.15it is easy to see that_{u /}0 and

u4t λd1ut−d2u

t, t∈0,1;

u0 0, u1

m−2

i1

αiu

ηi

;

u”’0 0, u1

m−2

i1

αiu

ηi

.

2.19

DenoteL−1_u ₋_u _for_u _∈_Z_{. Then there exist two complex numbers}_r

1 andr2 such

that

u4t−λd1ut−d2u

tL−1−r2I

L−1−r1I

ut 0. 2.20

Now if there exists somerii1,2such that

L−1−riI

ut 0, 2.21

then byLemma 2.2we knowris_k2 μkfor somek∈N, and consequently

ut cosskt 2.22

is a nontrivial solution. Substituting2.22into2.19, we have

λ μ

2

k

d1d2μk.

2.23

On the other hand, suppose, for example,

L−1₋_r 1I

ut/0, L−1₋_r 2I

L−1₋_r 1I

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Let wt : L−1 ₋ _r

1Iut. Then L−1 − r2Iwt 0. Reasoning as previously

mentioned, we haver2 s2_k for somek ∈ N, and consequently wt acosskta /0is a

nontrivial solution. Therefore,

L−1−r1I

ut acosskt. 2.25

Ifr1s2_k, then the general solution of the diﬀerential equation2.25, satisfyingu0

0, is

ut Ccosskt− a

2skt

sinskt, 2.26

which is similar to2.12. Reasoning as in the proof ofLemma 2.2we can get a contradiction. Thusr1/s2_kand the general solution of2.25, satisfyingu0 0, is

ut ut acosskt s2

k−r1

, 2.27

where utis the general solution of homogeneous diﬀerential equation corresponding to

2.25

L−1−r1I

ut 0. 2.28

Notice the termacosskt/s2_k−r1in2.27satisfies the boundary condition of1.1at

t1, soutalso satisfies

u0 0, u1

m−2

i1

αiu

ηi

. 2.29

Therefore, byLemma 2.2we knowut Ccossjtfor somej∈N, and consequently

r1s2j/s2k, ut Ccossjta

cosskt

s2

k−s2j

. 2.30

By substituting this into2.19, we have

aλd1d2μk

aμ2_k, Cλd1d2μj

Cμ2j. 2.31

Sinceμj/μk, if there exists someλsuch that2.31holds, then

d1d2μk

d1d2μj

μ2

k

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which implies

d1d2/0, d1 1 μk 1 μj

−d2, 2.33

a contradiction withd1>0 andd2>0.

Consequently,2.24does not hold. This together with2.20–2.23andLemma 2.2

guarantee that the generalized eigenvalues ofLdare given by

0< λ1Ld< λ2Ld<· · ·< λkLd−→∞ k−→∞, 2.34

whereλkLd μ2_k/d1 d2μk. The generalized eigenfunction corresponding toλkLdis

φkt cosskt.

Now we are in a position to show the generalized eigenvalues ofLdare simple.

Clearly, from above we know forλk:λkLd,I−λkLdφk0 and dimNI−λkLd 1.

Suppose there exists anv∈C2_{such that}

I−λkLdv

1

μkφkt.

2.35

This together with2.3and2.15guarantee thatv ∈ Y. If we letut Lvtas above, then we have

u4t−λk

d1ut−d2u

tcosskt, t∈0,1, 2.36

u0 0, u1

m−2

i1

αiu

ηi

; u”’0 0, u1

m−2

i1

αiu

ηi

. 2.37

Consider the following homogeneous equation corresponding to2.36:

u4t− μ

2

k

d1d2μk

d1ut−d2u

t0. 2.38

The characteristic equation associated with2.38is

λ4− μ

2

k

d1d2μk

d1−d2λ2

0. 2.39

Then there exists a real numberηsuch that

λ2μk

λ2−ηλ4− μ

2

k

d1d2μk

d1−d2λ2

0. 2.40

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First we consider the cased1 >0. In this case the general solution of2.38is

c1e

√_ηt

c2e−√ηtc3cossktc4sinskt. 2.41

After computation we obtain that the general solution of2.36is

ut c1e

√_ηt

c2e−√ηtc3cossktc4sinsktatsinskt, 2.42

wherea −d1d2μk/2sk2d1μkd2μ2_k. From boundary conditionu0 u”’0 0 in

2.37it follows that

ηc1−c2skc40;

ηηc1−c2−s3_kc4 0.

2.43

Byη >0 andμk>0, we knowc1−c20 andc40. Then2.42can be rewritten as

ut c1

e√ηte−√ηtc3cossktatsinskt. 2.44

Notice that the termc3cossktsatisfies2.37. From the boundary condition

u1

m−2

i1

αiu

ηi

, u1

m−2

i1

αiu

ηi

,

tsinskt

2skcosskt−s2ktsinskt

2.45

we have

c1

e√ηe−√ηasinsk m−2

i1

αi

c1

e√ηηi_e−√ηηi_aη

isinskηi

, 2.46

c1η

e√ηe−√η−as2_ksinsk m−2

i1

αi

c1η

e√ηηi_e−√ηηi−_aη

is2ksinskηi

. 2.47

Multiply2.46bys2_kand then add to2.47to obtain

c1

ηs2_ke√ηe−√ηc1

ηs2_k

m−2

i1

αi

e√ηηi _e−√ηηi_. _2.48

On the other hand, from1.2it can be seen that

e√ηe−√η>

m−2

i1

αi

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This together with2.48guarantee thatc1 0. Therefore,2.42reduces to

ut c3cossktatsinskt. 2.50

Similar to2.12, a contradiction can be derived.

Next consider the cased10. Thenη0 from above. In this case the general solution

of2.38is

c1c2tc3cossktc4sinskt. 2.51

By a similar process, one can easily get a contradiction.

To sum up, the generalized eigenvalues ofLdare simple, and the proof of this lemma

is complete.

3. Main Results

We now list the following hypotheses for convenience.

H1There existsa a1, a2∈R×R\ {0,0}such that

fx, ya1x−a2yox, y, asx, y−→0, 3.1

wherex, y∈R×R, and|x, y|:max{|x|,|y|}.

H2There existsb b1, b2∈R×R\ {0,0}such that

fx, yb1x−b2yox, y, asx, y−→ ∞. 3.2

H3There existsR >0 such that

_f

x, y< R M, for

x, y∈x, y:|x| ≤MR,y≤R, 3.3

whereMis defined as inLemma 2.1.

H4There exist two constants r1 < 0 < r2 such thatfx,−r1 ≥ 0 and fx,−r2 ≤ 0

for x ∈ −Mr, Mr, and fx,−ysatisfies a Lipschitz condition iny forx, y ∈

−Mr, Mr×r1, r2, wherermax{|r1|, r2}.

Now we are ready to give our main results.

Theorem 3.1. Suppose (H1)-(H2) hold. Suppose there exists two integersi0≥0andk >0such that

either

μ2

i0k

a1a2μi0k

<1< μ

2

i01

b1b2μi01

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or

μ2_i₀_k b1b2μi0k

<1< μ

2

i01

a1a2μi01

3.5

holds. Then BVP1.1has at least2knontrivial solutions.

Theorem 3.2. Suppose (H1) and (H2) hold and one of (H3) and (H4) hold. Suppose there exists two

integersi0andj0such that

μ2

i0

a1a2μi0

<1, μ

2

j0

b1b2μj0

<1. 3.6

Then BVP1.1has at least2i0j0solutions.

To set it up we first consider global results for the equation

vλAv, 3.1λ

onY, whereλ∈R, and the operatorAis defined as in2.7. Under the conditionH1,3.1λ can be rewritten as

vλLavHaλ, v, 3.7

hereHaλ, v λAv−λLav,Lais defined as in2.12 replacingdwitha. Obviously, byH1

andLemma 2.1–2.3, it can be seen thatHaλ, visov1forvnear 0 uniformly on bounded

λintervals andLais a compact linear map onY. A solution of3.1λis a pairλ, v∈E. By

H1, the known curve of solutions{λ,0|λ∈R}will henceforth be referred to as the trivial solutions. The closure of the set on nontrivial solutions of3.1λwill be denoted byΣas in

Lemma 1.1.

IfHaλ, v≡0, then3.7becomes a linear system

vλLav. 3.8

ByLemma 2.3,3.8possesses an increasing sequence of simple eigenvalues

0< λ1< λ2<· · ·< λk<· · ·, with λk

μ2_k a1a2μk

ask−→∞. 3.9

Any eigenfunctionφkcossktcorresponding toλkis inT_k.

A similar analysis as in6, Proposition 4.1yields the following results.

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Lemma 3.4. Assume that (H1) holds andλkis defined by3.9. Then for each integerk >0and each

ν , or−, there exists a continuaCν

kof solutions of 3.1λinΦνk∪ {λk,0}, which meets{λk,0}

and∞inΣ.

Proof. Consider 3.7 as a bifurcation problem from the trivial solution. From Lemma 1.1

and conditionH1it follows that for each positive integer k ∈ N,Σcontains a component

Ck ⊆E R×Y which can be decomposed into two subcontinuaC_k,C−_k such that for some

neighborhoodBofλk,0,

λ, v∈C_kC−_k∩B, λ, v/ λk,0 3.10

implyλ, v λ, αφkw, whereα >0α <0and|λ−λk|o1,w1o|α|atα0.

By3.7and the continuity of the operatorA:Y → Z, the setC_kνlies inR×Zand the injectionCν

k → R×Zis continuous. Thus,Cνkis also a continuum inR×Z, and the above

properties hold inR×Z.

SinceTkis open inZandφk∈T_k, we know

v α φk

w α ∈T

k 3.11

for 0/αsuﬃciently small. Then there existsε0>0 such that forε∈0, ε0, we have

λ, v∈Φk,

Ck

{λk,0}

∩Bε⊂Φk, 3.12

whereBεis an open ball inR×Zof radiusεcentered atλk,0. It follows from the proof of

6, Proposition 4.1that

λ, v∈Ck∩R×∂Tk ⇒u0, 3.13

which meansCk\ {λk,0} ∩∂Φk∅. Consequently,Cklies inΦk∪ {λk,0}.

Similarly we have thatCν_klies inΦν_k∪ {λk,0} ν or−.

Next we show alternative ii of Lemma 1.1 is impossible. If not, without loss of generality, assume thatC_k meetsλi,0with λk/λi ∈ σLa. Then there exists a sequence

ξm, zm∈C_kwithξm → λiandzm → 0 asm → ∞. Notice that

zmξmLazmHξm, zm. 3.14

Dividing this equation by zm1 and usingLemma 2.1andHξm, zm ozm1asm →

∞, we may assume without loss of generality thatzm/zm1 → zasm → ∞. Thus from

3.14it follows that

zλiLaz. 3.15

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T_i−are open inZ. Therefore,zm ∈T_iorT_i−formsuﬃciently large, which is a contradiction

withzm∈T_km≥1,i /k. Hence alternativeiiofLemma 1.1is not possible.

Finally it remains to show alternativeiiiofLemma 1.1is impossible. In fact, notice thatT_k− −T_k, andT_k−∩T_k ∅. Ifu ∈T_k, then−u ∈ T_k−. This guarantees thatCν_kdoes not contain a pair of pointsλ, v,λ,−v,_{v /}0.

Therefore alternativeiofLemma 1.1holds. This implies that for each integerk ∈N

and eachν , or−, there exists a continuaCν_kof solutions of3.1λinΦν_k∪ {λk,0}, which

meets{λk,0}and∞inΣ.

Under the conditionH2,3.1λcan be rewritten as

vλLbvHbλ, v, 3.16

hereHbλ, v λAv−λLbv,Lbis defined as in2.12 replacingdwithb.

Lethx, y: fx, y−b1xb2y. Then fromH2it follows that lim|x,y| → ∞hx, y/

|x, y|0. Define a function

hr:maxhx, y:x, y≤r. 3.17

Thenhris nondecreasing and

lim

r→ ∞

hr

r 0. 3.18

Obviously, by 3.18 and Lemma 2.1, it can be seen thatHbλ, vis ov1 for v near ∞

uniformly on boundedλintervals andLbis a compact linear map onY.

Similar to 3.8, by Lemma 2.3, Lb possesses an increasing sequence of simple

eigenvalues

0< λ1< λ2<· · ·< λk<· · ·, with λk

μ2_k b1b2μk

ask−→∞. 3.19

Noteφkcossktis an eigenfunction corresponding toλk. Obviously, it is inT_k.

Lemma 3.5. Assume that (H1)-(H2) holds. Then for each integerk >0and eachν , or−, there

exists a continuaDν

kofΣinΦνk∪ {λk,∞}coming from{λk,∞}, which meetsλk,0or has an

unbounded projection onR.

Proof. From2.7,3.16, and3.18 it follows thatHbλ, v is continuous onE,Hbλ, v

ov1atv∞uniformly on boundedλintervals. Moreover, as in the proof of12, Theorem

2.4, one can see thatv21Hbλ, v/v21is compact. FromLemma 2.3we knowλkis a simple

characteristic value ofLb for each integerk ∈ N. Thus by Lemmas1.2and1.3,Σcontains a

componentDkwhich can be decomposed into two subcontinuaD_k,D_k−which meet{λk,∞}.

Now we show that for a smaller neighborhoodI ⊂℘ofλk,∞,λ, v∈D_kD_k−∩I

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exists a neighborhood I ⊂ ℘ofλk,∞satisfying λ, v ∈ D_kD−_k∩Iand λ, v/ λk,∞

implyλ, v λ, αvkwwhereα >0α <0and|λ−λk|o1,w1 o|α|at|α|∞.

As in the proof ofLemma 3.4,Dν_kis also a continuum inR×Z, and the above properties hold inR×Z. SinceTν

k is open inZandw/αis smaller compared toφk ∈Tknearα ∞,

φkw/αand thereforevαφkw∈T_kforαnear∞. Here and in the following the same

argument works ifis replaced by−.

Therefore,D_k∩I⊂R×T_k∪λk,∞. Now we have two cases to consider, that is,D_k\I

is bounded or unbounded. First supposeD_k\Iis bounded. Then there existsλ, v∈ ∂D_k

with v ∈ ∂T_k. If_{v /}0, by Lemma 3.3we know v ∈ T_jν for some positive integer_{j /}k and

ν ∈ {,−}. As in the proof ofLemma 3.4, we get a contradiction, which meansv 0. Thus there exists a sequenceξm, zm∈D_kwithzm → v≡0 asm → ∞. This together withH1

guarantee thatξm, zmsatisfies3.14.

As in the proof of Lemma 3.4, we may assume without loss of generality that

zm/zm1 → zandξm → ξasm → ∞. Then we have

zξLaz. 3.20

Since_{z /}0,_{ξ /}0 is an eigenvalue of operatorLa. From this,3.9, and3.19it follows that

ξ λj for some positive integerj. Then by Lemma 2.3,zbelongs toT_j orT_j−. Notice that

zm−z1 → 0 and sozm−z2 → 0 as in the proof ofLemma 3.4. Thuszm ∈T_jorT_j− for

msuﬃciently large sinceT_j andT_j−are open. This together withzm∈T_k m≥1guarantee

thatkj. This meansD_kmeetsλk,0ifD_k\Iis bounded.

Next supposeD_k\Iis unbounded. In this case we showD_k \Ihas an unbounded projection onR. If not, then there exists a sequence ζm, ym ∈ D_k \I with ζm → ζand

ym1 → ∞asm → ∞. Letxm:ym/ym1,m≥1. From the fact that

ymζmLbymHb

ζm, ym

3.21

it follows that

xmζmLbxm

Hb

ζm, ym

_y_m

1

. 3.22

Notice thatLb : Y → Y is completely continuous. We may assume that there existsw ∈Y

withw11 such thatxm−w1 → 0 asm → ∞.

Letting m → ∞in 3.22and noticingHbζm, ym/ym₁ → 0 asm → ∞ one

obtains

wζLbw. 3.23

Since_{w /}0,_{ζ /}0 is an eigenvalue of operator Lb, that is, ζ λk0 for some positive integer

k0/k. Then byLemma 2.3wbelongs toT_k₀orT_k−₀. Notice the fact thatxm−w1 → 0 and

soxm−w2 → 0 as in the proof ofLemma 3.4. Thusxm∈T_k₀orT_k−₀ formsuﬃciently large

sinceT_k₀ andT_k−₀ are open. This is a contradiction withxm∈T_km≥1. ThusD_k\Ihas an

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Proof ofTheorem 3.1. Suppose first that

μ2

i0k

a1a2μi0k

<1< μ

2

i01

b1b2μi01

. 3.24

Using the notation of3.9and3.19, this meansλi0k<1< λi01and so fromLemma 2.3we

know

λi01< λi02<· · ·< λi0k<1< λi01 < λi02<· · ·< λi0k. 3.25

Consider3.7as a bifurcation problem from the trivial solution. We need only show that

Cν_i₀_j{1} ×Y/∅, j1,2, . . . , k; ν ,−. 3.26

Suppose, on the contrary and without loss of generality, that

C_i₀_i{1} ×Y ∅, for somei, 1≤i≤k. 3.27

ByLemma 3.4we know thatC_i₀_ijoinsλi0i,0to infinity inΣandλ, v 0,0is the unique

solution of3.1λ in whichλ0inE. This together withλi0i<1 guarantee that there exists

a sequence{ζm, ym} ⊂ C_i₀_i such thatζm ∈ 0,1andym₁ → ∞asm → ∞. We may

assume thatζm → ζ ∈ 0,1asm → ∞. Letxm : ym/ym₁,m ≥ 1,then3.22holds.

Similarly, we may assume that there existsw ∈ Y withw1 1 such thatxm−w1 → 0

asm → ∞and 3.23holds. From the proof of Lemma 3.5one can see ζ λi0i, which

contradictsλi0i>1. Thus3.27is not true, which means3.26holds.

Next suppose that

μ2_i₀_k b1b2μi0k

<1< μ

2

i01

a1a2μi01

. 3.28

This means

λi01< λi02<· · ·< λi0k<1< λi01 < λi02<· · ·< λi0k. 3.29

Consider3.16as a bifurcation problem from infinity. As above we need only to prove that

Dν_i₀_j{1} ×Y/∅, j 1,2, . . . , k; ν ,−. 3.30

FromLemma 3.5, we know thatDν

i0jcomes from{λi0j,∞}, meetsλi0j,0or has an

unbounded projection onR. If it meetsλi0j,0, then the connectedness ofD

ν

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guarantees that3.30is satisfied. On the other hand, ifDν_i₀_jhas an unbounded projection on

R, notice thatλ, v 0,0is the unique solution of3.1λ in whichλ 0inE, so3.30 also holds.

Proof ofTheorem 3.2. First suppose thatH3holds. Then there existsε >0 such that

1εfx, y< R M, for

x, y∈x, y:|x| ≤MR, y≤R. 3.31

Letλ, vbe a solution of 3.1λsuch that 0 ≤ λ <1εandv1 ≤R. Then by2.7,

3.1λ,3.31andLemma 2.1it is easy to see

v₁λAv₁λLFv₁ ≤λMFvMmax

t∈0,1λfLvt,−vt< M

R

M R. 3.32

Therefore,

Σ∩0,1ε×∂BR

∅. 3.33

This together with3.32and Lemmas3.4and3.5implies that

Cν_k∩0,1ε×BR

⊂0,1ε×BR, k1,2, . . . , i0; 3.34

Dν_j ∩0,1ε×∂BR

∅, j1,2, . . . , j0; 3.35

whereBR {v∈Y | v1 < R}andBR {v∈Y | v1 ≤R},CkνandDjνare obtained from

Lemmas3.4and3.5, respectively. SinceCν

kis a unbounded component of solutions of3.1λjoiningλk,0inE, it follows

from3.33and3.34thatCν_kcrosses the hyperplane{1} ×Ywith1, vν_{such that}_vν

1< R

ν or−,k1,2, . . . , i0. This means BVP2.5has 2i0nontrivial solutions{vνi} i0

1 in which

v₁ andv₁−are positive and negative solutions, respectively.

On the other hand, by3.33,3.35, andLemma 3.5one can obtain

Dν_j ∩{1} ×Y\BR

/

∅, j 1,2, . . . , j0. 3.36

This means BVP2.5has 2j0nontrivial solutions{wiν} j0

1 in whichw1andw1−are positive and

negative solutions, respectively.

Now it remains to show this theorem holds when the conditionH4is satisfied. From the above we need only to prove that

iforλ, v∈Cν_i ν or−,i1,2, . . . , i0,

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iiforλ, v∈Dν_j ν or−,j1,2, . . . , j0, we have that either

max

t∈0,1vt> r2, t∈0,1 3.38

or

min

t∈0,1vt< r1, t∈0,1. 3.39

In fact, like in13, suppose on the contrary that there existsλ, v∈Cν_i D_jνsuch that either

max{vt:t∈0,1}r2 3.40

or

min{vt:t∈0,1}r1 3.41

for somei, j.

First consider the case max{vt:t∈0,1}r2. Then there existst∈0,1such that

vt r2. Let

τ0 : inf

t∈0, t:vs≥ fors∈t, t,

τ1 : sup

t∈t,1:vs≥0 fors∈t, t.

3.42

Then

max{vt:t∈τ0, τ1}r2, 3.43

0≤vt≤r2, t∈τ0, τ1. 3.44

Therefore,vtis a solution of the following equation

−vt λfLvt,−vt, t∈τ0, τ1 3.45

withvτ0 vτ1 0 if 0< τ0< τ1<1 andvτ0 0 ifτ0 0.

By H4, there existsM ≥ 0 such that fx,−y My is strictly increasing iny for

x, y∈r1, r2×−Mr, Mr, wherermax{|r1|, r2}. Then

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UsingH4andLemma 2.1again, we can obtain

−r2−vt

λMr2−vt

−λfLvt,−vt Mvt−Mr2

−λfLvt,−vt Mvt−fLvt,−r2 Mr2

−λfLvt,−r2

≥0, t∈τ0, τ1

3.47

and ifτ11, by1.2we knowv1< r2. Therefore,

r2−vτ0>0, r2−vτ1>0 if 0< τ0< τ1<1;

r2−vτ00 ifτ00;

r2−vτ1>0 ifτ11.

3.48

This together with3.47and the maximum principle14imply thatr2−vt> 0 in

τ0, τ1, which contradicts3.43.

The proof in the case min{vt:t∈0,1}r1is similar, so we omit it.

Acknowledgments

The Project Supported by NNSF of P. R. China10871120, the Key Project of Chinese Ministry of EducationNo: 209072, and the Science & Technology Development Funds of Shandong Education CommitteeJ08LI10.

References

1 W. Feng and J. R. L. Webb, “Solvability ofm-point boundary value problems with nonlinear growth,”

Journal of Mathematical Analysis and Applications, vol. 212, no. 2, pp. 467–480, 1997.

2 C. P. Gupta, “A generalized multi-point boundary value problem for second order ordinary diﬀerential equations,”Applied Mathematics and Computation, vol. 89, no. 1-3, pp. 133–146, 1998.

3 R. Ma and N. Castaneda, “Existence of solutions of nonlinearm-point boundary-value problems,”

Journal of Mathematical Analysis and Applications, vol. 256, no. 2, pp. 556–567, 2001.

4 R. Ma and D. O’Regan, “Nodal solutions for second-orderm-point boundary value problems with nonlinearities across several eigenvalues,”Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 7, pp. 1562–1577, 2006.

5 R. Ma, “Nodal solutions for a second-order m-point boundary value problem,” Czechoslovak Mathematical Journal, vol. 56131, no. 4, pp. 1243–1263, 2006.

6 B. P. Rynne, “Spectral properties and nodal solutions for second-order,m-point, boundary value problems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 12, pp. 3318–3327, 2007.

7 J. Sun, X. Xu, and D. O’Regan, “Nodal solutions for m-point boundary value problems using bifurcation methods,”Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 10, pp. 3034– 3046, 2008.

8 J. R. L. Webb, “Positive solutions of some three point boundary value problems via fixed point index theory,”Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4319–4332, 2001.

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10 Z. Wei and C. Pang, “Multiple sign-changing solutions for fourth orderm-point boundary value problems,”Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 4, pp. 839–855, 2007.

11 P. H. Rabinowitz, “Some global results for nonlinear eigenvalue problems,”Journal of Functional Analysis, vol. 7, no. 3, pp. 487–513, 1971.

12 P. H. Rabinowitz, “On bifurcation from infinity,”Journal of Diﬀerential Equations, vol. 14, no. 3, pp. 462–475, 1973.

13 R. Ma, “Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems,”Applied Mathematics Letters, vol. 21, no. 7, pp. 754–760, 2008.