SHEN Wenguo

(Department of Basic Courses, Lanzhou Institute of Technology, Lanzhou 730050, China)

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Unilateral global bifurcation for fourth-order boundaryvalue problem with non-asymptotic nonlinearity at 0

SHEN Wenguo

(Department of Basic Courses, Lanzhou Institute of Technology, Lanzhou 730050, China)

fourth-orderproblems;unilateralglobalbifurcation;nodalsolutions;non-asymptoticnon-linearityat0

浙江大學學報(理學版),2016,43(5):525-531

0　Introduction

LetEbearealBanachspacewiththenorm‖·‖.Considertheoperatorequation

u=λBu+H(λ, u),

(1)

whereBisacompactlinearoperatorandH:R×E→EiscompactwithH=o(‖u‖)atu=0uniformlyonboundedλintervals.

Rabinowitz’sglobalbifurcationtheorem[1]hasshownthatifthecharacteristicvalueμofBisofoddmultiplicityand

thenthereexistsacomponentCμofSthatcontains(μ, 0),whichsatisfies:

Recently,SHEN[3-4]studiedtheexistenceofnodalsolutionsofthefollowingboundaryvalueproblem:

(2)

whereris a positive parameter, under the assumptions:

(A1) One of the following conditions holds:

(A2) h(t) ∈C([0, 1], [0, ∞))withh(t)?0onanysubintervalof[0, 1].

Lemma3[3-4](i)Thelineareigenvalueproblem

(3)

hasauniqueinfinitenumberofpositiveeigenvalues

0<λ1<λ2<…<λk<…→∞,ask→∞,

andtheeigenfunctionψkcorrespondingtoλkhasexactlyk-1zerosin(0, 1).

(ii)Foreachk∈N,thealgebraicmultiplicityofλkis1.

Meanwhile,RABINOWITZ[1]establishedunilateralglobalbifurcationtheory(theorem1.27andtheorem1.40of[1]).However,aspointedoutbyDANCER[2,5]andLPEZ-GMEZ[6],theproofsofthesetheoremscontaingaps.Fortunately,DANCER[2]gaveacorrectedversionoftheunilateralglobalbifurcationtheoremfortheproblem(1)whichhasbeenextendedtotheone-dimensionalp-LaplacianproblembyDAIetal.[7].In2013,DAIetal.[8]establishedaDancer-typeunilateralglobalbifurcationresultforfourth-orderproblemsofthedeformationsofanelasticbeaminequilibriumstatewhichbothendsaresimplysupported.

Motivatedbytheabovepapers,weshallestablishaDancer-typeunilateralglobalbifurcationresultaboutthecontinuumofsolutionsforthedeformationsofanelasticbeaminequilibriumstatewithfixedbothendpointswhichcanbedescribedbythefourth-orderproblems:

x′?+kx″+lx=λh(t)x+g(t,x,λ), 0

x(0)=x(1)=x′(0)=x′(1)=0,

(4)

wherehsatisfies (A2), andg:(0, 1)×R2→Rsatisfies the Carathéodory condition in the first two variables, such that

(5)

uniformly fort∈ (0, 1) andλon bounded sets.

Remark 1Since the problem (2) cannot transform into a system of second-order equation, the treatment method of second-order system does not apply to study the problem (2). Thus, existing literature on the problem (2) by bifurcation theory is limited[3-4,9].

Remark 2For other results on the existence and multiplicity of positive solutions and nodal solutions for other boundary value problems of fourth-order ordinary differential equations based on bifurcation techniques[10-11].

The rest of this paper is arranged as follows. In section 1, we establish the Dancer-type unilateral global bifurcation theory for problem (4). In section 2, we prove the existence of nodal solutions for the problem (2) under the linear growth condition onf.

1　Unilateral global bifurcation results

WedefinethelinearoperatorL:D(L)?E→Y,

Lx=x′?+kx″+lx, x∈ D(L)

withD(L)= {x∈C4[0, 1]|x(0)=x(1)=x′(0)=x′(1)=0}.ThenLisaclosedoperatorandL-1:Y→Eiscompletelycontinuous.

Define the operatorH:R×E→Eby

H(λ,x)(t):=λL-1(hx)+L-1(g(t,x,λ))=

Tλ(x)+L-1(g(t,x,λ)).

It is easy to show thatI-Tλis a nonlinear compact perturbation of the identity. Thus the Leray-Schauder degree deg(I-Tλ,Br(0),0) is well-defined for arbitraryr-ballBr(0) andλ≠λk.

Lemma 4For anyr>0, we have

deg(I-Tλ,Br(0),0)=

ProofSinceTλis compact and linear, by theorem 8.10 of [12],

deg(I-Tλ,Br(0), 0)=(-1)m(λ),

wherem(λ) is the sum of algebraic multiplicity of the eigenvaluesλof (3),satisfyλ-1λk<1.

Ifλ∈ [0,λ1), by lemma 3, then there are no suchλat all, then

deg(I-Tλ,Br(0), 0)=(-1)0=1.

Ifλ∈(λk,λk+1) for somek∈N, then

(λj)-1>1,j∈{1, 2,…,k}.

By lemma 3, we obtain

deg(I-Tλ,Br(0), 0)=(-1)k.

Furthermore, it is clear that problem (4) can be equivalently written as

x=H(λ,x)(t).

Clearly,His completely continuous fromR×E→EandH(λ, 0)=0, ?λ∈R.

Let

(6)

(7)

uniformlyfort∈ (0, 1)andλonboundedsets.

Theorem1Assume(A1), (A2)and(5)hold.Then

(i) (λk, 0)isabifurcationpointoftheproblem(4).

Proof(i)Supposethat(λk, 0)isnotabifurcationpointofproblem(4).Thenthereexistε> 0,ρ0>0suchthatfor|λ-λk|≤εand0<ρ<ρ0thereisnonontrivialsolutionoftheequation

x-H(λ, x)=0

with‖x‖=ρ.Fromtheinvarianceofthedegreeunderacompacthomo-topology,weobtainthat

deg(I-H(λ, ·),Bρ(0), 0)≡constant

(8)

forλ∈[λk-ε,λk+ε].

By takingεsmaller if necessary, we can assume that there is no eigenvalue of (3) inλ∈(λk,λk+ε]. Fixλ∈(λk,λk+ε]. We claim that the equation

x-(λL-1(hx)+τL-1(g(t,x,λ)))=0

(9)hasnosolutionxwith‖x‖=ρforeveryτ∈[0, 1]andρsufficientlysmall.Supposeonthecontrary,let{xn}bethesolutionof(9)with‖xn‖→0asn →+∞.

(10)

By(7), (10)andthecompactnessofL-1,choosingasubsequenceandrelabelingifnecessary,itfollowsthatyn→y0asn→∞.Thus

Ly0=λhy0and‖y0‖E=1.

Thisimpliesthatλisaneigenvalueof(3).Thisisacontradiction.Fromtheinvarianceofthedegreeunderhomo-topologyandlemma4,thenobtain

deg(I- H(λ, ·),Bρ(0), 0)=

deg(Ψλ,Bρ(0), 0)=(-1)k.

(11)

Similarly, forλ∈[λk-ε,λk), we find that

deg(I-H(λ, ·),Bρ(0), 0)=(-1)k-1.

(12)

Relations (11) and (12) contradicts (8) and hence (λk, 0) is a bifurcation point of problem (4).

(ii) By (7), we have that

(13)

uniformlyt∈(0, 1) andλon bounded sets. Furthermore, by (ii) of lemma 3, applying lemma 2, we can obtain the result.

(14)

By(7), (14)andthecompactnessofL-1,weobtainthatforsomeconvenientsubsequenceym→y0asm→+∞.Nowy0verifiestheequation

Ly0=λjhy0and‖y0‖E=1.

Hencey0∈SjwhichisanopensetinE,andasaconsequenceforsomemlargeenough, ym∈Sj,andthisisacontradiction.

Lemma6If(λ, u)isasolutionof(4)andx∈?Sk,thenx≡0.

ProofBytheproofoftheorem3.1in[13] (seecorollary1.12andtheproofoftheorem2.3togetherwiththeremarkfollowingthatproofin[1]),weeasilyobtaintheresult.

Bytheorem1andlemma5,wecaneasilydeducethefollowingDancer-typeunilateralglobalbifurcationresult.

(15)

(16)

By(7), (16)andthecompactnessofL-1,weobtainthatforsomeconvenientsubsequencezm→z0asm→+∞.Nowz0verifiestheequation

Lz0=λjhz0and‖z0‖E=1.

(17)

By(7), (17)andthecompactnessofL-1,weobtainthatforsomeconvenientsubsequenceyn→y0≠0asn→+∞.Nowy0verifiestheequation

Ly0=λ*h(t)y0(t), t∈(0,1)and‖y0‖E=1.

Inordertotreatthecasef0=∞,weshallneedthefollowinglemma.

Definition1[14]LetXbeaBanachspaceand{Cn|n=1,2,…}beafamilyofsubsetsofX.ThenthesuperiorlimitDof{Cn}isdefinedby

suchthatxni→x}.

(18)

Lemma 7[14]Each connected subset of metric spaceXis contained in a component, and each connected component ofXis closed.

Lemma 8[15]LetXbe a Banach space and let {Cn|n=1, 2,…} be a family of closed connected subsets ofX. Assume that

(i) there existszn∈Cn,n=1, 2,… andz*∈X, such thatzn→z*;

(ii)rn=sup{‖x‖|x∈Cn}=∞;

BR={x∈X|‖x‖≤R}.

ThenthereexistsanunboundedconnectedcomponentCinDandz*∈C.

2　Main results

Inordertoprovethemainresults,theconditions(A1), (A2)andthefollowingconditionsaresatisfiedinthefollowingpart:

(H1) f∈C(R, R)satisfiesf(s)s>0fors≠0.

Theorem3Let(A1), (A2), (H1), (H2)and(H3)hold.Assumefollowingconditionholdsforsomek∈N:

Theorem4Let(A1), (A2), (H1), (H2)and(H3)hold.Assumefollowingconditionholdsforsomek, n∈Nwithk≤n:

Proofoftheorem3Firstly,westudythebifurcationphenomenaforthefollowingeigenvalueproblem:

(19)

whereλ>0isaparameter.Itisclearthatanysolutionof(19)oftheform(1,x)yieldssolutionsxof(2).

Foreachn∈N,definef[n](s):R→Rby

Clearly,by(H2),wehave

Nowconsidertheauxiliaryfamilyoftheequations

(20)

Letζ[n]∈C(R, R)suchthat

(21)

Then

(22)

Letusconsider

(23)asabifurcationproblemfromthetrivialsolutionx≡0.

Equation(23)canbeconvertedtotheequivalentequation

x:=λL-1[a[n](·)x(·)](t)+

λL-1[ζ[n](·x(·))](t).

(24)

Clearly, ‖L-1[ζ[n](·,x(·))]‖E=o(‖x‖E),as‖x‖E→0.

Since

λn+‖xn‖→∞.

If

then

andmoreover,

AssumethatthereexistsaconstantnumberM>0suchthatforalln∈N,

λn∈ (0,M],

Inthiscase,itfollowsthat‖xn‖E→∞.

Letξ∈C(R, R)suchthat

f(x)= f∞x+ξ(x).

Then

Let

(25)

Wedividetheequation

(26)

since

Thus

y′?+ky″+ly=λra(t)f∞y.

Weclaimthat

Proofofthetheorem4Usingthesimilarproofwiththatoftheorem3,wecanobtaintheresult.

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10.3785/j.issn.1008-9497.2016.05.005

非線性項在零點非漸進增長的四階邊值問題單側(cè)全局分歧.

沈文國

(蘭州工業(yè)學院， 基礎學科部， 甘肅 蘭州 730050)

四階問題；單側(cè)全局分歧；結點解；非線性項在零點非漸進增長

O175.8

A

1008-9497(2016)05-525-07

date：August 1,2015.

Supported by the National Natural Science Foundation of China (11561038); the Gansu Provincial Natural Science Foundation(145RJZA087).

About the author:SHEN Wenguo (1963-), ORCID:http://orcid.org/0000-0001-7323-1887, Doctor, Professor, the field of interest is nonlinear functional differential equations,E-mail: shenwg369@163.com.