SOLVABILITY FOR SECOND-ORDER MULTI-POINT BOUNDARY VALUE PROBLEMS ON A HALF-LINE

In this paper, we are concerned with the existence of a second-order differential equation with m-point boundary conditions on the half line. The existence results are proved by the Leray-Scha¨uder point theorem in a special Banach space. Green function plays an important role in the proof and its discussion is very interesting. The results obtaine

Ann.ofDi.Eqs.28:2(2012),157-163

SOLVABILITYFORSECOND-ORDER

MULTI-POINTBOUNDARYVALUEPROBLEMS

ONAHALF-LINE

YuDuan,

HairongLian,

XinYu

(SchoolofSciences,ChinaUniversityofGeosciences,Beijing100083,E-mail:martinduan87@gmail.com(Y.Duan);lianhr@126.com(H.Lian))

Abstract

Inthispaper,weareconcernedwiththeexistenceofasecond-orderdierentialequationwithm-pointboundaryconditionsonthehalfline.TheexistenceresultsareprovedbytheLeray-Sch¨auderpointtheoreminaspecialBanachspace.Greenfunctionplaysanimportantroleintheproofanditsdiscussionisveryinteresting.Theresultsobtainedgeneralizethoseinthepreviousreferences.

Keywordsmulti-pointboundaryvalueproblem;Greenfunction;Leray-Sch¨auderxedpointtheorem;half-line

2000MathematicsSubjectClassication34B10;34B40

1Introduction

Multi-pointboundaryvalueproblems(BVPsforshort)ofdierentialequationsariseinavarietyofdierentareasofappliedmathematicsandphysics.Forexample,thevibrationsofaguywireofauniformcross-sectioncomposedofNpartsofdierentdensitiescanbesetupasamulti-pointboundaryvalueproblem[12].Manyscholarshavestudiedmoregeneralnonlinearmulti-pointboundaryvalueproblem,wereferthereadersto[3-6,8-10,12].

Boundaryvalueproblemsoninniteintervalsandtheirapplicationshavereceivedmanyattentioninrecentyears[1].Duetothenoncompactnessofinniteintervals,thediscussiononinniteintervalsproblemismorecomplicated,see[1,7,11].Tothebestknowledgeoftheauthors,fewworksweredoneintheliteratureontheexistenceofsolutionstosecond-ordermulti-pointBVPsoninniteintervals.Recently,LianandGe[7]consideredtheexistenceofsolutiontoathree-pointboundaryvalueproblem

{′x′(t)+f(t,x(t),x′(t))=0,0<t<+∞,

x(0)=αx(η),

t→+∞

limx′(t)=0.

Inthispaper,wediscussthefollowingsecond-orderdierentialequation

′x′(t)+f(t,x(t),x′(t))=0,

t∈(0,+∞),(1)

subjecttothegeneralmulti-pointboundaryvalueconditions

SupportedbytheFundamentalResearchFundsfortheCentralUniversities(2011YXL044)and

theNNSFofChina(11101385).

ManuscriptreceivedApril26,2011;RevisedSeptember14,2011Correspondingauthor.

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