))−1f(t,u)≤f(t,Mu)≤(g())−1f(t,u),∀M1,(t,u)∈(0,1)[0,∞)u1u1u1)≤h()f(t,u)≤r→0∑∑i1,⎪∑i1ii).⎪u''(0)0,u''(1)⎨⎪u(0)0,u(u(1)∑ii),⎧Theexistenceandincomparabilityofpositivesolutionstoaclassoffourthordersuper-linearm-pointsingularboundaryvalueproblems#51015DuXinsheng*(SchoolofMathematicsSciences,QufuNormalUniversity,ShanDongQuFu273165)Abstract:Thispaperinvestigatestheexistenceofpositivesolutionstoaclassoffourthordersingularsuper-linearm-pointboundaryvalueproblems.Anecessaryandsufficientconditionfortheexistenceofpositivesolutionsisgivenbymeansofthefixedpointtheoremsofconeexpansionandcompressionwithnormtype.Wealsoinvestigatetheincomparabilityofpositivesolutions.Keywords:Singularboundaryvalueproblems;Superlinearity;Positivesolutions1IntroductionDuringthepasttenyears,fourthordersingularboundaryvalueproblemshavereceivedconsiderableattention;see[1,2,3,4]andthereferencestherein.But,onlyfewresultshaveexistedforsuper-linearboundaryvalueproblems(see[3-4]andsomereferencestherein).Motivatedbytheworksof[3]and[4],thispapershallconsidertheexistenceofpositivesolutionstothefollowingsingularfourthordersuper-linearm-pointboundaryvalueproblemsu(4)(t)f(t,u(t))t∈(0,1)(1.1)20⎪m−2i1m−2u''((1.2)Where0i1,0i1,i1,2,K,m−2,012Lm−21areconstants,m−2i1m−2i1i1,m≥3,andfsatisfiedthefollowinghypothesis(H)f(t,u):(0,1)[0,∞)→[0,∞)iscontinuous,thereexisttwofunctionsg,h:(0,1)→[0,∞),withh(r)r,limh(r)r0suchthat25g(r)f(t,u)≤f(t,ru)≤h(r)f(t,u),∀r∈(0,1),(t,u)∈(0,1)[0,∞).(1.3)Remark1.1.Forany0≤u1u2,t∈(0,1),wehavef(t,u1)≤f(t,u2).Infact,forany0≤u1u2,t∈(0,1),by(1.3)weobtainf(t,u1)f(t,Remark1.2.Itisobviousthatu2u2u2u22f(t,u2).(1.4)30(h(11MM(1.5)Foundations:SRFDP(NO.20103705120002)Briefauthorintroduction:DuXinsheng,(1977-),male,associateprofessor,majorresearchdirection:Nonlinearfunctionanalysis.duxinsheng@qfnu.edu-1-(h(r))−1f(t,u)≤f(t,u)≤(g(r))−1f(t,u),LetXC[0,1],||u||max{|u|02|},where,|u(1.1)and(1.2)tohaveC[0,1]positivesolutionsisthatthefollowingintegralhold:0∫t(1−t)f(t,t)dt∞.(1.1)and(1.2)tohaveC[0,1]positivesolutionsisthatthefollowingintegralhold:0∫f(t,t)dt∞.problem(1.1)and(1.2)cannothavetwocomparableC[0,1]positivesolutions.Lemma2.1.Suppose(H)holds,letu(t)beaC[0,1]positivesolutionof(1.1)andu(t)∫G(t,s)[−u''(s)]ds1−∑ii∑∫G(,s)[−u''(s)]ds,m−2i0G(t,s)⎨1r∀r∈(0,1),(t,u)∈(0,1)[0,∞)(1.6)2|u|0sup|u(t),|u|2sup|u''(t).t∈[0,1]t∈[0,1]ThenXisaBanachspacewithnorm||.||.(1.7)3540Bysingularwemeanthefunctionsfin(1.1)areallowedtobeunboundedatthepointsatt0and/ort1.A(positive)functionu(t)∈C2[0,1]∩C4(0,1)iscalledaC2[0,1](positive)solutionof(1.1)and(1.2)ifitsatisfies(1.1)and(1.2).AC2[0,1](positive)solutionof(1.1)and(1.2)iscalledaC3[0,1](positive)solutionifu(3)(0)andu(3)(1−)bothexist.Supposeu1(t),u2(t)∈C[0,1].Ifu1(t)≤u2(t),∀t∈[0,1]oru1(t)≥u2(t),∀t∈[0,1]hold,thenwecallu1(t),u2(t)arecomparable,otherwise,u1(t),u2(t)arecalledincomparable.2MainresultsandseverallemmasForconvenience,welistthemainresultsinthispaper.Theorem2.1.Suppose(H)holds,th...