【作者】 杜新生；

【导师】 赵增勤；

【作者基本信息】 曲阜师范大学 ， 应用数学， 2002， 硕士

【摘要】 非线性泛函分析是现代分析数学的—个重要分支。以非线性泛函分析为基础而发展起来的奇异微分方程理论因其能很好的解释自然界中的各种各样的自然现象而倍受关注。有关奇异微分方程边值问题正解的存在性、唯一性、正解存在的充分必要条件近年些来获得了广泛的研究。 本文第一节中，我们将使用算于的迭代技巧、上下解方法以及Schauder不动点定理去研究一类四阶奇异边值问题： x(4)(t)=f(t,x(t)) (1) x(0)=x(1)=x″(0)=x″(1)=0 其中f满足条件： (H)：f(t，x)在(0，1)×[0，∞)上非负连续，对固定的t∈(0，1)，f(t，x)关于x单调递增，存在q∈(0，1)使得f(t，ax)≥aqf(t，x)。对任意的a∈(0，1)成立，(t，x)∈(0，1)×[0，∞)，f可能在t=0，t=1处奇异。 我们得到了如下结果： 定理1.1：设条件(H)满足，则方程存在C3[0，1]正解的充分必要条件为： 0＜integral from n=0 to 1(f(t,e(t))dt)＜∞ 其中e(t)=t(1-t)。 定理1.2：假设f满足条件(H)，且满足下面的不等式： 0＜integral from n=0 to 1(e(s)f(s,I)ds)＜∞ 则方程(1)至少存在一个C2[0，1]正解。 注＊k在定理1．二的证明过程中，我们得到了炉p，11正解的迭代勋 及0仙，11正解的髓估计，龊以往的文献所没有的，且证明摊与以往的 对于口腑异边值问题的研究有本质的不同． 注＊：韦在【纲中要求m足条件p．0、3．3）而由文献【叫可以知道 如果（3．1）满足，则（3．2）（3．3）自动满足． 在第二节中我们利用锥拉伸与压缩不动点定理、格林函数的住质以及紧 算子一髓近技巧考察了如下的…＊，1）共轭奇异陇问题．： 0 u’”’＋aXt）人。）＋坷t）g（。）一 0 t e（0，二） LJW0】＝0 0＜k＜n一2 o） in（l）＝0 无论是在次线性还是在超线住以及超线住、次线住混合的憎形下得到了该问 题正解以及多解的存在性定理． 为了叙述的方便，列出将本文将要用到的条件如下： （A） j［0，co）＋ ［0，co），010，co）＞ ［0，co） i＄＃． 饵）aN，1）－队co），b＊，1）－＋ 10，co）连续，且在 IO，11的任何子区间上不恒 为0，a＊），bk）可能在t二0，二处奇异． （CI）Ic＝0，IOO＝co，切＝0，gOO＝co； （q几二co，儿二0，m二co，＃o二0； （q几二0，人＝co，m二co，ao二0； （C4）fo＝co，fOO＝0，go＝0，gOO＝co； 其中 人二l。叭、。些；jOO二h叭、一呷．对于。，肋的定义相同． 我们的到了匆下的结果： 定NZ．1：假设八B，CI或人，B，CZ满足，且： J G（S），S）MS）＋b（s） 血＜ OO 则方程p）至少存在一个正解． ． 2 J 定理2．2：假设人BQ或八Bq满足，又设当f〔叮1］时，八O兰 ／（l），g（t）三g（），且 dll Itrl了u．SllllSI十OISjllSW＿ 其中 M二 max汀O），gm｝则问题p）至少存在两个正解． 注2．L如果在定理二中令N0、o，且叶o非奇异，则定理2．且既为文 献卜］中的主要结果，因此本定理为文［43］中主要结果的推广． 注2．2：由定理2的证明过程可以知道在定理2中可仅要求J和J满 足： fOO二OO，go＝OO O fo二0，gOO＝0． 注2J：目前还很少有文献涉及…一1，1）共轭奇异边值问图的多解的存 在性定理，因此本节中定理p习的躲路的． 在第三节中我们利用锥拉伸与压缩不动点定理以及格林函数的性质考察 如下的b，。一P）栅奇异边值问题： D（一1）”－＊”’（O一叭t）八L，叭t》t E（0，1） y’”’（0＝0 0 5 i S P－1（3） Iy‘”’（1 ＝＝ 0 0 ＜ i ＜ fi＊P－1 得到了该问题正解存在的充分条件． 为了叙述的方便列出本文用到的条件： （H）＆（t）EC《0，1），p，叫），f＊C仰，l）xN，co），N，co》，j（，y）5 P（t＊y）其中更多还原

【Abstract】 Nonlinear functional Analysis is an important branch of morderm math-matics .The singular differential equation has developed on the basis of nonlinear functional analysis .because it can explain a lot of natural phenomenal ,more and more mathematicans are devoting their tunes to the study of sigular differential equation.In the first chapter we will exploit the skill of operator iterate , the lower and upper solutions and Schauder fixed point theory to study a class of fourth ordered singular differential equation:where f satisfy:(H):f(t,x) G C((0, l),[0,)), for any fixed t (0,1), f(t,x) is increase function about x,there exist a q 6 (0,1) s.t for any a G (0,1), f(t, ax) > aqf(t, x), (t, x) e (0,1) x [0, ), / may be singular at t=0 or t=lwe obtained the fowUing results:Theoreml.l: Suppose (H) holds,then a necessary and sufficient conditions for problem(l)to have C3[0, 1] positive solutions is that the following inequality holds:where e(t)=t(l-t)Theorem 1.2: Suppose (H) holds then a sufficient conditions for problem (1) to have C2[0, 1] positive solutions is that the following inequality holds:Remark 1.1: In the process of the proof of theorem 1.1 ,we obtained the iterative approximate of the positive solution and the estimation of the error of the positive solutin, They are not found in the former literature.also the method has essent difference to the former mehond to fourth ordered singular differential equation.Remark 1.2:The literature [28] .where f satisties (3.1)-(3.3) we can acquire from literature [24] that if (3.1)satisfies ,then (3.2) (3.3) are also satisfies.In the second chapter ,we exploit the fixed point theory of cone expansion and compression ,the quality of Green function and the skill of operator union approximate to study the following (n-1,1) conjugate singular boundary value problem:In the case of sublinear or the case of suplinear and suplinear,sublinear mixed .we obtained the sufficient condition for problems (2) to have one positive solution and twine positive solutions.For convenience we list the fowlling assumptions:(A)f[0,oo) [0, ), 0[0, oo) [0,oo)f,g are continuous.(B)a(t) : (0,1) (0, ), b(t) : (0,1) [0,)is continuous,and does not vanished identically in any subinterval of [0,1], a(t) b(t) may be singular atwhere:obtained the fowlling results:Theorem 2.1: Supposed, B, C\ or A, B, C2 holds,and the following inequity holds:Then equation (2)has at least one positive solution.Theorem2.2: Suppose A, B,(73,orA,B, C\ holds ,as well we suppose:(2) has at least two positive solutions.Remark 2.1:In therom 2.1 if we suppose b(t)=0 ,a(t)does not singular at t=0,t=l,then theorem 2.1 is the main results in literature [43],therefore the results are new.Remark 2.2: We can learn from the process of the prove of theorem 2.2 that we can only require f and g satisfies :remark 2.3: Now ,As we learn there are very few people study twine positive solutions of problem (2) therefore the result of theroem 2.2 is new.In the third chapter ,we exploit the fixed point theorem of cone expansion and cone compression and the qulity of Green function to study the following (p,n-p) conjugate boundary value problem:Obtained the sufficien conditions for problem (3) to have positive solutions. For convenience we list the assumption:I unionsatisfies;We obtained the following results:Theonn 3.1: Suppose (H\), (H2) holds ,as well the fowlling inequlity holds:then equation (3) has at least one positive solution. Theorem 3.2: Suppose (Hi), (H2)ho\ds ,as well the fowlling inequlity holds:then equation (3) has at least one positive solution.Remark 3.1:In literature [8]/(t,w) = /(u(t)),also / is continous,and / is not singular,thus the main results in this section is the generalize of the main results in literature [8].Remark 3.2:In the former literatures require f(t,x) monotone,in this paper we get rid of this require ,and the method has essential difference to the former literature.the result in thi更多还原