【作者】 周金艳；

【导师】 闫宝强；

【作者基本信息】 山东师范大学 ， 应用数学， 2008， 硕士

【摘要】 半无穷区间上二阶边值问题起源于对非线性椭圆微分方程对称径向解以及半直线上中间多漏洞的煤气压力模型的研究．现在人们越来越关注半无穷区间边值问题正解的存在性，并取得了许多优秀成果．有限区间上二阶线性微分方程多点边值问题是由V．A．Il’in和E．I．Moiseev提出的．然后C．P．Gupta．针对非线性二阶三点边值问题进行了研究．此后，许多学者对非线性有限区间多点边值问题相继发表了大量的研究成果，采用的方法主要有Leray-Schauder连续定理和迭合度理论等．但目前关于半直线上多点边值问题的文章很少，且研究的大多是非线性项，f奇异时解的存在性，非线性项既依赖于x′又奇异变号的文章尚不多见．针对这个现状，本文主要利用不动点指数理论研究了无穷区间上三点边值问题正解的存在性．全文共分二章．在第一章中，我们用锥上的不动点指数理论考虑半直线上非线性二阶三点边值问题正解的存在性．其中0<α<1，η∈（0，+∞），f在x=0和x′=0奇异且变号．在文献[20]中，Hairong Lian，Weigao Ge研究了半直线上三点边值问题其中α≠1，0<η<+∞．在|f（t，u，v）|≤p（t）|u|+q（t）|v|+r（t）条件下，通过建立Green函数，利用Leray-Schauder连续定理，得到了其解的存在性．本文则增加了非线性项奇异且变号，且对于f的限制条件进一步减弱，利用不动点指数理论得到了正解的存在性，解决的问题更广泛，所以本文是对半直线上三点边值问题正解存在性理论的发展，具有重要的理论意义和应用价值．在研究上述问题正解存在性时，主要参考了文[20，21，24，25]．我们首先研究了问题（1）的正解不存在时的情况，然后分别讨论了f在x′=0奇异但在x=0不奇异时正解的存在性，以及在x′=0和x=0都奇异时正解的存在性．研究方法是首先构造适当的积分算子，然后提出新的条件克服奇异和变号，利用不动点指数理论得到所研究方程的近似解，其极限就是原方程的解．在第二章中，我们给出了半直线上微分方程三点边值问题正解存在的简单判定准则，其中0<α<1，η∈（0，+∞）．在文[19]中Bing Liu研究了如下方程一个及多个正解的存在性．这里我们将问题推广到无穷区间来研究．为了克服区间的无界性给我们造成的困难，我们构造了极限函数f₀，f_∞和f⁰，f^∞．对于非线性项奇异所造成的困扰，我们则提出新的条件加以解决．值得指出的是引理2．1．4的证明为我们以下的讨论提供了重要的理论条件．首先，我们在f⁰=0，f_∞或者f₀=∞，f^∞=0的条件下，得到了问题（2）至少一个正解的存在性．然后，f₀=f_∞=∞或者，f⁰=f^∞=0的前提下，给出了问题（2）两个正解的存在性条件．接着，我们给出了f₀，f_∞，f⁰，f^∞，（?）{0，∞}时，问题（2）正解存在性的判别准则．最后，我们给出例子来进行说明．在研究上述问题时，我们主要利用的是锥上的Krasnoselskii不动点定理．更多还原

【Abstract】 Second-order boundary value problems （BVPs） on infinite intervals , arising from the study of radially symmetric solutions of nonlinear elliptic equation and models of gas pressure in a semi-infinite porous medium, have received much attention, see and the references therein.Multi-point boundary value problems of second-order linear differential equations on a finite interval were initiated by V.A. Il’in and E.I. Moiseev and three-point, BVPs of nonlinear differential equations were studied by C.P. Gupta.Since then, more general nonlinear multi-point BVPs on finite intervals have been discussed extensively . The methods therein mainly depend on the Leray-Schaudcr continuation theorem, coincidence degree theory . However, few works are done for second-order multi-point BVPs on an infinite interval and these results are mostly on f without singularity.Few papers study existence of positive solutions for three-point boundary value problems on infinite intervals when non-linearity depends on x′and may change sign and may be singular. To fill the gap in this area, we discuss the existence of positive solutions for three-point boundary value problems on half-line, using the fixed point index theory. This paper is divided into two chapters.In chapter 1, we study the existence of positive solutions forthree-point boundary value problems for differential equations on half-line as followswhere 0 <α< 1,η∈（0, +∞）,f may change sign and may be singular at·x = 0 and x′= 0.In [20] Hairong Lian, Weigao Ge studied the following second-order three-point BVP on a half-linewhereα≠1,η∈（0,+∞）. With the help of the established Green function and the Leray-Schauder continuation theorem suitable conditions imposed on f（where |f（t,u,v）|≤p（t）|u| + q（t）|v| + r（t））are presented for the existence of solutions. In this papar, we consider the case that f may be change sign and singular. Using the fixed point index theory on a cone, we discuss the existence of positive solutions with less conditions imposed on f. So it’s used more widely than the passage above and it is the improvement of theory for existence of positive solutions for three-point boundary value problems.It has a great role in theory and practice.Some ideas come from [20,21,24,25]and references therein.The paper is organized as follows. Firstly, we study the nonexistence of positive solutions of （1）, then we discuss the existence of positive solutions when f is singular at x = 0 but not at x′= 0 and singular both at x = 0 and x′=0.The method we used is firstly constructing proper integral operators and use some new conditions to overcome the singularity and sign changing/Finally, using the fixed point index theory on a cone, we consider the set of the approximate solutions and obtain a convergent subsequence. The limit is a positive solution for equation （1）.In chapter 2, we establish some simple criterions for the existence of single and two positive solutions of the three-point BVP for differential equations on half linewhere 0 <α< 1,η∈（0, +∞） .In[19], Bing Liu studied the existence of single and multiple pos itive solutions to the three-point boundary value problem as followsIn this papar, we discuss the case in an infinite interval.In order to overcome the difficulty from the unboundedness of the interval, we constructed the limit funtions f₀,f_∞and f⁰,f^∞,and then we use some new conditions to deal with the singularity caused by the nonlinearity. It is worth noting that the provement of lemma 2.1.4 provided an important theoretical conditions for the following discussion .Firstly, we discuss the existence of single positive solution for the BVP（2） under f⁰ = 0, f_∞, or f₀ =∞, f^∞= 0. Then, we establish the existence conditions of two positive solutions for the BVP（2） under f₀ = f_∞=∞or f⁰ = f^∞= 0.After that, we obtain some existence results for positive solutions of the BVP（2） under f₀, f_∞, f⁰, f^∞（?） {0,∞}. Finally, we give some examples to illustrate our results. When we study the above problems,we mainly used the Krasnosel-skii’s fixed point theorem in. a cone.更多还原