【作者】 邓晓波；

【导师】 刘衍胜；

【作者基本信息】 山东师范大学 ， 基础数学， 2007， 硕士

【摘要】 常微分方程边值问题在经典力学和电学中有着极为广泛的应用，它是常微分方程学科的重要组成部分。基于丰富的实际应用背景，抽象空间中非线性常微分方程边值问题正解的存在性问题，在整个常微分方程研究领域，显得尤为重要。对于经典的边值问题(比如：Drichlct型两点边值问题，Robin型两点边值问题等)近30年来，已取得了深入而系统的结果。但是，有关三点边值问题(尤其是高阶微分方程多点边值问题)的研究却相对较少，如文[1]-[5]，[13]-[16]和[18]。在上述研究基础之上，本文主要研究Banach空间中的几类非线性微分方程多点边值问题一解及多解的存在性；并鉴于半正、脉冲、奇异的重要性，探讨了半正、脉冲、奇异对非线性微分方程的解产生的影响，且获得了一些较好的结果。因此本文研究的内容具有重要的理论意义和应用价值。全文共分四章。本文第一章利用关于严格集压缩算子的锥拉伸与压缩不动点定理，讨论了Banach空间E中一类四阶非线性微分方程三点边值问题多个正解的存在性，其中η∈(0，1)，并给出有限维空间中的例子说明我们的条件是合理的。第二章通过建立特殊的锥，利用不动点定理研究了半正边值问题正解的存在性，其中非线性项f(t，x)可能在某些t和x处为负，且f∶[0，1]×[0，+∞)→R满足Carathéodory条件，λ是一个正参数，η∈(1／2，1)，随后通过例子验证条件的合理性。第三章利用有界正线性算子的谱理论及锥上的不动点指数理论，得到了奇异半正边值问题分别在满足超线性和次线性条件下正解的存在性，并举例说明条件的合理性，其中f(t，x)可能在t=0，t=1和x=0点有奇异性，且非线性项f(t，x)可能在某些t和x处为负。最后一章利用M(?)nch不动点定理，讨论了Banach空间中一类带奇异性的脉冲微分方程多点边值问题解的存在性，其中β∈R，η∈(0，1)。最后通过无穷维空间的例子说明条件合理。更多还原

【Abstract】 The boundary value problems are an important part in the field of differential equations, which have comprchensive application in the classical physics and electricity. On the base of the richful pratical and applied background, the cxistence of positive solutions for boundary valuc problems of nonlincar differential equtions in abstract space bccome cspccially important in the area of the research about differcatial equations. Classical boundary value problems (for example: Drichlet two-point boundary value problem, Robin two-point boundary value problem, and so on) had been studied widely for last thirty years and there are many excellent results. However, relatively few results about three-point boundary value problems (especially multiple points boundary value problem for high-order differential equations) had been obtained, for example, the results of reference [1]-[5], [13]-[16] and [18].On the basis of above discussions, this paper mainly deals with the existence of one solution and multiple solutions of some nonlinear differential equations with multiple-point boundary value problem in Banach space; because of the importance of scmipositone property, impulse and singularity, we study their effects on the solutions of nonlindar differential equations, and we obtain some useful results. So the content that we studied has important theoretical and applicable value.There are four chapters in the dissertation.In the first chapter, by using cone compression and expansion fixed point theorem of strict set contraction operator, we deal with the existence of multiple positive solutions of the following fourth-order three-point boundary value problem in Banach space andη∈(0, 1), and an example in finite-dimensional space is worked out to indicate our conditions are reasonable.In the second chapter, by using cone fixed point, theorem, we investigate the cxistcnce of positive solutions of the following semipositone boundary value problem and nonlinearity f(t, x) maybe negative at some t and x, and that nonlinearity satisfies Caratheodory conditions,λis a positive paramcter,η∈(0, 1), then an cxample is considcrcd to indicate our conditions are reasonable.In the third chapter, by using spedtrum thcory of bounded positive operator and fixed point index theorem of cone, this chapter deals with the existence of positive solutions for the following singular semipositone boundary value problem which respectively satisfies superlinear and sublinear in Banach space and cxamples are worked out to indicatc our conditions are reasonable, where f(t, x) maybe singular at t=0, t=1 and x=0, f(t, x) maybe negative at some t and x.In the last chapter, by using Monch fixed point theorem, we consider the cxistence of solutions of the following second-ordcr multiplc-point singular boundary value problem of impulsive ditferential equations in Banach space andβ∈R,η∈(0, 1). In the end, an example in infinite-dimensional space is worked out to indicatc our conditions are reasonablc.更多还原