非线性奇异微分方程无穷边值问题的正解

【作者】 王颖；

【导师】 刘立山；

【作者基本信息】 曲阜师范大学 ， 应用数学， 2007， 硕士

【摘要】 作为现代分析数学的一个重要分支，非线性泛函分析近年来发展迅速，并广泛应用于物理，生物，化学，计算机信息等诸多领域，受到了越来越多的数学工作者的关注。其中，非线性无穷边值问题来源于应用数学和物理的多个分支，特别是在研究椭圆方程的径向对称解，大气压强等方面具有广泛的应用，是目前分析数学中最为活跃的领域之一。本文通过利用锥理论，不动点指数原理，Krasnosel-skii不动点定理等研究了几类奇异微分方程无穷边值问题正解的存在性情况，得到了一些新成果。根据内容本文分为以下三章：本文第一章中，通过构造一个特殊的Banach空间，利用锥上的不动点原理，在较弱条件下讨论了下列奇异微分方程无穷边值问题正解的存在性。其中λ＞0是参数，k∈（-∞，+∞），f：（O，+∞）×[0，+∞)→[0，+∞)是连续函数并且允许在t=0点具有奇异性；p∈C[0，+∞)∩C¹（0，+∞）且p在（0，+∞）上大于0，integral from n=0 to∞1／（p（s））ds＜+∞；α_i，β_i≥0，（i=1，2）。且满足ρ=α₂β₁+α₁β₂+α₁α₂B（0，∞）＞0，B（t，s）=integral from n=t to s 1／（p（v））dv。并获得了当特征值λ在某一范围内取值时，边值问题至少存在一个正解的结论。作者的结果包含，推广并改进了许多已知结果。了一类二阶非线性奇异微分方程无穷边值问题一个和多个正解的存在性。其中p，α，β≥0，α²+β²＞0，q＞0，f允许在t=0处奇异。在第三章中，我们利用锥上的不动点指数原理和Krasnosel-skii不动点理论，讨论了一类奇异微分方程无穷边值问题正解及多重正解的存在性。其中k∈（-∞，+∞）；f_i是非负连续函数；φ_i在t=0处奇异；p∈C[0，+∞)∩C¹（0，+∞）并且p在（0，+∞）上大于0，integral from n=0 to∞1／（p（s））ds＜+∞；α_i，β_i≥0（i=1，2），ρ=α₂β₁+α₁β₂+α₁α₂B（0，∞）＞0，B（t，s）=integral from n=t to s 1／（p（v））dv。更多还原

【Abstract】 As an important branch of modern analysis mathematics, nonlinear functional analysis arises quite naturally and quickly in recent years, and what’s more it applied extensively in the field of physics, chemistry and information technology, more and more mathematicians are devoting their time to it. Among them, the nonlinear infinite boundary value problem comes from a lot of branches of applied mathematics and physics, especially in the study of radial solutions of nonlinear elliptic equations and models of gas pressure in a semi-infinite porous medium, it is at present one of the most active field that is studied in analytical mathematics. In this paper, by using the cone theory, fixed point index theory and Krasnosel-skii fixed point theorem and so on, we investigate the existence of positive solutions of infinite boundary value problems of several kinds of singular differential equations. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into three sections according to contents.In chapter 1, by constructing a special Banach space, the author is concerned with the existence of positive solution for the following nonlinear sin-gujar boundary value problems on the half-line:whereλ> 0 is a parameter, k∈（-∞, +∞）, f : （0, +∞）×[0, +∞)→[0, +∞) is a continuous function and may be singular at t = 0; p∈C[0, +∞)∩C¹（0, +∞） with p > 0 on （0, +∞）,∫₀^∞1/p（s） ds < +∞;α_i,β_i≥0 （i = 1,2） withρ=α₂β₁ +α₁β₂ +α₁α₂B（0,∞） > 0 in which B（t,s） =∫_t^s 1/p（v） dv. The author derive an explicit interval for A such that for anyλin this interval, the existence of at least one positive solution to the boundary value problem is guaranteed. The results significantly extend and improve many known results even for non-singular cases.In Chapter 2, by introducing a special cone and applying fixed index theory in cone, the existence of multiple positive solutions of infinite boundary value problem of a class of singular second order differential equation is guaranteed.where p,α,β≥0,α² +β² > 0, q > 0, f can be singular at t = 0.In Chapter 3, we discuss the existence of positive solution and multiple positive solutions of infinite boundary value problem of a class of singular differential equation by using the fixed point theorem and the Krasnosel-skii theorem on cone expansion and compression in a special space and cone.where k∈（-∞, +∞）, f_i is continuous function;φ_i can be singular at t = 0; p∈C[0,+∞)∩C¹（0, +∞） with p> 0 on （0, +∞）,∫₀^∞1/p（s） ds < +∞;α_i,β_i≥0, （i = 1,2） withρ=α₂β₁ +α₁β₂ +α₁α₂B（0,∞） > 0, B（t, s） =∫_t^s 1/p（v） dv.更多还原