带有扰动项的非线性微分方程解的研究

发布时间：2018-06-22 00:15

  本文选题：非线性分数阶微分方程 + m点边值问题　； 参考：《曲阜师范大学》2017年硕士论文

【摘要】：本文研究的是带有扰动项的非线性微分方程的解.通过专家学者对非线性整数阶常微分方程的不断研究,我们对其已经非常了解,并且在物理学,生物学,经济学等许多领域得到了广泛的应用.随着科学的发展和不断地深入研究,我们通过对非线性常微分方程加以推广与改造,在带有扰动项的非线性微分方程方面进行了深入的研究,并且也取得了突破性的研究成果,如文献[1]运用Schauder不动点定理证明了带有扰动项的非线性微分方程解的存在性.文献[2],[3]都是运用锥拉伸与压缩不动点定理得到了所研究方程解的存在性.本文受文献[1]-[5]的启发,对带有扰动项的非线性微分方程进行了研究.根据内容,本文分为以下三章:第一章:主要是介绍本文将要用到的一些基本定义和一些与本文证明有关的引理.第二章:考虑带有扰动项的非线性分数阶微分方程边值问题解的存在性,其中D0α+u(t)是Riemann-Liouville分数阶导数,α≥ 2,1 ≤α-β ≤n - 1,n - 1 ≤ α ≤ n,ηi ≥ 0(i = 1,2,...,m - 2),0 ξ_1 ξ_2 … ξ_(m-2) 1,f :(0,1) × (0,∞) × (0,∞)→(0,∞)连续,e(t)∈L1([0,1],R)可能变号.运用Schauder不动点定理,得到解的存在性,最后给出应用.第三章:考虑带有扰动项的非线性分数阶微分方程的特征值问题其中λ是正实参数,D1qx(t)是Riemann-Liouville分数阶导数,q ≥ 2,n-1 ≤q ≤n,i ∈ N, 0 ≤ i ≤ n - 2, a_j ≥ 0(j = 1, 2,..., m - 2), 0 b_1 b_2 … b_(m-2) 1, (?),f ∈ C(0,1) × (0,∞) → [0,∞)并且在t = 0,1处奇异,e(t) ∈ L~1([0,1],R)可能变号.运用锥拉伸与压缩不动点定理,得到解的存在性,作为应用给出相应的例子.
[Abstract]:In this paper, we study the solutions of nonlinear differential equations with perturbed terms. Through the continuous study of nonlinear integer order ordinary differential equations by experts and scholars, we have been very familiar with them, and have been widely used in many fields such as physics, biology, economics and so on. With the development of science and the research of the nonlinear ordinary differential equation, we have made a thorough research on the nonlinear differential equation with perturbation term, and have also made a breakthrough research result. For example, in reference [1], the existence of solutions for nonlinear differential equations with perturbed terms is proved by using Schauder fixed point theorem. In references [2] and [3], the existence of solutions to the equations studied is obtained by using the fixed point theorems of cone stretching and compression. In this paper, the nonlinear differential equations with perturbed terms are studied, inspired by references [1]-[5]. According to the content, this paper is divided into the following three chapters: the first chapter: mainly introduces some basic definitions and some Lemma related to the proof of this paper. Chapter 2: consider the existence of solutions for the boundary value problems of nonlinear fractional differential equations with perturbed terms, where D 0 伪 u (t) is the fractional derivative of Riemann-Liouville, 伪 鈮，

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