本文選題：非線性分?jǐn)?shù)階微分方程 + m點(diǎn)邊值問(wèn)題　； 參考：《曲阜師范大學(xué)》2017年碩士論文

【摘要】：本文研究的是帶有擾動(dòng)項(xiàng)的非線性微分方程的解.通過(guò)專家學(xué)者對(duì)非線性整數(shù)階常微分方程的不斷研究,我們對(duì)其已經(jīng)非常了解,并且在物理學(xué),生物學(xué),經(jīng)濟(jì)學(xué)等許多領(lǐng)域得到了廣泛的應(yīng)用.隨著科學(xué)的發(fā)展和不斷地深入研究,我們通過(guò)對(duì)非線性常微分方程加以推廣與改造,在帶有擾動(dòng)項(xiàng)的非線性微分方程方面進(jìn)行了深入的研究,并且也取得了突破性的研究成果,如文獻(xiàn)[1]運(yùn)用Schauder不動(dòng)點(diǎn)定理證明了帶有擾動(dòng)項(xiàng)的非線性微分方程解的存在性.文獻(xiàn)[2],[3]都是運(yùn)用錐拉伸與壓縮不動(dòng)點(diǎn)定理得到了所研究方程解的存在性.本文受文獻(xiàn)[1]-[5]的啟發(fā),對(duì)帶有擾動(dòng)項(xiàng)的非線性微分方程進(jìn)行了研究.根據(jù)內(nèi)容,本文分為以下三章:第一章:主要是介紹本文將要用到的一些基本定義和一些與本文證明有關(guān)的引理.第二章:考慮帶有擾動(dòng)項(xiàng)的非線性分?jǐn)?shù)階微分方程邊值問(wèn)題解的存在性,其中D0α+u(t)是Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù),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,∞)連續(xù),e(t)∈L1([0,1],R)可能變號(hào).運(yùn)用Schauder不動(dòng)點(diǎn)定理,得到解的存在性,最后給出應(yīng)用.第三章:考慮帶有擾動(dòng)項(xiàng)的非線性分?jǐn)?shù)階微分方程的特征值問(wèn)題其中λ是正實(shí)參數(shù),D1qx(t)是Riemann-Liouville分?jǐn)?shù)階導(dǎo)數(shù),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處奇異,e(t) ∈ L~1([0,1],R)可能變號(hào).運(yùn)用錐拉伸與壓縮不動(dòng)點(diǎn)定理,得到解的存在性,作為應(yīng)用給出相應(yīng)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 偽 u (t) is the fractional derivative of Riemann-Liouville, 偽 鈮，

本文編號(hào)：2050668