本文選題：邊值問題 + 壓縮映射原理　； 參考：《山東科技大學(xué)》2017年碩士論文

【摘要】：非線性微分方程邊值問題是微分方程理論中一個(gè)頗為重要的研究分支,在建筑工程、航天工程、生物工程、物理學(xué)等領(lǐng)域都有著廣泛的應(yīng)用。因此,通過研究微分方程的邊值問題將會(huì)對(duì)于解決這些領(lǐng)域中常見的難題提供幫助。而且在實(shí)際的生活和科研中,許多問題所構(gòu)成的數(shù)學(xué)模型通常都是微分方程�？梢�,研究非線性微分方程的邊值問題顯得格外的重要。第一章主要介紹微分方程邊值問題研究的歷史及發(fā)展現(xiàn)狀,敘述了本文的研究目的和意義,并說明對(duì)微分方程邊值問題解的研究不僅具有重要的理論意義,而且具有重要的實(shí)際意義。第二章討論了奇異邊值問題解的唯一性,其中h(t)在t=1處奇異。本章的新穎之處在于Lipschitz系數(shù)與相應(yīng)線性算子的第一特征值有關(guān),我們通過u0-范數(shù)以及壓縮映射原理來給出上述邊值問題的唯一解。第三章討論了分?jǐn)?shù)階微分方程邊值問題至少存在一個(gè)正解,其中2p≤3是一個(gè)實(shí)數(shù).本章通過運(yùn)用相應(yīng)線性算子的第一特征值及其相關(guān)性質(zhì),以及不動(dòng)點(diǎn)指數(shù)等理論得到其至少存在一個(gè)正解的證明.第四章應(yīng)用單調(diào)迭代法以及上下解方法研究四階兩點(diǎn)邊值問題解的存在性,其中f:[0,1]×R→R為連續(xù)函數(shù)。本章的新穎之處在于,對(duì)此邊值問題利用線性算子的性質(zhì)建立一個(gè)比較結(jié)果,從而研究其極值解的存在性。
[Abstract]:Boundary value problem of nonlinear differential equation is an important branch of differential equation theory. It is widely used in architectural engineering, aerospace engineering, biological engineering, physics and so on. Therefore, studying the boundary value problems of differential equations will be helpful to solve the common problems in these fields. Moreover, in real life and scientific research, the mathematical models of many problems are usually differential equations. Therefore, it is very important to study the boundary value problem of nonlinear differential equation. The first chapter mainly introduces the history and development of the boundary value problem of differential equation, describes the purpose and significance of this paper, and shows that the study of the solution of the boundary value problem of differential equation is not only of great theoretical significance. And has important practical significance. In chapter 2, we discuss the uniqueness of the solution of the singular boundary value problem, where ht) is singular at t ~ (1). The novelty of this chapter is that the Lipschitz coefficient is related to the first eigenvalue of the corresponding linear operator. We obtain the unique solution of the above boundary value problem by using u0-norm and contraction mapping principle. In chapter 3, we discuss the existence of at least one positive solution for the boundary value problem of fractional differential equations, where 2p 鈮，

本文編號(hào)：1843197