【摘要】：二十世紀(jì)五十年代,非線性泛函分析已初步形成完整的理論體系,作為其重要組成部分,非線性微分積分問(wèn)題受到了國(guó)內(nèi)外數(shù)學(xué)界乃至整個(gè)自然科學(xué)界的高度重視,因?yàn)樗?能良好解釋眾多自然現(xiàn)象的功效性".從發(fā)展上看,非線性微分-積分問(wèn)題來(lái)源于應(yīng)用數(shù)學(xué)和物理學(xué)的多個(gè)方面,在應(yīng)用數(shù)學(xué)物理學(xué)和工程學(xué)等應(yīng)用學(xué)科上有著極為重要的應(yīng)用價(jià)值,研究此類(lèi)問(wèn)題的意義就在于此.隨著物理學(xué)、航空航天技術(shù)、生物技術(shù)等分支領(lǐng)域中實(shí)際問(wèn)題的不斷出現(xiàn),非線性泛函分析已成為解決這些非線性問(wèn)題的重要理論工具,其中非線性微分積分方程解的存在性與多重性已成為重要的研究課題之一,它能清晰地刻畫(huà)在物理、化學(xué)、經(jīng)濟(jì)等應(yīng)用學(xué)科中出現(xiàn)的各種非線性問(wèn)題.近些年來(lái),由于邊值問(wèn)題在化學(xué)工程,熱彈力學(xué),人口動(dòng)態(tài),熱傳導(dǎo),等離子物理中的各種應(yīng)用,帶有積分邊界條件的邊值問(wèn)題更是得到廣泛的關(guān)注.微分方程邊值問(wèn)題的存在性研究一般是將微分方程結(jié)合邊界條件,轉(zhuǎn)化為積分方程,尋求積分方程的不動(dòng)點(diǎn).本文研究了兩類(lèi)Banach空間上的帶積分邊值條件的非線性微分方程,豐富了微分方程邊值問(wèn)題正解的研究理論.本文的結(jié)構(gòu)安排如下:第一章,主要介紹了積分邊值微分方程的研究背景和研究現(xiàn)狀,并且給出了一些關(guān)于積分邊值問(wèn)題的基本定義、基本性質(zhì)以及一些重要的不動(dòng)點(diǎn)定理.第二章,研究了如下具有單調(diào)同態(tài)和積分邊值條件的三階非線性脈沖微分-積分方程在實(shí)Banach空間中正解的存在性.利用不動(dòng)點(diǎn)指數(shù)定理和Guo-Krasnosel'skill不動(dòng)點(diǎn)定理給出了一些積分邊值問(wèn)題正解的存在性的一些充分條件,并用實(shí)例驗(yàn)證了相應(yīng)的結(jié)果的合理性.第三章,通過(guò)構(gòu)造格林函數(shù),研究了如下p-Laplace微積分方程的積分邊值問(wèn)題在實(shí)Banach空間中正解的存在性：其中，，φ_p是p-Laplace算子，即φ_p(u)=|u|~(p-2)u(p1)，φ_p~(-1)(u)= φ_q(u)(1/p+1/q = 1).利用不動(dòng)點(diǎn)指數(shù)定理和Guo-Krasnosel'skill不動(dòng)點(diǎn)定理給出了一些積分邊值問(wèn)題正解的存在性的一些充分條件，并用實(shí)例驗(yàn)證了相應(yīng)的結(jié)果的合理性。
[Abstract]:In the 1950s, nonlinear functional analysis had formed a complete theoretical system. As an important part of it, nonlinear differential and integral problems were highly valued by the mathematics field at home and abroad and even the whole natural science field. Because of its "good explanation of the efficacy of many natural phenomena." From the point of view of development, nonlinear differential-integral problems come from many aspects of applied mathematics and physics, and have extremely important application value in applied subjects such as applied mathematics, physics and engineering, so the significance of studying such problems lies in this. With the emergence of practical problems in the fields of physics, aerospace technology and biotechnology, nonlinear functional analysis has become an important theoretical tool to solve these nonlinear problems. Among them, the existence and multiplicity of solutions of nonlinear differential integral equations have become one of the important research topics, which can clearly describe all kinds of nonlinear problems in physics, chemistry, economics and other applied disciplines. In recent years, boundary value problems with integral boundary conditions have been paid more and more attention due to their applications in chemical engineering, thermoelastic mechanics, population dynamics, heat conduction and plasma physics. In general, the existence of boundary value problems for differential equations is transformed into integral equations by combining boundary conditions, and the fixed points of integral equations are found. In this paper, we study two kinds of nonlinear differential equations with integral boundary value conditions on Banach spaces, which enrich the research theory of positive solutions for boundary value problems of differential equations. The structure of this paper is arranged as follows: in the first chapter, the research background and present situation of integro-boundary value differential equations are introduced, and some basic definitions, basic properties and some important fixed point theorems are given. In chapter 2, we study the existence of positive solutions for the third order nonlinear impulsive differential-integral equations with monotone homomorphism and integral boundary value conditions in real Banach spaces. By using the fixed point exponent theorem and Guo-Krasnosel'skill fixed point theorem, some sufficient conditions for the existence of positive solutions of some integral boundary value problems are given, and the rationality of the corresponding results is verified by an example. In chapter 3, by constructing Green's function, we study the existence of positive solutions of integral boundary value problems for p-Laplace calculus equations in real Banach spaces: where 蠁 _ p _ p is a p-Laplace operator, that is, 蠁 _ P _ p (u) = u ~ (p-2) u (p _ (1), 蠁 p ~ (-1) (u) = 蠁 Q (u) (1 / p ~ (1 / Q = 1). By using the fixed point exponent theorem and Guo-Krasnosel'skill fixed point theorem, some sufficient conditions for the existence of positive solutions of some integral boundary value problems are given, and the rationality of the corresponding results is verified by an example.
【學(xué)位授予單位】：昆明理工大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：O175.8

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