【作者】 张峰；

【导师】 赵增勤；

【作者基本信息】 曲阜师范大学 ， 应用数学， 2011， 硕士

【摘要】 随着科学技术的不断发展,各种各样的非线性问题已日益引起人们的广泛关注,非线性分析及应用已成为现代数学的重要研究方向之一.在物理学,机械学,生物学,车辆交通,经济学,地质学,工程技术及应用数学等领域中出现的很多问题都可以归结数学模型由非线性积分方程来解决.非线性积分方程理论是非线性分析及应用的一个重要分支,也是对分析学的众多领域和科学的其它分支有着重要应用的快速发展的领域,对应用数学有着举足轻重的作用.因此,研究非线性泛函积分方程解的存在性,进而研究解的性质不仅科学意义,而且也具有现实的意义.本文主要利用非紧性测度、弱非紧性测度、Schauder不动点定理、Krasnosel’skii型不动点定理和弱序列连续等理论、概念、方法研究了几类非线性泛函积分方程解的存在性及性质.所得结果或是新的,或是采用新方法在更弱的条件下推广和改进了以前的结果.根据内容本文分为以下四章：第一章绪论,主要介绍了本文的研究课题.第二章在本章中,主要讨论了如下非线性泛函积分方程其中,函数g：R+×R→R连续,f：R+×R×R→R连续,u：R+×R+×R→R连续,及α,β,r,η：R+→R十连续.于所有在无穷区间上连续,有界实函数组成的Banach空间中运用非紧性测度理论和Schander不动点定理来研究方程(2.1.1)解的存在性及局部吸引性.所得结论推广了相关文献中的一些结果.第三章在本章中,研究了如下分数阶非线性摄动积分方程其中,α∈(0,1)为一固定实数,r(·)表示Gamma函数,函数g：R+×R→R连续,u:R+×R+×R→R连续为给定的函数及算子A:BC(R+)→BC(R+)(BC(R+)为所有在无穷区间上连续,有界的实函数构成的Banach空间).于空间BC(R+)内运用非紧性测度理论及Schauder不动点定理探究了方程(3.1.1)解的存在性.进一步,通过适当的假设证明解是局部一致吸引及局部一致渐进吸引的.所得结论推广了相关文献中的一些结果.第四章在本章中,研究了如下非线性泛函积分方程其中,g,f：[0,1]×R→R满足Caratheodory条件,g,f和u是给定的可测函数,φ1,φ2：[0,1]→[0,1]是增的、绝对连续函数,x∈L1[0,1]是未知函数(L1[0,1]为在区间[0,1]上所有lebesgue可积函数所组成的Banach空间).本章利用弱非紧性测度理论,弱序列连续和Krasnosel’skii型不动点定理在空间L1[0,1]中来探讨方程(4.1.1)单调可积解的存在性.所得结论推广了相关文献中的一些结果.更多还原

【Abstract】 Along with the science and technology development, various nonlinear problems have aroused people’s widespread attention day by day. So the nonlinear analysis and its applications has become one of important research directions in modern mathemat-ics. Many problems arising in physical sciences, mechanics, biology, vehicular traffic theory, economics, geology, engineering and applied mathematics can be attributed to mathematical models solved by nonlinear integral equations. The theory of nonlinear integral equations is an important branch of nonlinear analysis and its applications, at the same time, a fast growing field with important applications to a number of areas of analysis as well as other branches of science. Therefore, it becomes of great scientific and real significance to study the existence and properties of solutions for nonlinear integral equations.The present paper employs the theories, concepts, methods such as measure of noncompactness theory, measure of weak noncompactness theory, Schauder fixed point theorem, Krasnosel’skii fixed point theorem, weakly sequentially continuous and so on, to investigate the existence and properties of solutions for some classes of nonlinear integral equations. The obtained results are either new or intrinsically generalize and improve the previous relevant ones.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the main contents of this paper.Chapter 2 We consider the following nonlinear functional integral equation where, the functions g:R+×R→R, f:R+×R×R→R, u:R+×R+×R→R, andα,β,γ,η:R+→R+ are continuous. We work in the Banach space BC(R+) consisting of real functions which are continuous and bounded on unbounded interval R+. By using measures of noncompactness and Schauder fixed point theorem to study the existence and local attractivity of solutions for equation (2.1.1) in BC(R+). The result obtained in this paper generalizes some previous related ones. Chapter 3 we investigate the following nonlinear perturbed integral equation of fractional orderα∈(0,1) is a fixed number andГ(·) denotes the Gamma function. Here, the given functions g:R+×R→R and u:R+×R+×R→R are continuous. A:BC(R+)→BC(R+)(BC(R+):the Banach space consisting of real functions which are continuous and bounded on an unbounded interval R+.) is an operator. By using measures of noncompactness and Schauder fixed point theorem, we prove the existence of solutions for equation (3.1.1) in the space BC(R+). Moreover, we show that those solutions are uniformly locally attractive and uniformly locally asymptotically attractive under appropriate assumptions. The results obtained in this paper generalize and improve some previous relevant ones.Chapter 4 We discuss the following nonlinear functional integral equation where, the functions g, f:[0,1]×R→R satisfy the Caratheodory conditions, g, f and u are given Lebesgue integrable functions,φ1,φ2:[0,1]→* [0,1] are increasing, absolutely continuous functions, x∈L1[0,1] is an unknown function(L1 [0,1] the Banach space of lebesgue integrable functions on the interval [0,1]). By using the measures of weak noncompactness, weakly sequentially continuous and a Krasnosel’skii type fixed point theorem to prove the existence of monotonic solutions for equation (4.1.1) in L1[0,1]. The results presented in this paper extend the some previous relevant ones.更多还原