【作者】 沈钦锐；

【导师】 周宗福；

【作者基本信息】 安徽大学 ， 基础数学， 2012， 硕士

【摘要】 泛函微分方程主要描述的是带有时滞现象的数学模型.带有周期时滞的泛函微分方程在生物学、经济学、生态学和人口动力系统等实际问题中有着广泛的应用,因此,对带有周期时滞的泛函微分方程周期解存在性的研究就更具有现实意义.本文主要讨论几类三阶时滞泛函微分方程周期解的存在唯一性问题,全文共分为五章.第一章介绍泛函微分方程周期解的背景知识和有关的研究动态,并介绍本文的主要结果.第二章利用重合度理论,研究一类具有偏差变元的三阶时滞泛函微分方程的T-周期解问题,获得了上述方程周期解存在唯一性的新结果.第三章利用重合度理论,在第二章的基础上,研究一类三阶多时滞泛函微分方程的T-周期解问题,获得了保证其周期解存在唯一性的充分性条件.第四章采用不同于第三章的研究方法,利用K-集压缩算子抽象连续性定理和一些新的分析技巧,研究一类三阶多时滞泛函微分方程的T-周期解问题,获得了上述方程T-周期解存在和唯一性的若干新结果.第五章采用重合度理论中的延拓定理,运用一些新的分析方法,讨论了如下一类三阶带分布时滞的p-Laplacian方程(φp((x(t)-cx(t-σ))"))’+f1(x(t))x’(t)+f2(x’(t))x"(t)+g(t,x(t),x(t-τ-(t)),(?),x(t+s)dm(s))=e(t)的T-周期解问题,得到了上述方程存在T-周期解的若干新结果.更多还原

【Abstract】 Functional differential equation is a mathematical model describing the phe-nomenon with time delays.The functional differential equations with periodic delays represent a natural framework for mathematical modeling of many real world phe-nomena such as biology, economy, ecology, the population dynamic system and so on.Therefore, the researches on existence and uniqueness of periodic solutions for functional differential equations with periodic delays have practical significance.This paper mainly discusses the problems on existence and uniqueness of peri-odic solutions for some classes of three order functional differential equations with delays. The full text is divided into five chapters.In chapter Ⅰ, the background knowledge of the subjects relevant to this dis-sertation and the study of dynamic are introduced.Then, the main results of the thesis are introduced.In Chapter Ⅱ,using the theory of coincide degree,a type of third order func-tional differential equation with delays is considered.Some new results on the existence and uniqueness of periodic solutions are obtained.In Chapter Ⅲ,using the theory of coincide degree,on the based of the second chapter,a type of third order functional differential equation with more delays is considered.Some sufficient conditions that guarantee the existence and uniqueness of periodic solution of the equation are obtained.In Chapter IV,comparing the fourth chapter,a different research methods is used to study this equation.Specifically,using the continuation theory for K-set con-tractive operator,a type of third order functional differential equation with more delays is considored.The sufficient conditions for the existance of unique T-periodic solution are obtained.In Chapter V,by using continuation theorem in coincidence degree theory, a type of third order p-Laplacian Equation with a deviating argument ((φp((x(t)-cx(t-σ))"))’+f1(x(t))x’(t)+f2(x’(t))x"(t)+g(t, x(t), x(t-τ(t)),(?)x(t+s) dm(s))=e(t) is considered.The sufficient condition for the existance of T-periodic solution is obtained.更多还原