【作者】 蒋振；

【导师】 申建华；

【作者基本信息】 湖南师范大学 ， 应用数学， 2010， 硕士

【摘要】 本论文主要讨论了一类无限时滞中立型Volterra型积分微分方程周期解的存在性与唯一性、一类中立型Duffing型微分方程的周期解、一类泛函微分方程正周期解的存在性与多解性.全文共分为四章.第一章简述了泛函微分方程的周期解存在性、唯一性的历史与研究现状,以及本文的主要工作.第二章讨论了一类具有无限时滞中立型积分微分系统的周期解.利用线性系统的指数型二分性理论,Schauder不动点定理,得到了这一类方程周期解的存在性与唯一性理论.第三章讨论了一类中立型Duffing型微分方程aχ"(t)+cχ’(t-τ)+bχ(t)+g(χ(t-τ)))=p(t)的周期解,利用迭合度方法得到了这一类方程周期解存在的充分条件.第四章讨论了一类脉冲泛函微分方程χ’(t)=A(t,χ(t))χ(t)-λf(t-τ(t)),t≠τk,k∈Nχ(τk+)=χ(τk)+Ek(χ(τk)),t=τk的正周期解的存在性与多解性.利用Krasnoselskii不动点定理,获得了判断这一类方程正周期解的存在性与多解性的一些结论.更多还原

【Abstract】 This thesis is composed of four chapters, which mainly studied the ex-istence and uniqueness of periodic solutions for one kind of neutral Volterra integro-differential equations with infinite delay、periodic solution for one kind of neutral Duffing equations and existence and multiplicity of positive periodic solutions for one kind of functional differential equations with impulses.As the introductions, in Chapter 1, the background and history of the existence and uniqueness or multiplicity of Periodicity solutions problems, and the main work of this paper are given.In Chapter 2, we studied the existence, uniqueness and stability of neutral Integro-differential equations By using the theory of exponential dichotomies of linear system and Schauder fixed point theorem, we obtain the existence and uniqueness theorem of peri-odic solutions for this kind of functional differential equations.In Chapter 3, we study a class of neutral Duffing equation, which have periodicity solutions ax"(t)+cx’(t-τ)+bx(t)+g(x(t-τ)))= p(t). By using the theory of coincidence degree, we obtain a sufficient theorem is obtained for the existence of a periodic solution of this kind of equations.In the last Chapter, we study the existence of multiple positive periodic solutions for functional differential equations with impulsesχ(t)= A(t,x(t))x(t)-λf(t-τ(t)),t≠Tk,k∈N x(τk+)=χ(Tk)+Ek(x(Tk)), t= Tk By using Krasnoselskii fixed point theorem, we obtain the existence of multiple positive periodic solutions for this kind of functional differential equations with impulses. Our results improve some known results or are new.更多还原