脉冲微分系统解的存在性问题
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摘要
本论文主要讨论了脉冲微分方程边值问题和周期解的存在性,一类脉冲捕食者-食饵生态微分方程正周期解的存在性和全局渐近稳定性.全文共分为五章.
第一章简述了脉冲微分方程周期解和边值问题的历史与研究现状,及本文的主要工作.
第二章考虑了脉冲微分方程周期边值问题解的存在.通过使用脉冲不等式和分析方法,得到了一阶脉冲微分方程周期边值问题新的比较结果,借助所给的新的上下解,结合单调迭代技巧,建立了一阶脉冲微分方程周期边值问题极值解的存在结果.同时,研究了二阶脉冲微分方程周期边值问题,修正了已有文献中比较定理的不妥之处,并提出了新的上下解.
第三章研究了脉冲微分方程两点边值问题解的存在性,利用Leray-Schauder度理论,得到了一类带导数项的脉冲微分方程两点边值问题解的存在性与多解性结果,指出了已有文章中错误的地方;利用五泛函不动点定理,获得了一类测度链上的具有p-Laplacian算子的脉冲微分方程两点边值问题至少存在三个解的充分条件.
第四章讨论了一类含有参数的脉冲微分方程正周期解的存在性.通过将脉冲微分方程转换成非脉冲微分方程,利用Krasnoselskill不动点定理和Leggett-Williams不动点理论,得到了这类脉冲微分系统正周期解的存在个数与参数有着密切的关系,特别是给出了系统至少存在三个正周期解的充分条件.
第五章研究了一类脉冲捕食者-食饵生态微分方程正周期解的存在性和全局渐近稳定性.运用Mawhin度理论与解的先验估计,获得了周期解的存在性结果.推广和改进了已有的相应的非脉冲微分方程的结论;利用Lyapunov泛函,得到了该系统正周期解是全局渐近稳定的,从而得到了周期解的存在唯一性,通过数值模拟来验证了所得到的结果.
This thesis mainly studies the existence of periodic solutions and boundary value problems for impulsive differential equations,the existence and global attractivity of positive periodic solutions for impulsive predator-prey model with disperson and time delays.It is consists of five chapters.
As the introductions,in Chapter 1,the background and history of periodicity and boundary value problems for impulsive differential equations are briefly addressed,and the main work of this paper are given.
Chapter 2 concerns the existence of solutions for impulsive differential equations with periodic boundary value.By using an impulsive differential inequality and analysis technique,we obtain some new comparison results on a first order impulsive periodic boundary value problems,and using the method of upper and lower solutions coupled with monotone iterative technique,we obtain the existence of extremal solution about periodic boundary value problems.Meanwhile, we recorrect the wrong comparison result of second order impulsive periodic boundary value problems in the previous paper,and present new definitions of upper and lower solutions.
In chapter 3,we discuss the existence of solutions for impulsive differential equations with two-point boundary condition.In the view of the existence of upper and lower solutions,by using Leray-Schauder degree theory,we obtain the existence result and multiplicity of solutions about two-point boundary value problems. By mean of five functionals fixed point theorem,we establish the existence of at least three positive solutions of boundary value problems for p-Laplacian impulsive functional dynamic equations on time scales.
In Chapter 4,we consider the existence of positive periodic solutions for impulsive differential equations with a parameter.Converting it to non-impulsive differential equation and using Krasnoselskill's and Leggett-Williams fixed theorems, we obtain that the existence and amount of positive solutions is concerned with the parameter.Especially,we establish sufficient conditions for the existence of triple positive periodic solutions.
In Chapter 5,we focus on an impulsive predator-prey model with dispersion and time delays.The existence of positive periodic solution is obtained by the help of Mawhin continuation coincidence degree theory and the global attractivity of positive periodic solution is presented by Lyapunove method.
第一章简述了脉冲微分方程周期解和边值问题的历史与研究现状,及本文的主要工作.
第二章考虑了脉冲微分方程周期边值问题解的存在.通过使用脉冲不等式和分析方法,得到了一阶脉冲微分方程周期边值问题新的比较结果,借助所给的新的上下解,结合单调迭代技巧,建立了一阶脉冲微分方程周期边值问题极值解的存在结果.同时,研究了二阶脉冲微分方程周期边值问题,修正了已有文献中比较定理的不妥之处,并提出了新的上下解.
第三章研究了脉冲微分方程两点边值问题解的存在性,利用Leray-Schauder度理论,得到了一类带导数项的脉冲微分方程两点边值问题解的存在性与多解性结果,指出了已有文章中错误的地方;利用五泛函不动点定理,获得了一类测度链上的具有p-Laplacian算子的脉冲微分方程两点边值问题至少存在三个解的充分条件.
第四章讨论了一类含有参数的脉冲微分方程正周期解的存在性.通过将脉冲微分方程转换成非脉冲微分方程,利用Krasnoselskill不动点定理和Leggett-Williams不动点理论,得到了这类脉冲微分系统正周期解的存在个数与参数有着密切的关系,特别是给出了系统至少存在三个正周期解的充分条件.
第五章研究了一类脉冲捕食者-食饵生态微分方程正周期解的存在性和全局渐近稳定性.运用Mawhin度理论与解的先验估计,获得了周期解的存在性结果.推广和改进了已有的相应的非脉冲微分方程的结论;利用Lyapunov泛函,得到了该系统正周期解是全局渐近稳定的,从而得到了周期解的存在唯一性,通过数值模拟来验证了所得到的结果.
This thesis mainly studies the existence of periodic solutions and boundary value problems for impulsive differential equations,the existence and global attractivity of positive periodic solutions for impulsive predator-prey model with disperson and time delays.It is consists of five chapters.
As the introductions,in Chapter 1,the background and history of periodicity and boundary value problems for impulsive differential equations are briefly addressed,and the main work of this paper are given.
Chapter 2 concerns the existence of solutions for impulsive differential equations with periodic boundary value.By using an impulsive differential inequality and analysis technique,we obtain some new comparison results on a first order impulsive periodic boundary value problems,and using the method of upper and lower solutions coupled with monotone iterative technique,we obtain the existence of extremal solution about periodic boundary value problems.Meanwhile, we recorrect the wrong comparison result of second order impulsive periodic boundary value problems in the previous paper,and present new definitions of upper and lower solutions.
In chapter 3,we discuss the existence of solutions for impulsive differential equations with two-point boundary condition.In the view of the existence of upper and lower solutions,by using Leray-Schauder degree theory,we obtain the existence result and multiplicity of solutions about two-point boundary value problems. By mean of five functionals fixed point theorem,we establish the existence of at least three positive solutions of boundary value problems for p-Laplacian impulsive functional dynamic equations on time scales.
In Chapter 4,we consider the existence of positive periodic solutions for impulsive differential equations with a parameter.Converting it to non-impulsive differential equation and using Krasnoselskill's and Leggett-Williams fixed theorems, we obtain that the existence and amount of positive solutions is concerned with the parameter.Especially,we establish sufficient conditions for the existence of triple positive periodic solutions.
In Chapter 5,we focus on an impulsive predator-prey model with dispersion and time delays.The existence of positive periodic solution is obtained by the help of Mawhin continuation coincidence degree theory and the global attractivity of positive periodic solution is presented by Lyapunove method.
引文
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