几类时滞微分差分方程的周期解和稳定性
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摘要
本文利用不动点理论、重合度理论、k-集压缩算子的抽象连续定理和Lya-punov泛函方法,对几类非线性时滞微分(差分)方程周期解的存在性以及神经网络模型的全局指数稳定性进行了研究。全文由六章构成。
第一章是概述,简要地介绍本文相关研究问题的背景、本文的主要工作及有关预备知识。
第二章应用锥上的Deimling不动点指标定理,结合分析技巧,研究了一类一阶时滞微分方程的周期解的存在性,得到了其周期解存在的充分条件。
第三章应用Mawhin连续性定理、分析技巧及不等式技巧,研究了两类具复杂偏差变元二阶微分方程的周期解,得到了具偏差变元的中立型微分方程的周期解存在的新结论及具多个偏差变元的Duffing型微分方程周期解的存在唯一性的充分条件。
第四章应用Manasevich-Mawhin连续性定理及分析技巧,研究了具偏差变元的Rayleigh型p-Laplacian方程及具多个p-Laplacian算子Rayleigh型微分方程周期解的存在性,获得了周期解存在的新的充分性条件；研究了具变时滞非自治Rayleigh方程,应用周期解的新的先验估计得到了方程周期解存在性的新结果。
第五章研究了基于比率的n-种群离散型捕食者-食饵模型的正周期解,通过应用不等式技巧获得了一个周期解的新的先验估计,基于更精确的先验估计和Mawhin连续性定理建立了一个更易验证的关于正周期解存在的充分性条件；应用k-集压缩算子的抽象连续定理和一些分析技巧研究了多时滞中立型对数人口模型的正周期解,得到了正周期解存在的新结果。
第六章首先研究了具有Lipschitz连续激活函数的连续型双向联想记忆神经网络,在无需假设激活函数和信号传播函数有界的条件下,建立了该网络模型存在唯一全局指数稳定的平衡点的新判据；基于Lyapunov泛函和线性矩阵不等式研究了具变时滞的离散时滞BAM神经网络,得到了一个与时滞相关的指数稳定性判据。由于去掉了对时滞函数不合理的约束条件,我们的结果能应用于具有更一般时滞函数的BAM神经网络,且易于验证。
Based on fixed point theorem, coincidence degree theorem, the abstract con-tinuation theorem of k-set contractive operator and Lyapunov functional, periodic solution of several kinds of nonlinear delayed differential and difference equations and global exponential stability of bidirectional associated memory neural networks are studied. This dissertation is divided into six chapters.
In Chapter one, a general introduction to the dissertation and some relevant background knowledge are given.
In Chapter two, by Deimling fixed point index theorem and some analysis skill, periodic solutions of a kind of first order delayed differential equation is investigated and some sufficient conditions on the existence of periodic solution are obtained. Our results improve and expand the existing results.
In Chapter three, based on Mawhin's continuation theorem、some analysis skill and inequality techniques, the existence of periodic solutions to two types of second order functional differential equation with complex deviating argument is studied. Some new results on existence of periodic solutions of second order neutral functional differential equation with deviating argument are obtained and some sufficient conditions on the existence and uniqueness of periodic solutions for a kind of second order Duffing-type differential equation with multiple deviating arguments are established.Our results extend the existing results.
In Chapter four, by Manasevich-Mawhin continuation theorem and some anal-ysis skill, the existence of periodic solutions for a kind of Rayleigh type p-Laplacian equation with deviating argument and a kind of Rayleigh type multi-p-Laplacian equation with deviating argument is considered. Some new sufficient conditions on the existence of periodic solutions are presented. Then, combined Mawhin's continuation theorem and inequality techniques with a new priori estimate of a periodic solution, some new results on the existence of periodic solutions of non-autonomous Rayleigh type equation with varying delay are obtained.
In Chapter five, the positive periodic solution for ratio-dependent n-species discrete time system is investigated. By using of inequality techniques, a sharp priori estimate of the periodic solutions is given. Based on the more precise priori estimate and the continuation theorem, an easily verifiable sufficient criterion of the existence of positive periodic solutions is established. Results obtained greatly improve the existing results. Then, based on an abstract continuous theorem of k-set contractive operator and some analysis skill, a neutral multi-delay logarithmic population model is studied, a new result is obtained for the existence of positive periodic solutions.
In Chapter six, the existence, uniqueness, and global exponential stability of the equilibrium point of bidirectional associative memory (BAM) neural networks with time delays is studied. By applying Young's inequality and Holder's inequal-ity techniques together with the properties of monotonic continuous functions, without assuming that the activation functions and signal propagation functions are bounded, some new sufficient conditions for ascertaining global exponential stability, have been derived for BAM neural networks, improving and extending previous work. Then, the global exponential stability of discrete-time BAM neural networks with variable delays is dealt. A delay-dependent exponential stability criterion has been derived by means of a Lyapunov functional and inequality tech-niques. Unrealistic constraints on the delay functions have been removed. As this criterion has no extra constraints on the variable delay functions, it can be applied to quite general BAM neural networks with a broad range of time delay functions and is easily checked in practice.
第一章是概述,简要地介绍本文相关研究问题的背景、本文的主要工作及有关预备知识。
第二章应用锥上的Deimling不动点指标定理,结合分析技巧,研究了一类一阶时滞微分方程的周期解的存在性,得到了其周期解存在的充分条件。
第三章应用Mawhin连续性定理、分析技巧及不等式技巧,研究了两类具复杂偏差变元二阶微分方程的周期解,得到了具偏差变元的中立型微分方程的周期解存在的新结论及具多个偏差变元的Duffing型微分方程周期解的存在唯一性的充分条件。
第四章应用Manasevich-Mawhin连续性定理及分析技巧,研究了具偏差变元的Rayleigh型p-Laplacian方程及具多个p-Laplacian算子Rayleigh型微分方程周期解的存在性,获得了周期解存在的新的充分性条件；研究了具变时滞非自治Rayleigh方程,应用周期解的新的先验估计得到了方程周期解存在性的新结果。
第五章研究了基于比率的n-种群离散型捕食者-食饵模型的正周期解,通过应用不等式技巧获得了一个周期解的新的先验估计,基于更精确的先验估计和Mawhin连续性定理建立了一个更易验证的关于正周期解存在的充分性条件；应用k-集压缩算子的抽象连续定理和一些分析技巧研究了多时滞中立型对数人口模型的正周期解,得到了正周期解存在的新结果。
第六章首先研究了具有Lipschitz连续激活函数的连续型双向联想记忆神经网络,在无需假设激活函数和信号传播函数有界的条件下,建立了该网络模型存在唯一全局指数稳定的平衡点的新判据；基于Lyapunov泛函和线性矩阵不等式研究了具变时滞的离散时滞BAM神经网络,得到了一个与时滞相关的指数稳定性判据。由于去掉了对时滞函数不合理的约束条件,我们的结果能应用于具有更一般时滞函数的BAM神经网络,且易于验证。
Based on fixed point theorem, coincidence degree theorem, the abstract con-tinuation theorem of k-set contractive operator and Lyapunov functional, periodic solution of several kinds of nonlinear delayed differential and difference equations and global exponential stability of bidirectional associated memory neural networks are studied. This dissertation is divided into six chapters.
In Chapter one, a general introduction to the dissertation and some relevant background knowledge are given.
In Chapter two, by Deimling fixed point index theorem and some analysis skill, periodic solutions of a kind of first order delayed differential equation is investigated and some sufficient conditions on the existence of periodic solution are obtained. Our results improve and expand the existing results.
In Chapter three, based on Mawhin's continuation theorem、some analysis skill and inequality techniques, the existence of periodic solutions to two types of second order functional differential equation with complex deviating argument is studied. Some new results on existence of periodic solutions of second order neutral functional differential equation with deviating argument are obtained and some sufficient conditions on the existence and uniqueness of periodic solutions for a kind of second order Duffing-type differential equation with multiple deviating arguments are established.Our results extend the existing results.
In Chapter four, by Manasevich-Mawhin continuation theorem and some anal-ysis skill, the existence of periodic solutions for a kind of Rayleigh type p-Laplacian equation with deviating argument and a kind of Rayleigh type multi-p-Laplacian equation with deviating argument is considered. Some new sufficient conditions on the existence of periodic solutions are presented. Then, combined Mawhin's continuation theorem and inequality techniques with a new priori estimate of a periodic solution, some new results on the existence of periodic solutions of non-autonomous Rayleigh type equation with varying delay are obtained.
In Chapter five, the positive periodic solution for ratio-dependent n-species discrete time system is investigated. By using of inequality techniques, a sharp priori estimate of the periodic solutions is given. Based on the more precise priori estimate and the continuation theorem, an easily verifiable sufficient criterion of the existence of positive periodic solutions is established. Results obtained greatly improve the existing results. Then, based on an abstract continuous theorem of k-set contractive operator and some analysis skill, a neutral multi-delay logarithmic population model is studied, a new result is obtained for the existence of positive periodic solutions.
In Chapter six, the existence, uniqueness, and global exponential stability of the equilibrium point of bidirectional associative memory (BAM) neural networks with time delays is studied. By applying Young's inequality and Holder's inequal-ity techniques together with the properties of monotonic continuous functions, without assuming that the activation functions and signal propagation functions are bounded, some new sufficient conditions for ascertaining global exponential stability, have been derived for BAM neural networks, improving and extending previous work. Then, the global exponential stability of discrete-time BAM neural networks with variable delays is dealt. A delay-dependent exponential stability criterion has been derived by means of a Lyapunov functional and inequality tech-niques. Unrealistic constraints on the delay functions have been removed. As this criterion has no extra constraints on the variable delay functions, it can be applied to quite general BAM neural networks with a broad range of time delay functions and is easily checked in practice.
引文
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