几类非线性微分方程与差分方程的定性分析

发布时间：2018-04-12 19:44

本文选题：振动性 + 黎曼-斯蒂尔切斯　；参考：《曲阜师范大学》2017年博士论文

【摘要】：众所周知,对于大多数整数阶和分数阶的微分方程、差分方程来说,寻求其通解是非常困难的,有时甚至是不可能的.因而数学工作者们只能从方程本身去分析它的解可能具有的某些性质,比如:存在性、有界性、振动性、渐近性、稳定性等,这些问题的研究促进了方程定性理论的发展.本文主要研究了几类整数阶微分方程解的振动性,脉冲微分系统正周期解的存在性,分数阶微分方程和分数阶差分方程解的有界性等问题,推广并改进了文献中的相关结果.主要内容如下:第一章简要概述了整数阶微分方程解的振动性,具有脉冲效应的捕食系统正周期解的存在性,分数阶微分方程和分数阶差分方程解的有界性等问题的研究背景与发展概况,同时介绍了本文的主要工作.在第二章中,利用广义Riccati技巧、积分平均技巧以及微分不等式理论,我们讨论了含有Riemann-Stieltjes积分项和变指数增长条件的二阶强迫微分方程解的振动性.在2.1节,研究了一类含有Riemann-Stieltjes积分项和变指数增长条件的二阶强迫微分方程解的振动性,建立了方程振动的El-Sayed型准则和Kamenev型准则,改进了文献中的已有结论.在2.2节,研究了一类含有p-Laplacian以及Riemann-Stieltjes积分项和变指数增长条件的二阶强迫微分方程解的振动性,建立了方程振动的El-Sayed型准则和Kamenev型准则,所得结果推广和改进了相应文献中的己有结论,并通过一些实例,说明了相应准则可应用于以前所不能处理的若干情形.在第三章中,我们利用重合度理论中的延拓定理(Mawhin连续性定理)研究了一个具有脉冲效应以及Holling-IV型功能反应函数和干扰系数的捕食系统正周期解的存在性问题,获得了一些新的结果.该结果表明在适当的线性周期脉冲扰动下,脉冲微分系统保持了原来相应的非脉冲微分系统的周期性.在第四章中,我们通过建立两类含弱奇异核的Gronwall-Bellman型积分不等式,推广和改进了相应文献中的己有结果,并用这两类不等式研究了两类分数阶微分方程解的有界性问题.在第五章中,我们对离散分数阶方程解的有界性问题展开研究.在5.2节,我们应用Riemann-Liouville型分数阶和分建立了一类非线性Gronwall-Bellman型分数阶和分不等式.在5.3节,我们应用Riemann-Liouville型分数阶和分建立一类非线性Volterra-Fredholm型分数阶和分不等式.在5.4节,利用上述不等式研究了一类分数阶差分方程解的有界性和唯一性问题以及一类Volterra-Fredholm型分数阶和分方程解的有界性问题.在第六章中,我们对今后的研究工作进行了展望.
[Abstract]:As we all know, for most differential equations of integer order and fractional order, it is very difficult, sometimes even impossible, to find the general solution of the difference equation.Therefore, the mathematical workers can only analyze some properties of its solution from the equation itself, such as existence, boundedness, oscillation, asymptotic property, stability and so on. The study of these problems has promoted the development of qualitative theory of equation.In this paper, we study the oscillation of solutions of several kinds of integer-order differential equations, the existence of positive periodic solutions of impulsive differential systems, the boundedness of solutions of fractional differential equations and fractional difference equations, and generalize and improve the related results in the literature.The main contents are as follows: in the first chapter, the oscillations of integer order differential equations and the existence of positive periodic solutions for predator-prey systems with impulsive effects are briefly summarized.The background and development of the research on the boundedness of the solutions of fractional differential equations and fractional difference equations are reviewed. The main work of this paper is also introduced.In chapter 2, by using generalized Riccati technique, integral averaging technique and differential inequality theory, we discuss the oscillation of solutions of second order forced differential equations with Riemann-Stieltjes integral term and variable exponential growth condition.In section 2.1, the oscillations of solutions of a class of second order forced differential equations with Riemann-Stieltjes integral term and variable exponential growth condition are studied. The El-Sayed type criterion and Kamenev type criterion for the oscillation of the equation are established, and the existing results in the literature are improved.In section 2.2, the oscillations of solutions of a class of second order forced differential equations with p-Laplacian and Riemann-Stieltjes integral terms and variable exponential growth conditions are studied. The El-Sayed type criteria and Kamenev type criteria for the oscillation of the equations are established.The results generalize and improve the existing conclusions in the relevant literatures and show that the corresponding criteria can be applied to some cases which can not be dealt with before by some examples.In chapter 3, we study the existence of positive periodic solutions of a predator-prey system with impulsive effect, Holling-IV type functional response function and interference coefficient by using continuation theorem Mawhin continuity theorem in coincidence degree theory.Some new results have been obtained.The results show that the impulsive differential system keeps the periodicity of the corresponding nonimpulsive differential system under the proper linear periodic impulsive perturbation.In chapter 4, by establishing two classes of Gronwall-Bellman type integral inequalities with weakly singular kernels, we generalize and improve the existing results in the corresponding literature, and study the boundedness of solutions of two kinds of fractional differential equations with these inequalities.In chapter 5, we study the boundedness of solutions of discrete fractional equations.In Section 5.2, we establish a class of nonlinear Gronwall-Bellman fractional order inequality by using the fractional sum of Riemann-Liouville type.In Section 5.3, we establish a class of nonlinear fractional order inequality of Volterra-Fredholm type by using the fractional order sum of Riemann-Liouville type.In Section 5.4, the boundedness and uniqueness of solutions for a class of fractional difference equations and the boundedness of solutions for a class of fractional and partial equations of Volterra-Fredholm type are studied by using the above inequalities.In the sixth chapter, we look forward to the future research work.
【学位授予单位】：曲阜师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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