时标上几类动力方程和动力包含解的存在性与振动性
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摘要
由Hilger引入的时标理论,不仅在连续分析和离散分析间架起了一座桥梁,而且能更好地洞察两者之间的本质差异.它更精确地描述了有时在连续时间有时在离散时间出现的现象.为了进一步发展时标理论与应用,本文的工作主要致力于动力方程和动力包含解的性质,包括以下三个方面：动力方程解的存在性,动力包含解的存在性,动力方程解的振动性.
在第一章中,我们给出了本文研究的背景,介绍了本文的结构安排,并且给出了处理动力方程和动力包含解的存在性和振动性的主要工具和时标上有关的基本运算公式.
第二章讨论了三类时标上动力方程解的存在性.我们选择了时标上四阶四点边值问题,并进一步研究了一阶和二阶脉冲动力方程.分别运用Krasnoselskii-Zabreiko不动点定理和Schaefer不动点定理,我们得到了这三类方程解的存在性.值得指出的是,对于四阶动力方程,我们得出了Green函数的表达式,这个计算方法适合时标上其它高阶动力方程.对于一阶和二阶脉冲动力方程,所得结果弥补并拓展了已有结论.
在第三章,利用多值映射的不动点定理和上(下)半连续的选择定理,我们考虑了时标上两类带有边值条件的动力包含解的存在性,得到了一些新的结论.我们给出了例子来说明主要结论的有效性.第三章的结果拓展了在现有文献中关于时标上二阶动力包含的结果.
第四章研究了时标上动力方程解的振动性.本章分两个小节.第一小节给出了二阶半线性动力方程解的振动性,而第二小节主要考虑了二阶非线性动力方程.主要工具包括Riccati转换技巧和一些不等式.第三章的结果统一并扩展了文献中的已有结论.
Time scales theory, introduced by Hilger, not only establishes a bridge be-tween the continuous and the discrete analysis, but also understands deeply the essential between them. The theorem on time scales describes more accurately the phenomena which happen sometimes in continuous time and sometimes in discrete time. In order to develop time scales theorem and application further, our attention focus on the properties of solutions for dynamic equations and dy-namic inclusions,including the three aspects of the time scales theory:Existence of solutions for dynamic equations, existence of solutions for dynamic inclusions, oscillation for dynamic equations.
In Chapter 1, we recall the background of our topics of this thesis, show the outline and give the main tools for dealing the existence and oscillations of solutions for dynamic equations and dynamic inclusions and some basic calculus on time scales related with this thesis.
Chapter 2 is dealing with the existence of solutions for three classes of dy-namic equations on time scales. Thus, by means of fixed-point theorems due to Krasnoselskii-Zabreiko, and Schaefer, we present the existence of solutions for fourth-order four-point boundary problems, a class of first-order impulsive dy-namic equations and a class of second-order dynamic equations on time scales, respectively. It is worth pointing out for the Forth-order dynamic equation, the Green function is new and this method can be applied to study some other boundary value problems for high-order dynamic equations on time scales. For the first-order or second-order impulsive dynamic equations, the obtained result complement and extend some known results.
In Chapter 3, combining fixed-point theorems on multi-valued mappings with continuous selection theorems for upper (or lower) semi-continuous, we in-vestigate existence of solutions for two classes of dynamic inclusions with bound-ary conditions on time scales. Some new criteria are given. We also provide examples to illustrate our main results. The obtained results fill the gap that there are only a few results about second-order dynamic inclusions on time scales in the literatures.
Chapter 4 is devoted to the oscillation of solutions for dynamic equations. We discuss the problems in two subsections. In section 1, sufficient criteria of oscillation for a half-linear second-order dynamic equation is established. In sec-tion 2, we are concerned with the oscillation of a nonlinear second-order dynamic equation. The main tools include Riccati transformation and some inequalities. The results in Chapter 3 generalize and extend some known results in the liter-atures.
在第一章中,我们给出了本文研究的背景,介绍了本文的结构安排,并且给出了处理动力方程和动力包含解的存在性和振动性的主要工具和时标上有关的基本运算公式.
第二章讨论了三类时标上动力方程解的存在性.我们选择了时标上四阶四点边值问题,并进一步研究了一阶和二阶脉冲动力方程.分别运用Krasnoselskii-Zabreiko不动点定理和Schaefer不动点定理,我们得到了这三类方程解的存在性.值得指出的是,对于四阶动力方程,我们得出了Green函数的表达式,这个计算方法适合时标上其它高阶动力方程.对于一阶和二阶脉冲动力方程,所得结果弥补并拓展了已有结论.
在第三章,利用多值映射的不动点定理和上(下)半连续的选择定理,我们考虑了时标上两类带有边值条件的动力包含解的存在性,得到了一些新的结论.我们给出了例子来说明主要结论的有效性.第三章的结果拓展了在现有文献中关于时标上二阶动力包含的结果.
第四章研究了时标上动力方程解的振动性.本章分两个小节.第一小节给出了二阶半线性动力方程解的振动性,而第二小节主要考虑了二阶非线性动力方程.主要工具包括Riccati转换技巧和一些不等式.第三章的结果统一并扩展了文献中的已有结论.
Time scales theory, introduced by Hilger, not only establishes a bridge be-tween the continuous and the discrete analysis, but also understands deeply the essential between them. The theorem on time scales describes more accurately the phenomena which happen sometimes in continuous time and sometimes in discrete time. In order to develop time scales theorem and application further, our attention focus on the properties of solutions for dynamic equations and dy-namic inclusions,including the three aspects of the time scales theory:Existence of solutions for dynamic equations, existence of solutions for dynamic inclusions, oscillation for dynamic equations.
In Chapter 1, we recall the background of our topics of this thesis, show the outline and give the main tools for dealing the existence and oscillations of solutions for dynamic equations and dynamic inclusions and some basic calculus on time scales related with this thesis.
Chapter 2 is dealing with the existence of solutions for three classes of dy-namic equations on time scales. Thus, by means of fixed-point theorems due to Krasnoselskii-Zabreiko, and Schaefer, we present the existence of solutions for fourth-order four-point boundary problems, a class of first-order impulsive dy-namic equations and a class of second-order dynamic equations on time scales, respectively. It is worth pointing out for the Forth-order dynamic equation, the Green function is new and this method can be applied to study some other boundary value problems for high-order dynamic equations on time scales. For the first-order or second-order impulsive dynamic equations, the obtained result complement and extend some known results.
In Chapter 3, combining fixed-point theorems on multi-valued mappings with continuous selection theorems for upper (or lower) semi-continuous, we in-vestigate existence of solutions for two classes of dynamic inclusions with bound-ary conditions on time scales. Some new criteria are given. We also provide examples to illustrate our main results. The obtained results fill the gap that there are only a few results about second-order dynamic inclusions on time scales in the literatures.
Chapter 4 is devoted to the oscillation of solutions for dynamic equations. We discuss the problems in two subsections. In section 1, sufficient criteria of oscillation for a half-linear second-order dynamic equation is established. In sec-tion 2, we are concerned with the oscillation of a nonlinear second-order dynamic equation. The main tools include Riccati transformation and some inequalities. The results in Chapter 3 generalize and extend some known results in the liter-atures.
引文
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