测度链上动力方程解的分类与存在性
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摘要
测度链上动力方程理论不但可以统一微分方程和差分方程,更好地洞察二者之间的本质差异,而且还可以更精确地描述那些有时随时间连续出现而有时又离散发生的现象.在探讨测度链上的动力方程的动力学行为时人们所熟悉的基本工具诸如Fermat定理,Rolle定理以及介值定理等不再成立,同时很难找到适应不同测度链的模拟程序,这些在给测度链理论研究带来诸多困难的同时,也更引起了广大学者的兴趣.
本文首先考虑了测度链上一阶和二阶动力方程组解的分类问题,根据所讨论问题的动力学渐近性,对其最终正解进行了细致的分类.通过构造不同的Banach空间,并借助于Schauder不动点定理,Knaster不动点定理等,获得了一阶和二阶的动力方程组各类具有特定渐近性的解存在的充分或者必要条件.这样的分类可以缩小定性研究的范围,也可以使人们对问题的动力学行为发展趋势进行预测,通过对参数的适当调整,以便对其发展进行干预、控制.
其次,研究了测度链上一类奇异p-拉普拉斯动力方程多点边值问题解的存在性.类似于微分方程和差分方程,测度链上非线性项可变号的动力方程边值问题同样是一个困难,但又重要的问题.与现有工作相比,我们所考虑的动力方程不但非线性项可变号,而且包含一阶导数项,所考虑的边值条件更具一般性,比如可以包含Dirichlet及Robin边值条件等.同时所考虑的奇异性更具体.借助于上下解方法和Schauder不动点定理,建立了测度链上非线性项变号的奇异p-拉普拉斯多点边值问题正解的存在性准则.探讨了证明过程中用到的上、下解的构造问题,并给出了一定条件下具体的上下解构造方法,最后举例说明了所得结果的应用.
最后,讨论了测度链上p-拉普拉斯动力方程两点边值问题的正对称解的存在性.首先利用Krasnosel'skii不动点定理研究了此问题分别在六种假设下至少存在一个或两个正对称解的充分条件.其次,分别借助于广义的Avery-Henderson不动点定理和Avery-Peterson不动点定理,考虑了此边值问题有三个及任意奇数个正对称解的存在性条件,并比较了两种方法的异同点及优劣.最后举例说明了结果的应用.
The theory of dynamic equations on time scales can not only unify differential and difference equations and understand deeply the essential difference between them,but also provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time.Hence,the studying of dynamic equations on time scales is worth with theoretical and practical values.At the same time,many difficulties occur when considering dynamic equations on time scales.For example,basic tools from calculus such as Fermat theorem,Rolle theorem and the intermediate value theorem may not necessarily hold and it is difficult to find a universal program for simulation in a model with various timescales,which attract attention of great deal researchers.
In this PhD thesis,we first consider classification schemes for positive solutions of the first and second order dynamic systems.In terms of their asymptotic magnitudes, classification schemes for positive solutions of the first-order and second-order nonlinear dynamic systems are given.By Schander's fixed point theorem,Knaster's fixed point theorem and constructing various Banach space,we obtained some sufficient and/or necessary conditions for the existence of solutions with designated asymptotic properties.Such schemes are important since further investigations of qualitative behaviors of solutions can then be reduced to only a number of cases, which make one can forecast and control the direction of development for the dynamic system by adjusting parameter.
Next,we study the existence of positive solution for a class singular p-Laplacian dynamic equation with multi-point boundary condition on time scales.As considering the difference equations and differential equations,it is very difficult and important in studying the singular boundary value problems on time scales with the nonlinear term changing sign.Compared with the known results,the nonlinearity is allowed to change sign and is involved with the first-order derivative explicitly.In particular,the boundary condition includes the Dirichlet boundary condition and Robin boundary condition.The singularity may occur at the clearer location.By using Schauder fixed point theorem and upper and lower solution method,we ob- tain some new existence criteria for positive solution to a singular p-Laplacian BVPs with sign-changing nonlinearity involving derivative.We consider how to construct a lower solution and an upper solution in certain conditions and obtained a specific method.As an application,an example is given to illustrate our results.
Finally,we discuss the existence of symmetric positive solutions for p-Laplacian two point boundary value problem on time scales.By using Krasnosel'skii's fixed point theorem in a cone,we first give the sufficient condition for single positive symmetric solutions of the problem under six possible cases.Using generalized Avery-Henderson fixed point theorem and Avery-Peterson fixed point theorem respectively, the existence criteria for at least three positive and arbitrary odd positive symmetric solutions of p-Laplacian dynamic equations are established and the difference between two methods is pointed out.Two simple examples are given to illustrate our results.
本文首先考虑了测度链上一阶和二阶动力方程组解的分类问题,根据所讨论问题的动力学渐近性,对其最终正解进行了细致的分类.通过构造不同的Banach空间,并借助于Schauder不动点定理,Knaster不动点定理等,获得了一阶和二阶的动力方程组各类具有特定渐近性的解存在的充分或者必要条件.这样的分类可以缩小定性研究的范围,也可以使人们对问题的动力学行为发展趋势进行预测,通过对参数的适当调整,以便对其发展进行干预、控制.
其次,研究了测度链上一类奇异p-拉普拉斯动力方程多点边值问题解的存在性.类似于微分方程和差分方程,测度链上非线性项可变号的动力方程边值问题同样是一个困难,但又重要的问题.与现有工作相比,我们所考虑的动力方程不但非线性项可变号,而且包含一阶导数项,所考虑的边值条件更具一般性,比如可以包含Dirichlet及Robin边值条件等.同时所考虑的奇异性更具体.借助于上下解方法和Schauder不动点定理,建立了测度链上非线性项变号的奇异p-拉普拉斯多点边值问题正解的存在性准则.探讨了证明过程中用到的上、下解的构造问题,并给出了一定条件下具体的上下解构造方法,最后举例说明了所得结果的应用.
最后,讨论了测度链上p-拉普拉斯动力方程两点边值问题的正对称解的存在性.首先利用Krasnosel'skii不动点定理研究了此问题分别在六种假设下至少存在一个或两个正对称解的充分条件.其次,分别借助于广义的Avery-Henderson不动点定理和Avery-Peterson不动点定理,考虑了此边值问题有三个及任意奇数个正对称解的存在性条件,并比较了两种方法的异同点及优劣.最后举例说明了结果的应用.
The theory of dynamic equations on time scales can not only unify differential and difference equations and understand deeply the essential difference between them,but also provide accurate information of phenomena that manifest themselves partly in continuous time and partly in discrete time.Hence,the studying of dynamic equations on time scales is worth with theoretical and practical values.At the same time,many difficulties occur when considering dynamic equations on time scales.For example,basic tools from calculus such as Fermat theorem,Rolle theorem and the intermediate value theorem may not necessarily hold and it is difficult to find a universal program for simulation in a model with various timescales,which attract attention of great deal researchers.
In this PhD thesis,we first consider classification schemes for positive solutions of the first and second order dynamic systems.In terms of their asymptotic magnitudes, classification schemes for positive solutions of the first-order and second-order nonlinear dynamic systems are given.By Schander's fixed point theorem,Knaster's fixed point theorem and constructing various Banach space,we obtained some sufficient and/or necessary conditions for the existence of solutions with designated asymptotic properties.Such schemes are important since further investigations of qualitative behaviors of solutions can then be reduced to only a number of cases, which make one can forecast and control the direction of development for the dynamic system by adjusting parameter.
Next,we study the existence of positive solution for a class singular p-Laplacian dynamic equation with multi-point boundary condition on time scales.As considering the difference equations and differential equations,it is very difficult and important in studying the singular boundary value problems on time scales with the nonlinear term changing sign.Compared with the known results,the nonlinearity is allowed to change sign and is involved with the first-order derivative explicitly.In particular,the boundary condition includes the Dirichlet boundary condition and Robin boundary condition.The singularity may occur at the clearer location.By using Schauder fixed point theorem and upper and lower solution method,we ob- tain some new existence criteria for positive solution to a singular p-Laplacian BVPs with sign-changing nonlinearity involving derivative.We consider how to construct a lower solution and an upper solution in certain conditions and obtained a specific method.As an application,an example is given to illustrate our results.
Finally,we discuss the existence of symmetric positive solutions for p-Laplacian two point boundary value problem on time scales.By using Krasnosel'skii's fixed point theorem in a cone,we first give the sufficient condition for single positive symmetric solutions of the problem under six possible cases.Using generalized Avery-Henderson fixed point theorem and Avery-Peterson fixed point theorem respectively, the existence criteria for at least three positive and arbitrary odd positive symmetric solutions of p-Laplacian dynamic equations are established and the difference between two methods is pointed out.Two simple examples are given to illustrate our results.
引文
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