【作者】 郭雅丽；

【导师】 张传义；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2008， 博士

【摘要】 丹麦数学家H. Bohr于1925-1926年间建立了概周期函数理论。概周期函数是拥有某种结构性质的连续函数,是周期函数的一般化。经过许多数学工作者的努力,概周期函数理论得到了长足的发展。一方面是关于概周期型函数的扩充和性质的研究,另一方面是概周期型函数理论在相应学科中的应用,特别是方程概周期型解的存在性条件。本文主要包括两部分内容的研究:一部分是关于几种一阶概周期型微分方程解的存在唯一性的研究讨论,另一部分是关于一类带梯度算子的二阶概周期型微分方程解的存在唯一性的讨论。对于带有严格单调算子的微分方程的概周期解的存在唯一性已经得到了一些数学工作者的研究,并且有了一些重要的结果。Sarason定义了缓慢振荡函数,但至今没有人将它应用到带有严格单调算子的微分方程中。1941年,Fre′chet最先研究引入了渐近概周期函数。渐近概周期函数是概周期函数的一个推广。但到目前为止,还无人对带有严格单调算子的微分方程的渐近概周期解的存在唯一性进行研究。带有梯度算子的二阶微分方程是一类特殊的非线性方程,这种方程在化工和电子等方面有重要的作用。九十年代初利用变分法人们解决了这种方程解的存在性问题。利用函数的凸性,方程的概周期解的存在性和唯一性问题也已经得到了解决,这以后人们深入地研究更广泛的带梯度算子的二阶方程,在它的概周期解的存在性和唯一性方面得到了一些很好的结果。近几年中,越来越多的人致力于带有半群无穷小生成元的微分方程的概周期解的研究,但对于其缓慢振荡解的研究是很少的。本文主要解决了以下几个问题:1.本文考虑将缓慢振荡函数和带有严格单调算子的微分方程相结合,利用反证法给出了带有严格单调算子的一阶微分方程缓慢振荡解存在和唯一的一个充分条件和一个必要条件,并且举例说明了,这两个条件都不可能成为充分必要条件。特别地对于此类方程的一类特殊情形,我们给出了缓慢振荡解存在和唯一的充分必要条件。2.本文利用渐近概周期函数的性质给出了带有严格单调算子的一阶微分方程渐近概周期解存在和唯一的一个充分条件和一个必要条件,并且举例说明了,这两个条件都不可能成为充分必要条件。特别地对于此类方程的一类特殊情形,我们给出了渐近概周期解存在和唯一的充分必要条件。并且我们把一些结果推广到了一类二阶微分方程。3.本文解决了带梯度算子的二阶方程渐近概周期解的存在和唯一性的问题。我们利用渐近概周期函数的性质可以得到这种方程的渐近概周期解在R+上的存在性。另一方面,我们可以把原方程转化,利用迭代法和相关的线性常微分方程的渐近概周期解的存在性和唯一性条件,得到此方程的渐近概周期解的存在和唯一性。4.本文利用Banach不动点定理考虑了带有半群无穷小生成元的一阶微分方程的缓慢振荡解存在唯一的条件。更多还原

【Abstract】 The theory of almost periodic functions was mainly created by the Danish math-ematician H. Bohr during 1925-1926. Almost periodic functions are the class of con-tinuous functions possessing certain structural properties and are a generalization ofpure periodicity. The theory was developed further by some mathematicians. Onedirection is the broader study of functions of almost periodic type, another directionis the application in certain subject, in particular, the existence of the almost periodictype solutions of equations.The paper consists of two parts:one part concerns the existence and uniquenessof the solutions of some one-order almost periodic type differential equations, theother part concerns the existence and uniqueness of the solutions of some second-order almost periodic type differential equations with gradient operators.Some people discussed the almost periodic solutions of differential equationswith strictly monotone operators, and they have given some important results. Sara-son defined slowly oscillating functions. To our knowledge, nobody has applied suchfunctions to the theory of differential equations with strictly monotone operators. In1941, asymptotically almost periodic functions was originally introduced by Fre′chet.Asymptotically almost periodic functions are a generalization of almost periodic func-tions. To our knowledge, nobody has studied the existence and uniqueness of asymp-totically almost periodic solutions of the differential equations with strictly monotoneoperators.Second-order equations with gradient operators is a kind of special nonlineardifferential equations. The equations have applications in certain chemical industryand electron. By means of variational principle, at the beginning of 1990s peopleprovided results about the existence of the solutions of such equations. Because ofthe convexity of functions, the existence and uniqueness of almost periodic solutionsof such equations have been solved. In the following years people deeply study widersecond-order equations with gradient operators and get many perfect results of theexistence and uniqueness of almost periodic solutions of such equations.In recent year, more and more people have given important contributions to the questions of almost periodic solutions of the differential equations with infinitesimalgenerators of semigroups. But slowly oscillating solutions of these differential equa-tions have seldom been discussion.In this paper, we mainly solve the following questions:Firstly, we give the relation of slowly oscillating functions and differential equa-tions with strictly monotone operators. By means of reduction to absurdity, the au-thors discuss a necessary and a sufficient conditions for the existence and uniquenessof slowly oscillating solutions of the equations. Then we give examples to show thatthey can’t be necessary and sufficient conditions. Particularly, as a special class, theauthors give necessary and sufficient conditions for the existence and uniqueness ofslowly oscillating solutions of the differential equations with gradient operators.Secondly, By means of the properties of asymptotically almost periodic func-tions, the authors discuss a necessary and a sufficient conditions for the existence anduniqueness of asymptotically almost periodic solutions of the equations with strictlymonotone operators. Then we give examples to show that they can’t be necessary andsufficient conditions. Particularly, as a special class, the authors give necessary andsufficient conditions for the existence and uniqueness of asymptotically almost peri-odic solutions for the differential equations with gradient operators. Then we extendthe results to some second-order equations.Thirdly, we solve the problem of existence and uniqueness of asymptoticallyalmost periodic solutions of the second-order equations with gradient operators. Bymeans of the properties of asymptotically almost periodic functions, we get the exis-tence on R+ of asymptotically almost periodic solutions of the equations. On the otherhand, we transform the equations. Thus we present existence and uniqueness theoremfor asymptotically almost periodic solutions of the equations by means of iterationand the conditions for the existence and uniqueness of almost periodic solutions ofcorrelative linear differential equations.Fourthly, we discuss some conditions for the existence and uniqueness of slowlyoscillating (mild) solutions of the differential equations with infinitesimal generatorsof semigroups by Banach fixed point theorem.更多还原