【作者】 张俊；

【导师】 梁进； 肖体俊；

【作者基本信息】 中国科学技术大学 ， 应用数学， 2009， 博士

【摘要】 本文主要讨论了伪概自守函数和相关函数的基本性质及其在发展方程中的应用。全文共分五章。第一章介绍了本文的研究背景和主要工作。第二章是预备知识,主要介绍了概周期函数、概自守函数、渐近概周期函数、渐近概自守函数、伪概周期函数、伪概自守函数等的概念和基本性质。此外,我们还介绍了C₀半群和cosine算子函数的一些定义和相关性质。第三章主要研究了概自守函数和伪概自守函数的基本性质。这些性质为研究自守函数在发展方程中的进一步应用奠定了基础。§3.1主要研究了概自守函数和具有零平均值的函数的一些基本性质。§3.2中我们主要讨论了在Lipschitz连续性假设下,函数f（t,x）与x（t）复合后保持伪概自守性质不变的条件,并获得了关于伪概自守函数的复合定理。§3.3讨论了伪概自守函数的分解的唯一性,同时证明了其在范数下的完备性。§3.4讨论了广义的伪概自守函数,即此时其扰动项并不是有界连续函数的情形。第四章主要研究了伪概自守函数在线性发展方程u^’（t）=Au（t）+f（t）, t∈R和半线性发展方程u^’（t）=Au（t）+f（t,u（t））, t∈R中的应用。我们分别针对线性算子A生成指数稳定的C₀半群和生成紧的C₀半群情形,进行了研究,给出了这两类发展方程的伪概自守的温和解的存在性定理,并在某些情形下得到了解的唯一性。我们在第五章研究了具有指数增长的渐近概周期函数的性质及在一阶微分方程和非完全二阶算子微分方程中的应用。更多还原

【Abstract】 Properties of pseudo almost automorphic functions and related ones are discussedin this thesis. Applications of these properties to evolution equations are also presented. This thesis consists of five chapters.Chapter 1 presents the research background and main results of this thesis.Chapter 2 is about preliminaries of this thesis. Definitions and basic properties of almost periodic, almost automorphic, asymptotically almost periodic, asymptoticallyalmost automorphic, pseudo almost periodic and pseudo almost automorphic functions are stated. Moreover, definitions of C₀ semigroups and cosine operator functions are revoiewed.Chapter 3 is concerned with basic properties of almost automorphic functions and pseudo almost automorphic functions, which is a basis for exploring further applications of these functions to evolution equations.In Section 3.1, some basic properties of almost automorphic functions and functions with 0 mean value are investigated.In Section 3.2, we study the composition of pseudo almost automorphic functions f（t,x） and x（t） under Lipschitz conditions.In Section 3.3, we show the uniqueness of decomposition of pseudo almost automorphicfunctions, and then prove that pseudo almost automorphic functions form a Banach space under the normSection 3.4 is mainly about generalized pseudo almost automorphic functions, whose corrective terms are not bounded and continuous.Chapter 4 studies the applications of pseudo almost automorphic functions to linear evolution equations u^’（t） = Au（t）+f（t）, t∈R, and semilinear evolution equations u^’（t） = Au（t） + f（t,u（t））, t∈R. For the case of A generating an exponentially stable C₀ semigroup and the case of A generating a compact C₀ semigroup, respectively, we establish new existence and uniqueness theorems for mild pseudo almost automorphic solutions to these evolution equations.In Chapter 5, we investigate exponentially asymptotically almost periodic functions.Applications to first order and incomplete second order evolution equations are discussed.更多还原