微分方程的加权伪几乎自守性及其应用
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摘要
本文主要研究了加权伪几乎自守函数及其推广函数的性质,以及这些函数在微分方程中的应用.
全文共分成四章.
第一章,简述了几乎自守函数的历史背景,研究现状以及本文的主要工作.
第二章,对线性边界微分方程,建立了几乎自守性的Massera型准则,推广了经典的Massera关于纯量周期常微分方程的定理.另一方面,对于Lienard方程,研究了其解的有界性与几乎自守解存在性的关系,并利用这些结论讨论了Lienard方程伪几乎自守解的存在唯一性.
第三章,利用Hille-Yosida算子的相关理论和Banach压缩映像原理,研究了带有非稠定算子的中立型抽象泛函微分方程的加权伪几乎自守性.
第四章,系统研究了Banach空间中加权Stepanov伪几乎自守函数的性质,如完备性和平移不变性,并建立了加权Stepanov伪几乎自守函数的复合定理.作为应用,我们研究了其在中立型泛函微分方程、Volterra积分方程和分数阶微分方程中的应用,建立了存在唯一加权伪几乎自守解的若干判据.
最后,总结了论文的结果,提出论文下一步的工作以及本文所参考的主要文献.
This paper mainly discuss the properties of weighted pseudo almost automor-phic functions and related generalized functions. Applications of these functionsto various class of diferential equations are presented.
The paper is composed of four chapters.
In Chapter1, the historical background and recent developments of almostautomorphic functions and main results of this paper are introduced.
In Chapter2, for linear boundary diferential equations, a Massera type cri-terion is established for the existence of an almost automorphic solution, whichextends the classical Massera periodic theorem on scalar periodic ordinary dif-ferential equations. On the other hand, for Li′enard equation, the relationshipsbetween bounded solution and the existence of almost automorphic solution areinvestigated, then using these results, we study the existence and uniqueness ofpseudo almost automorphic solutions of Li′enard equation.
In Chapter3, by using the theory of Hille-Yosida operator and the Banachcontraction mapping principle, we investigate the weighted pseudo almost auto-moprhy for neutral abstract functional diferential equations with operator of nondense domain.
In Chapter4, we systematically explore the properties of the weighted Stepanov-like pseudo almost automorphic functions in Banach space including completenessand translation invariance, establish the composition theorem for Stepanov-likepseudo almost automorphic functions. As an applications, we establish some suf-cient criteria for the existence, uniqueness of the weighted pseudo almost automor-phic solution to a class of neutral functional diferential equations, Volterra integralequations and semilinear fractional diferential equation.
At the end of this paper, the remark of main findings and further study direc- tion are presented, many related references are listed.
全文共分成四章.
第一章,简述了几乎自守函数的历史背景,研究现状以及本文的主要工作.
第二章,对线性边界微分方程,建立了几乎自守性的Massera型准则,推广了经典的Massera关于纯量周期常微分方程的定理.另一方面,对于Lienard方程,研究了其解的有界性与几乎自守解存在性的关系,并利用这些结论讨论了Lienard方程伪几乎自守解的存在唯一性.
第三章,利用Hille-Yosida算子的相关理论和Banach压缩映像原理,研究了带有非稠定算子的中立型抽象泛函微分方程的加权伪几乎自守性.
第四章,系统研究了Banach空间中加权Stepanov伪几乎自守函数的性质,如完备性和平移不变性,并建立了加权Stepanov伪几乎自守函数的复合定理.作为应用,我们研究了其在中立型泛函微分方程、Volterra积分方程和分数阶微分方程中的应用,建立了存在唯一加权伪几乎自守解的若干判据.
最后,总结了论文的结果,提出论文下一步的工作以及本文所参考的主要文献.
This paper mainly discuss the properties of weighted pseudo almost automor-phic functions and related generalized functions. Applications of these functionsto various class of diferential equations are presented.
The paper is composed of four chapters.
In Chapter1, the historical background and recent developments of almostautomorphic functions and main results of this paper are introduced.
In Chapter2, for linear boundary diferential equations, a Massera type cri-terion is established for the existence of an almost automorphic solution, whichextends the classical Massera periodic theorem on scalar periodic ordinary dif-ferential equations. On the other hand, for Li′enard equation, the relationshipsbetween bounded solution and the existence of almost automorphic solution areinvestigated, then using these results, we study the existence and uniqueness ofpseudo almost automorphic solutions of Li′enard equation.
In Chapter3, by using the theory of Hille-Yosida operator and the Banachcontraction mapping principle, we investigate the weighted pseudo almost auto-moprhy for neutral abstract functional diferential equations with operator of nondense domain.
In Chapter4, we systematically explore the properties of the weighted Stepanov-like pseudo almost automorphic functions in Banach space including completenessand translation invariance, establish the composition theorem for Stepanov-likepseudo almost automorphic functions. As an applications, we establish some suf-cient criteria for the existence, uniqueness of the weighted pseudo almost automor-phic solution to a class of neutral functional diferential equations, Volterra integralequations and semilinear fractional diferential equation.
At the end of this paper, the remark of main findings and further study direc- tion are presented, many related references are listed.
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