Banach空间中泛函微分方程的解及其性质
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摘要
这篇博士论文共分五章,主要研究Banach空间中抽象半线性及非线性泛函微分方程解的基本理论,以及渐近非扩张型非线性算子半群的遍历理论。
第一章讨论了一类具非局部条件的半线性泛函微分方程,利用线性算子半群理论、连续函数空间中抽象非紧测度理论和Schauder不动点定理,在不需要半群的紧性和等度连续性的情况下,得到了具非局部条件的半线性泛函微分方程的mild解的存在性以及解集的紧性。
第二章考虑中立型泛函微分方程和积分微分方程,利用解析半群理论、Hausdor?非紧测度及Darbo-Sadovskii不动点定理,得到一类新的中立型泛函微分方程及积分微分方程mild解的存在性。所讨论的模型包含了常见几种类型的方程,可对中立型时滞方程、非局部方程等进行统一处理。所得结果改进和推广了一些已有结果。
第三章研究了具无穷时滞的半线性泛函微分方程的不变流问题。利用Caraththeodory型函数的Scorza Dragoni性质以及Lebesgue型导数,证明了当扰动函数为Caraththeodory型函数时,Banach空间中一类管道是无穷时滞半线性方程的解通道的充要条件为切条件成立。
第四章建立了一类具无穷时滞的非线性发展方程强解的存在性、唯一性及对初值的Lipschitz连续性。利用m-增生算子的Yosida逼近构造近似方程,用逼近的方法,证明了此类方程的强解是其逼近方程解的一致极限。
第五章研究非线性非Lipschitz算子半群的遍历性质,在一致凸Banach空间中建立了渐近非扩张型半群殆轨道的强遍历收敛定理,改进和推广了已有的相关结果,所建立的方法可对非Lipschtian算子和非Lipschitz半群等情形统一处理。
This work is devoted to study the abstract semi-linear and nonlinear functionaldi?erential equations in Banach spaces, followed by that of the ergodic theoryfor semigroups of nonlinear non-Lipschitzian operators。
There are five chapters in this dissertation. In Chapter 1 a class of semi-linear functional di?erential equations with nonlocal conditions is discussed. Byutilizing the theory of the measure of noncompactness in the space of continuousfunctions and Schauder’s fixed point theorem, existence of mild solutions andcompactness of solution set are obtained without the assumption of the com-pactness or equicontinuity of the associated semigroups.
Chapter 2 is concerned with neutral functional di?erential and integro-di?erential equations. By employing the theory of analytic semigroups, Haus-dor?’s measure of noncompactness and Darbo-Sadovskii’s fixed point theorem,existence of mild solutions to neutral functional di?erential and integro-di?erentialequations with nonlocal conditions. The models discussed in this chapter enableus to handle simultaneously several classes of equations, such as neutral di?er-ential equations with delay and nonlocal neutral di?erential equations.
Viability for a class of semilinear di?erential equations with infinite delay inBanach spaces is studied in Chapter 3. Based on the Scorda Dragoni’s propertyfor Caratheodory type functions and Lebesgue type derivative, it is verified thatthe necessary and su?cient condition for a tube in a Banach space to be viablefor a class of semilinear di?erential equations with infinite delay is the tangencycondition.
Chapter 4 is devoted to establish the existence of unique strong solution of aclass of nonlinear evolution equations. A straightforward approximation schemeis available to show that the solution of such equation is the uniform limit of thecontinuously di?erentiable solutions of approximation equations involving theYosida approximants of m?accretive operators.
The final chapter is concerned with the ergodic theorem for commutativesemigroups of non-Lipschitzian mappings. Strong ergodic theorem for semigroupof asymptotically non-expansive type in the intermediate sense mapping is es-tablished in a uniformly convex Banach space, which extends and unifies manypreviously known results.
第一章讨论了一类具非局部条件的半线性泛函微分方程,利用线性算子半群理论、连续函数空间中抽象非紧测度理论和Schauder不动点定理,在不需要半群的紧性和等度连续性的情况下,得到了具非局部条件的半线性泛函微分方程的mild解的存在性以及解集的紧性。
第二章考虑中立型泛函微分方程和积分微分方程,利用解析半群理论、Hausdor?非紧测度及Darbo-Sadovskii不动点定理,得到一类新的中立型泛函微分方程及积分微分方程mild解的存在性。所讨论的模型包含了常见几种类型的方程,可对中立型时滞方程、非局部方程等进行统一处理。所得结果改进和推广了一些已有结果。
第三章研究了具无穷时滞的半线性泛函微分方程的不变流问题。利用Caraththeodory型函数的Scorza Dragoni性质以及Lebesgue型导数,证明了当扰动函数为Caraththeodory型函数时,Banach空间中一类管道是无穷时滞半线性方程的解通道的充要条件为切条件成立。
第四章建立了一类具无穷时滞的非线性发展方程强解的存在性、唯一性及对初值的Lipschitz连续性。利用m-增生算子的Yosida逼近构造近似方程,用逼近的方法,证明了此类方程的强解是其逼近方程解的一致极限。
第五章研究非线性非Lipschitz算子半群的遍历性质,在一致凸Banach空间中建立了渐近非扩张型半群殆轨道的强遍历收敛定理,改进和推广了已有的相关结果,所建立的方法可对非Lipschtian算子和非Lipschitz半群等情形统一处理。
This work is devoted to study the abstract semi-linear and nonlinear functionaldi?erential equations in Banach spaces, followed by that of the ergodic theoryfor semigroups of nonlinear non-Lipschitzian operators。
There are five chapters in this dissertation. In Chapter 1 a class of semi-linear functional di?erential equations with nonlocal conditions is discussed. Byutilizing the theory of the measure of noncompactness in the space of continuousfunctions and Schauder’s fixed point theorem, existence of mild solutions andcompactness of solution set are obtained without the assumption of the com-pactness or equicontinuity of the associated semigroups.
Chapter 2 is concerned with neutral functional di?erential and integro-di?erential equations. By employing the theory of analytic semigroups, Haus-dor?’s measure of noncompactness and Darbo-Sadovskii’s fixed point theorem,existence of mild solutions to neutral functional di?erential and integro-di?erentialequations with nonlocal conditions. The models discussed in this chapter enableus to handle simultaneously several classes of equations, such as neutral di?er-ential equations with delay and nonlocal neutral di?erential equations.
Viability for a class of semilinear di?erential equations with infinite delay inBanach spaces is studied in Chapter 3. Based on the Scorda Dragoni’s propertyfor Caratheodory type functions and Lebesgue type derivative, it is verified thatthe necessary and su?cient condition for a tube in a Banach space to be viablefor a class of semilinear di?erential equations with infinite delay is the tangencycondition.
Chapter 4 is devoted to establish the existence of unique strong solution of aclass of nonlinear evolution equations. A straightforward approximation schemeis available to show that the solution of such equation is the uniform limit of thecontinuously di?erentiable solutions of approximation equations involving theYosida approximants of m?accretive operators.
The final chapter is concerned with the ergodic theorem for commutativesemigroups of non-Lipschitzian mappings. Strong ergodic theorem for semigroupof asymptotically non-expansive type in the intermediate sense mapping is es-tablished in a uniformly convex Banach space, which extends and unifies manypreviously known results.
引文
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