Banach空间中拟线性广义逆扰动及线性包含约束极值解

发布时间：2018-06-01 13:41

本文选题：Banach空间 + 拟线性广义逆　；参考：《哈尔滨工业大学》2016年博士论文

【摘要】：广义逆理论研究产生于求解线性不适定方程(其中方程包括线性代数方程、微分方程、偏微分方程和积分方程等)的过程。广义逆理论研究内容丰富,其中最为突出的是关于各类投影广义逆的理论及其应用的研究。自上世纪三十年代起,有关这方面的研究成果众多,当中最引人注意的是研究投影广义逆的扰动问题,其中包括Hilbert空间线性广义逆的扰动分析、Moore-Penrose广义逆扰动分析、Banach空间中有界线性算子的广义逆扰动分析和Banach空间中有界线性算子度量广义逆的扰动分析。算子广义逆的扰动分析结果在多个数学领域都有广泛应用,引起了学界普遍关注,特别在计算数学、非线性分析中分歧理论、Banach流形的广义截断性研究等方面。由于Banach空间中线性算子的线性广义逆不适合研究不适定线性算子方程极值解、最小范数解及最佳逼近解,所以研究Banach空间中的度量广义逆及其扰动分析就显得尤为重要。本文的首要目的是对于闭线性算子的投影广义逆的扰动分析建立统一理论,该类投影广义逆既包括Hilbert空间中的Moore-Penrose广义逆,又包括Banach空间中闭线性算子的线性斜投影广义逆,也包括Banach空间闭线性算子的度量广义逆。简而言之,本文要研究的是Banach空间中闭线性算子的Moore-Penrose拟线性投影广义逆的扰动分析,研究目的是构建统一的扰动分析理论。本文首先从两个方面对Moore-Penrose拟线性投影广义逆的扰动进行研究:一方面,对于稠定的闭线性算子T,在扰动算子δT是T-有界(δT本身可能无界)及满足特定条件下,利用把关于有界线性算子的Banach引理推广为关于有界齐性算子的广义Banach引理,建立本文独特的关于Banach空间中闭线性算子的Moore-Penrose拟线性投影广义逆的扰动定理及其扰动界的刻画。这样的扰动结果使以往的文献中熟知的结果成为特例。另一方面,对于稠定的闭线性算子T及扰动算子δT满足特定不等式的条件下,运用广义Neumann引理分别给出有界线性算子的Moore-Penrose度量广义逆及闭线性算子的Moore-Penrose拟线性投影广义逆的新扰动定理及有关误差估计的三个不等式。此扰动成果也推广了文献中的相对应的已知扰动结果。由以上两个结果,构建了关于闭线性算子的Moore-Penrose有界拟线性广义逆的扰动分析理论框架。本文然后运用Banach空间中多值线性算子的代数算子部分、度量广义逆和Banach空间的对偶映射给出了Banach空间中线性包含约束极值解的等价表示形式。将Banach空间中线性包含约束极值解的求解化为等价的同一空间中线性包含无约束极值解。这个结果既包含Hilbert空间中线性包含的最小二乘解(无约束及有约束的)又包含Banach空间中线性包含的极值解(无约束的)的已得结果。由此,看出Banach空间中多值线性算子度量广义逆的应用。
[Abstract]:The generalized inverse theory is derived from the process of solving linear ill-posed equations (including linear algebraic equations, differential equations, partial differential equations, integral equations, etc.). The research on generalized inverse theory is rich, and the most prominent one is the theory of projection generalized inverse and its application. Since the 1930s, there have been many achievements in this field, the most notable of which is the study of the perturbation of projective generalized inverses. It includes perturbation analysis of linear generalized inverse in Hilbert space and perturbation analysis of generalized inverse of bounded linear operator in Hilbert space and generalized inverse metric of bounded linear operator in Banach space. The results of perturbation analysis of generalized inverse of operators have been widely used in many fields of mathematics, especially in computational mathematics, bifurcation theory and generalized truncation of Banach manifolds in nonlinear analysis. Because the linear generalized inverse of linear operator in Banach space is not suitable to study the extremum solution, minimum norm solution and best approximation solution of ill-posed linear operator equation, it is particularly important to study the metric generalized inverse and its perturbation analysis in Banach space. The primary purpose of this paper is to establish a unified theory for the perturbation analysis of the projective generalized inverse of closed linear operators, which includes not only the Moore-Penrose generalized inverse in Hilbert space, but also the linear oblique projection generalized inverse of closed linear operator in Banach space. It also includes the metric generalized inverse of closed linear operators in Banach spaces. In short, this paper studies the perturbation analysis of the Moore-Penrose quasi-linear projection generalized inverse of closed linear operators in Banach spaces. The purpose of this study is to construct a unified perturbation analysis theory. In this paper, we first study the perturbation of Moore-Penrose quasi-linear projection generalized inverse from two aspects: on the one hand, for closed linear operator T, the perturbation operator 未 T is T- bounded (未 T itself may be unbounded) and satisfies certain conditions. By extending the Banach Lemma for bounded linear operators to generalized Banach Lemma for bounded homogeneous operators, this paper establishes a unique perturbation theorem for the Moore-Penrose quasi-linear projection generalized inverse of closed linear operators in Banach spaces and its characterization of perturbation bounds. Such perturbation results make known results in previous literature a special case. On the other hand, if closed linear operator T and perturbed operator 未 T satisfy certain inequalities, By using the generalized Neumann Lemma, the new perturbation theorems and three inequalities about the Moore-Penrose metric generalized inverse of the bounded linear operator and the Moore-Penrose quasi-linear projection generalized inverse of the closed linear operator are given respectively. This perturbation result also extends the corresponding known perturbation results in the literature. Based on the above two results, a theoretical framework for perturbation analysis of Moore-Penrose bounded quasilinear generalized inverses for closed linear operators is constructed. In this paper, the algebraic operator part of the multivalued linear operator in Banach space is used to measure the generalized inverse and the dual mapping of the Banach space. The equivalent representation of the constrained extremum solution of the linear inclusion in the Banach space is given. In this paper, the solution of constrained extremum solution of linear inclusion in Banach space is transformed into an equivalent solution of unconstrained extreme value of linear inclusion in the same space. This result includes not only the least square solutions (unconstrained and constrained) of linear inclusions in Hilbert spaces, but also the obtained results of the extremum solutions (unconstrained) of linear inclusions in Banach spaces. Therefore, the application of multivalued linear operators to measure generalized inverse in Banach spaces is shown.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O177.2

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