摘 要

這篇論文分成兩部分，第一部分討論factored inhomogeneous linear equations在 Banach 空間上。第二部分討論抽象 Cauchy problem 的擾動在不同的拓樸空間。論文的安排如下:
第一章我們討論factored inhomogeneous linear equations 並使用d’Alembert formula 找到它的解。第二章我們探討C-semigroups 的 relative bounded perturbation 在 Banach 空間上。 得到下面的結論若 A 是一個 C-semigroup 的生成元在 Banach (X) 空間上, B 是一個 relative A-bounded 的運算子, 那麼 可生成一個 C-semigroup 在 X上。 第三章我們將第二章的結論推廣到sequentially complete locally convex space 上。 第四章我們將第三章的理論應用到實際上的問題也就是光子在星際間的傳播。最後一章我們討論 semigroups 有下面的結果: 若A 是一個 semigroups 的生成元 forcing term function 滿足 local Lipschitz condition 那麼 the abstract semilinear initial value problem 有唯一解 。 在每章後面我們都附上一些partial differential equations的例子。

Abstract

This thesis consists of two parts. The first part is discussing the factored inhomogeneous linear equations in Banach space. The second part is concerning perturbations of abstract Cauchy problem in various topological spaces. We arrange this thesis as follows: In Chapter 1 we study the factored inhomogeneous linear equation. We get a generalized d’Alembert formula to get the solution of this factored equation. In chapter 2 we study the relative bounded perturbation of C-semigroups on Banach space. We show that if A generates a C-semigroup on a Banach space, B is a relative A-bounded operator, then also generates a (analytic) C-semigroup on the same space. In chapter 3 we generalized the results in chapter 2 to sequentially complete locally convex space. In chapter 4 we study the photon transport problem, in there we apply the results gotten from chapter 3 to solve this problem. In the last chapter, we study the semigroups. We show that if A generates a semigroup on a topological space and the forcing term function satisfies local Lipschitz condition, then the abstract semilinear initial value problem will has a unique solution. The applications of these results to certain partial .differential equations were given in each chapter.

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