關於函數及積分,演化方程式解之行為的探討

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陳正忠(Jeng-Chung Chen) 查詢紙本館藏

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數學系

論文名稱

關於函數及積分,演化方程式解之行為的探討
(Behavior of Functions and Solutions ofIntegral Equations and Evolution Equations)

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摘要(中)

這篇論文主要在研究向量值數列，向量值函數的行為以及Volterra 積分方
程式的解和演化方程式的解的漸近行為，得到一些新結果。
這些結果主要分成兩個部分，第一部份是從第二章到第五章，處理數列與函
數的行為；在第二章跟第三章，我們分別比較數列跟函數的Cesaro mean 和Abel
mean 的增長次序；在第四章，我們討論數列跟函數與Cesaro mean 和Abel mean
的比率極限定理(ratio limit theorem)和ratio Tauberian theorem；在第五
章，我們討論算子數列的超循環性質跟拓樸混合性質。
第二部份由第六，七章組成，主要分別考慮對Volterra 積分方程式的解和
演化方程式的解的漸近行為。在第六章中，我們導出一些關於Volterra 積分方
程式的(a,k)-正規豫解族在零點的逼近速率。在第七章，我們得到演化方程式的
幾乎週期解的存在性的各種充分條件，這些結果推廣了以前的結果到一個更一般
性關於C-半群的不規則方程式。

摘要(英)

This dissertation is concerned with behavior of vector-valued sequences and functions
and asymptotic behavior solutions of Volterra integral equations and of evolution
equations.
There are two parts. The first part which consists of Chapters 2-5 deals with
behavior of sequences and functions; in Chapter 2 and 3, we compare growth orders
of Ces`aro and Abel means of sequences and functions, respectively; in Chapter 4,
we present ratio limit theorems and ratio Tauberian theorems for Ces`aro means and
Abel means of sequences and functions; in Chapter 5, we discuss the notions of
hypercyclicity and topological mixing for operator sequences.
The two chapters in the second part of this dissertation will deal with asymptotic
behavior of solutions of Volterra equations and of evolution equations, respectively. In
Chapter 6, we deduce some approximation theorems with rates for (a, k)-regularized
resolvent family for Volterra integral equations. In Chapter 7, we obtain various suf-
ficient conditions for the existence of almost periodic solutions of evolution equations
which extend previous ones to a more general class of ill-posed equations involving
C-semigroups.

關鍵字(中)

★ 正規豫解族
★ 拓樸混合
★ 向量值數列
★ 向量值函數
★ 比率定理
★ 超循環算子
★ C半群
★ 演化方程

關鍵字(英)

★ topologically mixing
★ hypercyclic sequences of operators
★ k)-regularized resolvent family
★ (a ratio limit theorem
★ growth order
★ Abel mean
★ Cesaro mean
★ C-semigroups
★ evolution equations

論文目次

Acknowledgements ii
1 Introduction 1
2 Growth Orders of Means of Sequences in Banach Spaces 8
2.1 General Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Lemmas and Proof for General Estimates . . . . . . . . . . . . . . . . 10
2.3 Means of Powers of Operators . . . . . . . . . . . . . . . . . . . . . . 14
3 Growth Orders of Ces`aro and Abel Means of Functions in Banach
Spaces 20
3.1 Estimates of Growth Orders . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Growth orders of means of C0-semigroups . . . . . . . . . . . . . . . 28
4 Ratio Limit Theorems and Tauberian Theorems 34
4.1 Results for functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Results for sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Generalized limit and Tauberian theorems . . . . . . . . . . . . . . . 42
5 Topological Mixing and Hypercyclicity Criterion for Sequences of
Operators 44
5.1 Observations on Topological Mixing and Hypercyclicity Criterion . . 45
5.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6 Asymptotic Behavior of (a, k)-regularized Resolvent Families 54
6.1 Regularized Approximation Processes . . . . . . . . . . . . . . . . . . 55
6.2 Approximation Properties of Regularized Solution Families . . . . . . 58
7 C-Semigroups and Almost Periodic Solutions of Evolution Equations
65
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.1.2 Spectral theory of functions . . . . . . . . . . . . . . . . . . . 66
7.1.3 C-semigroups: Definition and basic properties . . . . . . . . . 68
7.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.2.1 Evolution C-semigroups . . . . . . . . . . . . . . . . . . . . . 70
7.2.2 Evolution C-semigroups in invariant subspaces of AP(X) . . . 73
7.2.3 The resonant case . . . . . . . . . . . . . . . . . . . . . . . . . 75
Bibliography 80

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指導教授

蕭勝彥(Sen-Yen Shaw)

審核日期

2005-5-31

推文