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。 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.
