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.

[1] L. Amerio, G. Prouse, Almost Periodic Functions and Functional Equations, Van
Nostrand Reinhold, New York, 1971.
[2] S. I. Ansari, Hypercyclic and cyclic vectors, J. Funct. Anal. 128 (1995), 374-383.
[3] W. Arendt, C.J.K. Batty, Almost periodic solutions of first and second oder
Cauchy problems, J. Differential Equations 137(1997), N.2, 363-383.
[4] W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace
Transforms and Cauchy Problems, Monographs in Mathematics Vol. 96, Birkh¨auser
Verlag, 2001.
[5] R. Aron and D. Markose, On universal functions, preprint, 2002.
[6] J. Banks, Topological mapping properties defined by digraphs, Discrete and Cont.
Dyn. Syst. 5 (1999), 83-92.
[7] H. Berens, Interpolationsmethoden zur Behandlung von Approximationsprozessen
auf Banachraumen, Lect. Notes Math. 64, Springer-Verlag, Berlin-Heidelberg-New
York, 1968.
[8] T. Berm´udez, A. Bonilla, J.A. Conejero and A. Peris, Hypercyclic, topologically
mixing and chaotic semigroups on Banach spaces, Studia Math., to appear.
[9] L. Bernal-Gonz´alez and K.-G. Grosse-Erdmann, The Hypercyclicity Criterion for
sequences of operators, Studia Math. 157 (2003),17-32.
[10] L. Bernal-Gonz´alez and A. Montes-Rodr´ıguez, Universal functions for composition
operators, Complex Variables Theory Appl. 27 (1995), 47-56.
[11] J. B`es and A. Peris, Hereditarily hypercyclic operators, J. Funct. Anal. 167
(1999), 94-112.
[12] J.-C. Chen, R. Sato, and S.-Y. Shaw, Growth Orders of Ces`aro and Abel Means
of Functions in Banach Spaces, preprint.
[13] G. Costakis andM. Sambarino, Topologically mixing hypercyclic operators, Proc.
Amer. Math. Soc. 132 (2003), 385-389.
[14] Y. Derriennic and M. Lin, On invariant measures and ergodic theorems for positive
operators, J. Funct. Anal. 13 (1973), 252-267.
[15] R. Emilion, Mean-bounded operators and mean ergodic theorems, J. Funct.
Anal. 61 (1985), 1-14.
[16] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,
Graduate Texts in Mathematics Vol. 194, Springer-Verlag, 2000.
[17] R. M. Gethner and J. H. Shapiro, Universal vectors for operators on space of
holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), 281-288.
[18] S. Grivaux, Some observations regarding the Hypercyclicity Criterion, preprint.
[19] K.-G. Grosse-Erdmann, Holomorphe Monster und Universelle Funktionen, Mitt.
Math. Sem. Giessen. 176 (1987).
[20] K.-G. Grosse-Erdmann, Universal families and hypercyclic operators, Bull.
Amer. Math Soc. 36 (1999), 345-381.
[21] G. Godefroy and J.H. Shapiro, Operators with dense, invariant, cyclic vector
manifolds, J. Funct. Anal. 98 (1991), no 2, 229-269.
[22] E. Hille and R.S. Phillips, Functional Analysis and Semigroups, Amer. Math.
Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957.
[23] C. Kitai, Invariant Closed Sets for Linear Operators, Ph.D. thesis, University of
Toronoto, (1982).
[24] C.-C. Kuo and S.-Y. Shaw, On -times integrated C-semigroups and the abstract
Cauchy problem, Studia Math. 142 (2000), 201-217.
[25] C-C, Kuo, S-Y. Shaw, Abstract Cauchy problems associated with local Csemigroups.
Preprint.
[26] B.M. Levitan, V.V. Zhikov, Almost Periodic Functions and Differential Equations,
Moscow Univ. Publ. House 1978. English translation by Cambridge University
Press 1982.
[27] C.-Y. Li, R. Sato, and S.-Y. Shaw, Ratio Tauberian theorems for positive functions
and sequences in Banach lattices, preprint.
[28] Y.-C. Li, R. Sato, and S.-Y. Shaw, Boundedness and growth order of means of
discrete and continuous semigroups of operators, preprint.
[29] Y.-C. Li and S.-Y. Shaw, N-times integrated C-semigroups and the abstract
Cauchy problem, Taiwanese J. Math. 1 (1997), 75–102.
[30] C. Lizama, On approximation and representation of k-regularized resolvent families,
Int. Eq. Operator Theory 41 (2001), 223-229.
[31] C. Lizama, On perturbation of K-regularized resolvent families, Taiwanese J.
Math. 7 (2003), 217-227.
[32] C. Lizama, Regularized solutions for abstract Volterra equations, J. Math. Anal.
Appl. 243 (2000), 278-292.
[33] H. P. Lotz, Uniform convergence of operators on L1 and similar spaces, Math,
Z. 190 (1985), 207-220.
[34] J.L. Massera, The existence of periodic solutions of systems of differential equations,
Duke Math. J. 17 (1950). 457–475.
[35] A. Montes-Rodr´ıguez, A note on Birkhoff open sets, Complex Variables Theory
Appl. 30 (1996), 193-198.
[36] S. Murakami, T. Naito, N.V. Minh, Evolution semigroups and sums of commuting
operators: a new approach to the admissibility theory of function spaces, J.
Differential Equations 164 (2000), 240-285.
[37] T. Naito, N.V. Minh, J. S. Shin, New spectral criteria for almost periodic solutions
of evolution equations, Studia Mathematica 145 (2001), 97-111.
[38] H. Oka, Linear Volterra equations and integrated solution families, Semigroup
Forum 53 (1996), 278-297.
[39] A. Peris and L. Saldivia, Syndetically hypercyclic operators, Integr. Equ. Oper.
Theory, 51 (2005), 275-281.
[40] J. Pr¨uss, Evolutionary Integral Equations and Applications, Birkh¨auser, Basel,
1993.
[41] Salas, H´ector N., Hypercyclic weighted shifts, Trans. Amer. Math. Soc. 347
(1995), no. 3, 993-1004. MR 95e:47042
[42] R. Sato, Ratio limit theorem and applications to ergodic theory, Studia Math.
66 (1980), 237-245.
[43] R. Sato, On ergodic averages and the range of a closed operator, Taiwanese J.
Math., to appear.
[44] J.H. Shapiro, Composition Operators and Classical Function Theory, Springer-
Verlag, NY (1993).
[45] S.-Y. Shaw and Y.-C. Li, On n-times integrated C-cosine functions, Evolution
Equations, Marcel Dekker, New York, 1994, pp. 393-406.
[46] S-Y. Shaw, C.C. Kuo, Generation of local C-semigroups and solvability of the
abstract Cauchy problems. Preprint.
[47] S.-Y. Shaw and H. Liu, Convergence rates of regularized approximation processes,
J. Approximation Theory 115 (2002), 21-43.
[48] M. Sova, Cosine operator functions, Rozprawy Math. 49 (1966), 1-47.
[49] A. Zygmund, Trigonometric Series. Vol. 1, Cambridge Univesity Press, Cambridge,
1959.