臺灣博碩士論文加值系統

摘要
假設 ， 且 為一閉線性算子。在適當的條件下，我們導出一些 次積分預解族的基本性質和如下非線性Volterra 方程解的存在定理：
for
應用這些結果，我們可以獲得有關 次積分預解族的攝動、乘法及其伴隨問題的定理。

Abstract
Let , , , and let be a closed linear operator in . Under suitable conditions, we obtain some basic properties of -times integrated solution families for and existence theorems of solutions for the following nonlinear Volterra equation：
for .
Applying these results, we shall deduce some perturbation and multiplication theorems of -times integrated solution families . Moreover, adjoint problems for -times integrated solution families are also discussed .

Contents
Abstract
§1 . Itroduction …………………………………………...1
§2 . Basic Properties and Existence Theorems …………..3
§3 . Perturbation Theorems ………..……………………27
§4 . Multiplication Theorems and Adjoint Problems …...33
References ………………………………………………..40

References
[1] W.Arendt, and H.Kellermann, Integrated solutions of Volterra integrodifferential equations and applications, “Volterra integrodifferential equations in Banach spaces and applications”, Pitman Res.Notes in Math., 190(1989),21-51.
[2] R.Grimmer and J.Pruss, On linear Volterra equations in Banach spaces, Comput. Math. Appl., (1985), 189-205.
[3] M.Hiber, Laplce transforms and -times integrated semigroups. Semigroup Forum, 3(1991), 595-612.
[4] C.-C. Kuo and S.-Y. Shaw, On -times integrated C-semigroups and the abstract Cauchy problem, Studia Math., to appear.
[5] C.-C. Kuo and S.-Y. Shaw, On strong and weak solutions of abstract Cauchy problems.
[6] Y.-C.Li and S.-Y.Shaw, On generators of integrated C-semigroups and C-cosine functions, Semigroup Forum, 47(1993), 29-35.
[7] Y.-C.Li and S.-Y.Shaw, N-times integrated C-semigroups and the abstract Cauchy problem, Taiwanese Math., 1(1997), 75-102.
[8] X.Liang, The Cauchy Problem for Higher Order Abstract Differential Equations. (1998).
[9] C.Lizama, Regularized Solutions for Abstract Volterra Equations, J.Math.Anal. Appl., 243(2000), 278-292.
[10] I.Miyadera, M.Okubo, and N.Tanaka., On Integrated Semigroups which are not Exponentially Bounded. Proc.Japan Acad ., 69(1993), 199-204.
[11] H.Oka, Integerated resolvent operators,J.Integral Equations Appl., 7 (1995),193-232.
[12] H.Oka,Second Order Linear Volterra Integrodifferential Equations, Semigroup Forum, 53 (1996), 25-43.
[13] H.Oka, Linear Volterra Equations and Integrated Solution Families, Semigroup Forum, 53 (1996), 278-297.
[14] A.Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equation. (1983).
[15] A.Rhandi Multiplicative Perturbations of Linear Volterra Equations, Proc. Amer. Math.Soc., 119(1993), 493-501.