本文是考慮抽象半線性微分方程式

d/dt u(t)=Au(t)+f(t,u(t) 0≤t0
u(t0)=u0 u0∈C(D(A))

其中A是一個在Banach空間Z上之C-半群的生成元，f:[to,T]x X 爲一個函數。我們給予函數f某些適當的條件，使得以上之抽象半線性方程式(0.1)有唯一的古典解、強解或弱解。我們也找出弱解存在的最大時間範圍，並探討此解在趨近邊界時的行爲；此外， 我們也證明了解對初値條件的連續性。爲了證明這些結果，我們先證明(0.1)所對應的 非齊次方程式

d/dt u(t)=Au(t) +f(t) 0≤t0
u(t0)=u0 U0 ∈ C(D(A))

在給予非齊次項函數C某些適當的條件，使得以上之抽象非齊次微分方程式(0.2)有唯一的古點解、強解或弱解。本文最大之特色是無需假設這個C-半群是指數有界(exponential bounded)

The main concern of this paper is under some suitable conditions on the forcing term and the operator A to find the unique classical solution, strong solution or mild solution for the abstract semilin-ear initial value problem:
d/dt u(t)=Au(t)+f(t,u(t) 0≤t0
u(t0)=u0 u0∈C(D(A))
where A is an infinitesimal generator of a C-semigroup {T(t):t ≥ 0}, f:[t0，T] x X->X and X is a Ba-nach space. We also discussed the maximum interval of the existence for the mild solutions and continuous dependence of initial data. The basic technique used in this paper is the fixed point theory for differential equations in Banach space. For this purpose, we prove firet that the corresponding inhomoge-neous equation
d/dt u(t)=Au(t) +f(t) 0≤t0
u(t0)=u0 U0 ∈ C(D(A))
has a unique classical solution, strong solution or mild solution. However, the most enjoy here is that we do not need to assume that the C-semigroup is exponentially bounded.