Differential equations in Banach spaces

Remarks

Example: if $A$ is a bounded operator on $X$, then $u(t) = e^{tA} u_0 = \sum_{k \ge 0} \frac{t^k}{k!} A^k u_0$ solves the abstract Cauchy problem on $\R$. The series converges in operator norm since $\norm{A^k} \le \norm{A}^k$, and term-by-term differentiation gives $u' = A u$. For $X = \R^n$ this is the matrix exponential solution of a linear system of ordinary differential equations.

Non-example: take $X = C([0, 1])$ with the supremum norm and $u(t) = (s \mapsto \abs{s - t})$ for $t \in [0, 1]$. For each fixed $s$, the number $u(t)(s)$ has a derivative in $t$ for $t \neq s$, equal to $-\sgn(s - t)$. But the difference quotients $\frac{u(t + h) - u(t)}{h}$ do not converge in the supremum norm: their pointwise limit $s \mapsto -\sgn(s - t)$ is not continuous for $0 < t < 1$. So $u$ is not differentiable as a map into $X$, and it is not a classical solution of any differential equation in $X$.

When $A$ is closed but unbounded, the Cauchy problem is well posed exactly when $A$ generates a strongly continuous semigroup $T(t)$, and then $u(t) = T(t) u_0$.