Abstract Volterra equation

Remarks

A mild solution is a function $u \in C(J; X)$ such that $\int_0^t a(t - s) u(s) \, ds \in D(A)$ for all $t \in J$ and $u(t) = f(t) + A \int_0^t a(t - s) u(s) \, ds$. Every strong solution is a mild solution, since $A$ is closed; compare the mild solution of an abstract Cauchy problem.

Special cases. For $a = 1$, a strong solution with $f \in C^1(J; X)$ satisfies $u' = A u + f'$ and $u(0) = f(0)$. For $a(t) = t$, differentiating twice gives $u'' = A u + f''$ with $u(0) = f(0)$ and $u'(0) = f'(0)$, when $f \in C^2(J; X)$.

Example: $X = \C$, $A = 1$, $a = 1$ and $f = 1$. The function $u(t) = e^t$ is a strong solution, since $1 + \int_0^t e^s \, ds = e^t$.

Non-example: in the same setting, $u(t) = 1 + t$ is not a solution, since $1 + \int_0^t (1 + s) \, ds = 1 + t + t^2/2$.