---
tag: 8V9J445R7N
kind: definition
name: Abstract Volterra equation
msc: [45N05, 45D05, 47D06]
defines: 1N7TZCPBF0
uses: [DND8CBEVXT, 06GZ956657, VM7B38PA87, X7GTF3D98F]
sources:
  - url: https://en.wikipedia.org/wiki/Volterra_integral_equation
---

## Definition

Let $X$ be a complex [[Banach space|06GZ956657]] and let $A \colon D(A) \subseteq X \to X$ be a [[closed linear operator|VM7B38PA87]].
Let $a \in L^1_{\mathrm{loc}}(\R_+)$ be a scalar [[locally integrable|X7GTF3D98F]] kernel on $\R_+ = [0, \infty)$, let $\tau > 0$, $J = [0, \tau]$, and let $f \in C(J; X)$.
The *abstract Volterra equation* with kernel $a$, operator $A$ and forcing $f$ is
$$u(t) = f(t) + \int_0^t a(t - s) \, A u(s) \, ds, \qquad t \in J.$$
A *strong solution* is a function $u \in C(J; X)$ such that $u(t) \in D(A)$ for all $t \in J$, $A u \in C(J; X)$, and the equation holds for all $t \in J$.

## 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|3VBRS94V1G]] 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 $1 + \int_0^t (1 + s) \, ds = 1 + t + t^2/2$.

## Explanations

### L1 human

It is a Volterra equation in which numbers are replaced by vectors of a Banach space and multiplication by a fixed number is replaced by an operator $A$.
The kernel $a$ weights the past, and the operator $A$ usually acts in the space variables, for instance as a Laplacian.