Relationship between integral equations and partial differential equations In my functional analysis class, we did a lot of problems involving integral equations such as proving existence & uniqueness using spectral theory and the Banach fixed point theorem.
I've never seen integral equations outside of functional analysis, but apparently they are useful for ordinary/partial differential equations. If someone familiar with integral equation methods could give some motivation, I would really appreciate it.
Also, are there any good textbooks that discuss integral equation methods for PDE's?
 A: Here's a classic motivating example. Let $H$ be the Heaviside unit step function
$$
H(x) = \begin{cases} 1 & x > 0 \\ 0 & x \leq 0 \end{cases},
$$
and suppose we want to solve the differential equation $$u' = H.$$ We attack each part of $H$ separately:
\begin{align*}
x > 0 &\implies u(x) = x + c_0 &\mbox{ for some } c_0 \in \mathbb R, \\
x \leq 0 &\implies u(x) = c_1 &\mbox{ for some } c_1 \in \mathbb R.
\end{align*}
Now our differential equation has $u'$ in it; this means that $u$ is differentiable, and hence $u$ is continuous. Continuity of $u$ at $0$ enforces the constraint $c_0 = c_1$. Hence, letting $c := c_0 = c_1$, we get
$$
u(x) = \begin{cases} x + c & x > 0 \\ c & x \leq 0 \end{cases}.
$$
We've run into the problem: we cannot differentiate $u$ at $0$, as we get $1$ from above and $0$ from below. Hence there is no differentiable function $u$ satisfying $u' = H$. Uh oh.
Let's see the magic that happens if we turn our differential equation into an integral equation by first multiplying by a "nice" test function $\varphi$ and integrating. By nice, we require that we can take derivatives of $\varphi$, and that $\varphi$ vanishes at infinity. Our integral equation is
$$
\int_{\mathbb R} \varphi u' = \int_{\mathbb R} \varphi H.
$$
Let's see if $u$ works in this equation. Starting with the LHS, and integrating by parts
\begin{align*}
\varphi u \big|_{-\infty}^\infty  - \int_{\mathbb R} u \varphi' &= - \int_{\mathbb R} u \varphi ' \\
&= - \int_{-\infty}^0 c \varphi' - \int_0^\infty (x + c) \varphi' \\
&=- \int_0^\infty x \varphi' \\
&=- x \varphi \big|_{0}^\infty + \int_0^\infty \varphi \\
&= \int_0^\infty \varphi \\
&= \int_{\mathbb R} \varphi H,
\end{align*}
where we've integrated by parts again and used our "nice" properties of $\varphi$.
The upshot: $u$ did not work for the differential equation, but it did work for the integral equation. This is the point of using integral equations instead: they allow for greater possibility of the existence of solutions because solutions aren't required to have such stringent regularity conditions (like differentiability and continuity). We "shift" those requirements onto the "nice test function" $\varphi$.
A: Contractible mapping principles and Banach fixed point theorems have their direct application to the systems of linear equation, partial differential equation and INTEGRAL equation. Their applications are interwoven, since every contraction mapping has a fixed point.
