Section 1.6 Fixed Point Methods
Now we come back to the reaction-diffusion equation ((1.2.1)). A solution is called a mild solution, if it satisfies the variation of constant formula ((1.2.2)). To show the existence of such a solution, we construct a Picard iteration and use a fixed point argument. Again, the question of the right function space arises. Let us assume \(X\) is the Banach space of interest. We take a given function \(v\in X\) and we define a nonlinear operator \(B\) as
\begin{equation*}
B(v) = e^{At} u_0 + \int_0^t e^{A(t-s)} f(v) ds.
\end{equation*}
If we can show that \(B:X\to X\text{,}\) then there is hope to apply a fixed point theorem. If \(B\) has a fixed point, \(u = B(u)\) in \(X\text{,}\) then this \(u\) is a mild solution of ((1.2.1)). We will learn in Chapter Chapter 7 a variety of fixed point theorems that apply to this situation.