Banach's Fixed Point theorem extra condition My question doesn't concern Banach's fixed point directly but rather an extra condition that I have seen pop up at different times, but I can't remember where.
Let $T:X\rightarrow X$ be the contraction from Banach's fixed point theorem where we assume $X$ is also a normed space with the norm ||.||.
Then the extra condition for $f\in X$:$||Tf||\leq||f||$ is often shown.
Of course if one verifies this condition one obtains that $T$ indeed maps $X$ into $X$, however if we already know this what does this condition ensure?
I'm asking from a differential equations viewpoint. If I remember correctly this condition helped with achieving a continuation of a solution in time, i.e. rule out blow-ups.
I am sorry if this question is vague, but I was just wandering what I had previously used this condition for?
 A: This question is probably motivated by a typical application of the contraction mapping principle to PDEs. Now I am going to just make up a toy model, to illustrate the main points of this argument; these points are marked by blackboard font. The answer to the question lies in the last point.

Consider the initial value problem
$$\tag{1}
\begin{cases} 
\partial_t u = \Delta u +u^2, & t\in (0, T), x\in \mathbb R^n, \\
u(0, x)=u_0(x).
\end{cases}
$$
We want to establish that, if $T$ is sufficiently small, then there is a solution $u\in C(0, T; L^{\infty}(\mathbb R^n))$. This is the Banach space of functions $u=u(t,x)$ that are bounded in $x$ for all $t\in (0, T)$, and such that the map
$$
t\in (0, T)\mapsto u(t, \cdot)\in L^\infty(\mathbb R^n)$$
is continuous.
This is not the only evolution space, but it certainly is the most natural; we can afford $u(t, x)$ to be defined for almost all $x$, but we must require it to be defined for all $t$, for the initial condition $u(0, x)=u_0$ to make sense. Thus, a space such as $L^p(0, T; L^{\infty}(\mathbb R^n))$ is not good, because its elements are defined only for almost all $t$. This is why we chose a space of continuous functions of $t$. With this argument, we have chosen the ambient Banach space.
Now we need to choose the candidate contraction mapping. Here, the input comes from the classical theory of ODEs; we define
$$
\Phi_{u_0}(u) = e^{t\Delta}u_0 +\int_0^t e^{(t-s)\Delta}(u^2(s, \cdot))\,ds,$$
because, if $v(t, x)$ is an arbitrary function, then $\Phi_{u_0}(v)$ solves the initial value problem
$$
\begin{cases} 
\partial_t u = \Delta u + v^2, \\ 
u(0, x)=u_0(x).
\end{cases}
$$
We have given a formal definition of our candidate contraction mapping.
To finish the setup of the functional framework, we need one more input. We gave a formal definition of $\Phi_{u_0}$, we established that it must be defined in some closed subset $X$ of the Banach space $E:=C(0, T;L^\infty)$, but we still haven’t specified such subset. To do so, we observe that, since $\lVert e^{t\Delta}f\rVert_\infty\le \lVert f\rVert_\infty$, then
$$\tag{2}
\lVert \Phi_{u_0}(u)\rVert_E \le \lVert u_0\rVert_\infty + T\lVert u\rVert_E^2.$$
And this suggests the definition of $X$. We let
$$
X=\{v\in E\ :\ \lVert v\rVert_E\le R\}.$$
I can now, finally, answer the question. Why is this a good choice for $X$? First, it is a complete metric space. Second, by (2) we see that
$$
\lVert\Phi_{u_0}(u)\rVert_E \le \lVert u_0\rVert_\infty +TR^2,\quad \forall u\in X, $$
thus if $R=2\lVert u_0\rVert_\infty$, then for all
$$
T\le \frac{1}{4\lVert u_0\rVert_\infty}, $$
we have that $\Phi_{u_0}$ maps $X$ into $X$. Notice that what we did with (2) is exactly what the question asks; controlling
$$
\lVert \Phi_{u_0}(u)\rVert_E \text{ in terms of }\lVert u\rVert_E.$$
We have found the correct domain for the mapping $\Phi_{u_0}$.
We are not done. We still need to prove that $\Phi_{u_0}$ is a contraction mapping. I am not going to do so; it may be that this requires taking a smaller $T$. If somebody is interested then they should look into some book in parabolic PDEs, such as the one of Quittner and Souplet, which is the one I used to write this answer. (P. Quittner, P. Souplet, “Superlinear parabolic problems”, Theorem 15.2, https://www.springer.com/gp/book/9783764384425).
