In many areas of analysis and applied mathematics, we encounter equations of the form $x = T(x)$. A solution to this equation is called a **fixed point** of the mapping $T$. The Contraction Mapping Principle (also known as the [Banach Fixed Point Theorem](/theorems/289)) is a powerful tool because it guarantees not only the *existence* and *uniqueness* of such a fixed point but also provides a constructive method to find it: simple iteration. This principle underpins the proofs of the Picard-Lindelöf theorem for differential equations, the [Inverse Function Theorem](/page/Inverse%20Function%20Theorem), and numerous iterative numerical methods.
## Formal Definition
We formulate the principle in the setting of [metric spaces](/page/Metric%20Space).
[definition: Contraction Mapping]
Let $(X, d)$ be a metric space. A mapping $T: X \to X$ is called a **contraction mapping** (or simply a contraction) if there exists a constant $k \in \mathbb{R}$ with $0 \le k < 1$ such that for all $x, y \in X$:
\begin{align*}
d(T(x), T(y)) \le k d(x, y).
\end{align*}
The constant $k$ is called the Lipschitz constant or contraction factor.
[/definition]
The main result states that on a complete metric space, every contraction has a unique fixed point.
[quotetheorem:71]
We note that some of the conditions can be relaxed slightly to give us
[quotetheorem:79]
The completeness of $X$ plays a crucial role in the contraction mapping principle. Indeed, contractions on incomplete metric spaces may fail to have fixed points. For example, define $f:X\rightarrow X$ as $x\mapsto \frac{x}{2}$. Then $f$ is a contraction on $X$ but has no fixed point in $X$.
## Applications
### In ODES
The most ovious is
[quotetheorem:69]
### In PDES
Let $\Omega \subset \mathbb{R}^3$ be a bounded domain with a smooth [boundary](/page/Boundary) $\partial\Omega$. For a fixed time $t_0 > 0$, we consider the semilinear parabolic problem. First, we explicitly define the nonlinearity $h$ and its regularity:
\begin{align*}
h: \mathbb{R} &\to \mathbb{R} \\
s &\mapsto \pm |s|^{p-1}s
\end{align*}
where $p \in [1, 5)$. The problem seeks an unknown function $u: [0, t_0] \times \Omega \to \mathbb{R}$ satisfying:
\begin{align*}
\partial_t u - \Delta u &= h(u) && \text{in } (0, t_0] \times \Omega \\
u &= 0 && \text{on } (0, t_0] \times \partial\Omega \\
u(0, x) &= u_0(x) && \text{for } x \in \Omega
\end{align*}
for a given initial datum $u_0 \in H_0^1(\Omega)$.
Before defining the solution concept for the semilinear problem, we recall the properties of the linear, non-homogeneous version. Consider the problem for $v: [0, t_0] \times \Omega \to \mathbb{R}$:
\begin{align*}
\partial_t v - \Delta v &= g && \text{in } (0, t_0] \times \Omega \\
v &= 0 && \text{on } (0, t_0] \times \partial\Omega
\end{align*}
where $g \in L^2(0, t_0; L^2(\Omega))$. Standard parabolic regularity theory ensures this problem admits a unique solution $v$ satisfying:
\begin{align*}
v \in C([0, t_0]; H_0^1(\Omega)) \cap L^2(0, t_0; H^2(\Omega))
\end{align*}
with the time [derivative](/page/Derivative) $\partial_t v \in L^2(0, t_0; L^2(\Omega))$. This motivates the following definition for the semilinear case.
[definition:Strong Solution]
We call a **strong solution** to the semilinear problem a [function](/page/Function) $u$ belonging to the class:
\begin{align*}
u \in C([0, t_0]; H_0^1(\Omega)) \cap L^2(0, t_0; H^2(\Omega))
\end{align*}
with $\partial_t u \in L^2(0, t_0; L^2(\Omega))$, such that $u(0) = u_0$ and the equation holds almost everywhere.
[/definition]
We then have the following theorem concerning the local existence and uniqueness of strong solutions, which will be discussed next.
## 1.3. Local Existence
We now establish the local existence of strong solutions for the semilinear problem defined in the previous section.
[theorem:LocalExistence]
For every radius $R \geq 0$, there exists a time parameter $t_0 = t_0(R) > 0$ such that the Cauchy problem (1.5) admits a unique strong solution, for every initial datum $u_0 \in H_0^1(\Omega)$ satisfying $\|u_0\|_{H^1} \leq R$.
[/theorem]
[proof]
**Step 1: Functional Framework**
For a time $t_0 > 0$ to be fixed subsequently, we define the solution space $X_{t_0}$ as:
\begin{align*}
X_{t_0} = C([0, t_0]; H_0^1(\Omega)) \cap L^2(0, t_0; H^2(\Omega))
\end{align*}
We consider the complete metric space defined by the closed ball $B \subseteq X_{t_0}$:
\begin{align*}
B = \{ v \in X_{t_0} : \|v\|_{X_{t_0}} \leq 4R \}
\end{align*}
We define the mapping $f$, which maps a function $w$ to the solution $u$ of the linearized problem:
\begin{align*}
f: B &\to X_{t_0} \\
w &\mapsto u
\end{align*}
where $u$ is the unique solution to the linear, non-homogeneous problem:
\begin{align*}
\partial_t u - \Delta u &= h(w) && \text{in } (0, t_0] \times \Omega \\
u &= 0 && \text{on } (0, t_0] \times \partial\Omega \\
u(0) &= u_0 && \text{in } \Omega
\end{align*}
To ensure $f$ is well-defined, we verify that for $w \in X_{t_0}$, the term $h(w)$ belongs to $L^2(0, t_0; L^2(\Omega))$. For $p \in [2, 5)$, we utilize the 3D Agmon inequality:
\begin{align*}
\|\varphi\|_{L^\infty}^2 \lesssim \|\varphi\|_{H^1} \|\varphi\|_{H^2}, \quad \forall \varphi \in H^2(\Omega) \cap H_0^1(\Omega)
\end{align*}
Combined with the Sobolev embedding $H_0^1(\Omega) \subseteq L^p(\Omega)$, we deduce that the linear problem admits a unique global solution. Thus, the map $f$ is well-defined. Our strategy relies on the Contraction Mapping Principle.
**Step 2: Contraction Estimate**
We demonstrate that $f$ is a contraction. Let $u, v \in B$ and define $w = u - v$. We analyze the difference $h(u) - h(v)$ using the [fundamental theorem of calculus](/theorems/632):
\begin{align*}
h(u) - h(v) = w \int_0^1 h'(\lambda u + (1 - \lambda)v) \, d\mathcal{L}^1(\lambda)
\end{align*}
We focus on the case $p \in [3, 5)$. Using the Hölder inequality, we estimate the $L^2$ norm of the difference. The Sobolev embedding $H_0^1(\Omega) \subseteq L^6(\Omega)$ and the Agmon inequality imply:
\begin{align*}
\|h(u) - h(v)\|_{L^2} \lesssim \|w\|_{H^1} (1 + \|u\|_{H^2} + \|v\|_{H^2})^{\frac{p-3}{2}} (1 + \|u\|_{H^1} + \|v\|_{H^1})^{\frac{p+1}{2}}
\end{align*}
Let $w_1, w_2 \in B$ and set $u_j = f(w_j)$. The difference $\bar{u} = u_1 - u_2$ solves:
\begin{align*}
\partial_t \bar{u} - \Delta \bar{u} &= h(w_1) - h(w_2)
\end{align*}
Taking the $L^2$ inner product with $\Delta \bar{u}$ yields the differential inequality:
\begin{align*}
\frac{d}{dt} \|\bar{u}\|_{L^2}^2 + \|\Delta \bar{u}\|_{L^2}^2 \lesssim \|\bar{w}\|_{X_{t_0}}^2 (1 + \|w_1\|_{H^2} + \|w_2\|_{H^2})^{p-3} (1 + R)^{p+1}
\end{align*}
Integrating with respect to time over $(0, t_0]$:
\begin{align*}
\|\bar{u}(t)\|_{L^2}^2 + \int_0^{t_0} \|\Delta \bar{u}(s)\|_{L^2}^2 \, d\mathcal{L}^1(s) \lesssim \|\bar{w}\|_{X_{t_0}}^2 (1 + R)^{p+1} \int_0^{t_0} (1 + \|w_1\|_{H^2} + \|w_2\|_{H^2})^{p-3} \, d\mathcal{L}^1(s)
\end{align*}
We apply the Hölder inequality with conjugate exponents $\frac{2}{5-p}$ and $\frac{2}{p-3}$ (noting $p < 5$) to the [integral](/page/Integral) term. This yields a bound involving $t_0^{5-p}$:
\begin{align*}
\int_0^{t_0} (1 + \|w_1\|_{H^2} + \|w_2\|_{H^2})^{p-3} \, d\mathcal{L}^1(s) \lesssim t_0^{\frac{5-p}{2}} Q(R + t_0)
\end{align*}
where $Q$ is an increasing positive function. Consequently, defining $\lambda(t_0)$ as:
\begin{align*}
\lambda(t_0) = C Q(R + t_0) t_0^{\frac{5-p}{4}}
\end{align*}
we obtain:
\begin{align*}
\|\bar{u}\|_{X_{t_0}} \leq \lambda(t_0) \|\bar{w}\|_{X_{t_0}}
\end{align*}
By choosing $t_0$ sufficiently small, we ensure $\lambda(t_0) \leq 1/2$.
**Step 3: Self-Mapping Property**
We now verify that $f$ maps $B$ into $B$. We apply the previous argument with $w_1 = w$ (where $w \in B$) and $w_2 = 0$. We observe that $f(0) = v$, where $v$ is the solution to the linear homogeneous problem (1.6) with $g=0$. Since $h(0) = 0$, we have:
\begin{align*}
\|v\|_{X_{t_0}} \leq 2R
\end{align*}
Using the triangle inequality and the contraction estimate (with $\lambda \leq 1/2$):
\begin{align*}
\|f(w)\|_{X_{t_0}} \leq \|f(w) - f(0)\|_{X_{t_0}} + \|f(0)\|_{X_{t_0}} \leq \lambda \|w\|_{X_{t_0}} + 2R \leq \frac{1}{2}(4R) + 2R = 4R
\end{align*}
Thus, we conclude $f(w) \in B$.
**Step 4: Conclusion**
We have demonstrated that $f: B \to B$ is a contraction mapping on a complete metric space. By the Contraction Mapping Principle, there exists a unique fixed point $\bar{u} = f(\bar{u})$, which corresponds to the unique strong solution of the Cauchy problem (1.5) on $[0, t_0]$.
[/proof]
