AFA The Brouwer and Schauder fixed-point theorems

Section 7.2 The Brouwer and Schauder fixed-point theorems

The Brouwer fixed point theorem is a central piece in algebraic topology in finite dimensions \(\RR^n\text{.}\) It is equivalent to the negative retraction principle, and this is where we start our arguments. The Brouwer fixed point theorem is a rather general result for finite dimensional spaces. We lift it to the infinite dimensional setting in Schauders fixed-point theorem, where we employ some sense of compactness.

Definition 7.2.1.

Let \(X\) be a topological space and \(T:X\to M\) a continuous map with \(M\subset X\text{.}\) \(T\) is called a retraction on \(M\) if \(T(x)=x\) for all \(x\in M\text{.}\) In this case \(M\) is called a retract of \(X\text{.}\)

Example 7.2.2.

For example \(X=\RR^n, M=B_R(0)\) and

\begin{equation} T(x) =\left\{\begin{array}{cc} x \amp \qquad \mbox{if} \quad x\in B_R(0), \\ R\frac{x}{|x|} \amp\qquad \mbox{if} \quad x\not\in B_R(0), \end{array}\right.\tag{7.2.1} \end{equation}

is a continuous retraction of \(X\) to \(M\text{.}\)

Theorem 7.2.3.

In \(\RR^n\) there is no continuous map \(T:\bar B_1(0) \to \partial B_1(0)\text{,}\) which leaves the boundary points fixed, i.e.

\begin{equation*} T(x) =x, \qquad \mbox{for all }\quad \quad x\in \partial B_1(0). \end{equation*}

Proof.

See textbooks on topology, for example \cite{runde}.

Theorem 7.2.4.

Each continuous map \(T:\bar B_1(0)\to \bar B_1(0) \) in \(\RR^n\) has a fixed point.

Proof.

Assume otherwise, i.e. \(T(x)\neq x\) for all \(x\in \bar B_1(0)\text{.}\) Then we construct a continuous map from the ball to the boundary \(r:\bar B_1(0) \to \partial B_1(0)\) as follows. For each \(x\in B_1(0)\) we follow the line segment from \(T(x)\) to \(x\) to the boundary. Since \(T(x)\neq x\) for all \(x\text{,}\) this line segment is well defined, and it has a unique intersection with the boundary, called \(r(x)\text{.}\) This map is continuous, since \(T\) is continuous, and it satisfies \(r(x) = x\) for all \(x\in \partial B_1(0)\text{.}\) This is a contradiction to the negative retract principle in Theorem Theorem 7.2.3.

Corollary 7.2.5.

If \(M\subset \RR^n\) is homeomorphic to \(\bar B_1(0)\text{,}\) then each continuous map \(T:M\to M\) has a fixed point.

Proof.

Let \(\phi:M\to \bar B_1(0)\) denote the homeomorphism and \(\phi^{-1}\) its continuous inverse. Then we apply Brouwers fixed point theorem to the conjugate map

\begin{equation*} f:=\phi\circ T \circ \phi^{-1}: \bar B_1(0) \to \bar B_1(0). \end{equation*}

Example 7.2.6.

The previous corollary includes nonempty convex compact sets in \(\RR^n\text{,}\) star-shaped domains, and \(p\)-norms.

Before we proceed to Schauder's fixed point theorem, we need a technical Lemma about convex combinations.

Proposition 7.2.7.

Let \(K\Subset X\) be a compact subset and let \(\ep>0\) be given. Let \(A\) denote a finite set \(A\subset K\) such that

\begin{equation*} K\subset \bigcup_{a\in A} B_\ep(a) \end{equation*}

is a finite open covering of \(K\text{.}\) We define a convex combination of the elements of \(A\) as a map \(\phi_A:K\to K\) as

\begin{equation} \phi_A(x) = \frac{\sum_{a\in A} m_a(x) \; a}{\sum_{a\in A} m_a(x)}, \quad \mbox{where}\quad m_a(x) =\left\{\begin{array}{ll} 0 \amp \quad \mbox{for } \|x-a\|\geq \ep \\ \ep - \|x-a\| \amp \quad \mbox{for } \|x-a\|<\ep\end{array}\right.. \tag{7.2.2} \end{equation}

The function \(\phi_A(x)\) denotes a weighted average of the anchor points \(a_i\in A\text{,}\) weighted by the distance to \(x\text{.}\) Then \(\phi_A(x)\) is continuous on \(K\) and for each \(x\in K\) we have

\begin{equation*} \|\phi_A(x) - x\|<\ep. \end{equation*}

Proof.

Note that \(m_a(x)\geq 0\) and for each \(x\in K\) there is at least one \(a\in A\) such that \(x\in B_\ep(a)\text{.}\) Hence \(\sum_{a\in A} m_a(x) >0\) and \(\phi_A(x)\) is well defined on \(K\text{.}\) The map \(m_a:K\to [0,\ep]\) is continuous, hence \(\phi_A\) is continuous on \(K\text{.}\) Now, if \(x\in K\) then

\begin{equation*} \phi_A(x) - x = \frac{\sum_{a\in A} m_a(x) (a-x)}{\sum_{a\in A} m_a(x) }, \end{equation*}

and for those \(m_a\) with \(m_a(x)>0\) we have \(\|x-a\|<\ep\text{.}\) Hence

\begin{equation*} \|\phi_A(x) - x\|\leq \frac{\sum_{a\in A} m_a(x) \; \ep }{\sum_{a\in A} m_a(x)} = \ep. \end{equation*}

Theorem 7.2.8.

Let \(X\) be a Banach space and \(M\subset X\) a nonempty, bounded, and convex subset. Consider a continuous map \(T:M\to M\text{.}\)

If \(M\) is compact, then \(T\) has a fixed point.
If \(T\) is compact, then \(T\) has a fixed point.

Proof.

We define \(K=\overline{T(M)}\text{.}\) In both cases the set \(K\) is compact and for each \(n\in \NN\) we can find a finite covering with balls of radius \(1/n\text{:}\)

\begin{equation*} K \subset \bigcup_{a\in A_n} B_{\frac{1}{n}} (a), \end{equation*}

where \(A_n\) is a finite set. We use the above function \(\phi_A\) to define \(\phi_{A_n}\text{.}\)

The element \(\phi_{A_n}(x)\in X\) is a convex combination of elements \(a\in A_n\text{,}\) hence

\begin{equation*} \phi_n(K)\subset \mbox{convex hull}(K) \subset M, \end{equation*}

since \(T:M\to M\) and \(M\) is convex. Then we define maps \(T_n = \phi_{A_n} \circ T: M\to M\) with

\begin{equation*} \|T_n(x) - T(x) \|\leq \frac{1}{n}, \end{equation*}

by Proposition Proposition 7.2.7.

We define \(M_n := M \cap \mbox{span}(A_n) \text{.}\) Since \(\mbox{span}(A_n)\) is a finite dimensional subspace, the sets \(M_n\) are a bounded, closed and convex subsets of a finite dimensional subspace, and \(T_n:M_n \to M_n \) is continuous. By the Corollary to Brouwers fixed point theorem (Corollary Corollary 7.2.5) we have a fixed point \(x_n\) for each \(n\text{:}\) \(T_n(x_n)=x_n\text{.}\)

If \(M\) is compact, then the bounded sequence \(\{x_n\}_n\) has a convergent subsequence
\begin{equation*} x_{n_j}\to x, \qquad \mbox{for}\quad j\to \infty, \end{equation*}
and since \(T\) is continuous we have \(T(x)=x\text{.}\)
If \(T\) is compact, then \(\{T(x_n)\}_n\) has a convergent subsequence, and the result is the same as under item 1.

Example 7.2.9.

[Elliptic equations] Schauder theory is an important tool to solve elliptic partial differential equations. Please see the comprehensive introduction \cite{GT}.