Two fixed point theorems in Banach space

Here is a famous fixed point theorem in finite dimension by Brouwer:

Theorem 1 (Brouwer fixed point theorem) Let {M \subset {\mathbb R}^n} be a convex compact set, for any continuous function {f : M \rightarrow M}, there exists a point {x_0 \in M} such that {f (x_0) = x_0}.

There are couple of ways to extend this theorem to Banach spaces. First recall that a mapping between two Banach spaces is called compact, if the mapping is continuous (not necessarily linear) and the images of bounded sets are pre-compact.

Lemma 2 Let {X} be a Banach space, {M \subset X} be a bounded nonempty set, and {T : M \rightarrow X} be a compact operator. Then there exists a sequence of continuous operator {T_k : M \rightarrow X} such that

\displaystyle \sup_{x \in M} \|T_k(x) - T(x)\| \le \frac{1}{k}.

Proof: Since {T(M)} is relatively compact, there exist {N = N(k)} balls {\{B(x_i)\}_{i = 1}^N} of radius {\frac{1}{k}} that cover {T(M)}, with {x_i \in T(M)} for all {i}. Consider {T_k : M \rightarrow X} defined by

\displaystyle T_k (x) := \frac{\sum_{i=1}^N dist (T(x) , T(M) \setminus B(x_i)) x_i}{\sum_{i=1}^N dist (T(x) , T(M) \setminus B(x_i))}.

One can easily check the denominator has a positive lower bound, and both numerator and denominator are continuous (since {T} is continuous and the composition of two continuous functions is continuous). Hence {T_k} is continuous. One can also easily check that the denominator is positive only when {T(x) \in B(x_i)} for some {i}, i.e. {\| T(x) - x_i\| \le \frac{1}{k}}. Therefore

\displaystyle |T_k(x) - T(x)| \le \frac{\sum_{i=1}^N dist (T(x) , T(M) \setminus B(x_i)) \| x_i - T(x)\|}{\sum_{i=1}^N dist (T(x) , T(M) \setminus B(x_i))} \le \frac{1}{k}.

\Box

Theorem 3 (Schauder fixed point theorem) Let {X} be a Banach space, {M} be a bounded closed convex set. If {T : M \rightarrow M} is a compact operator, then there exists an {x_0 \in M} such that {T(x_0) = x_0}.

Proof: Let {T_k} be the sequence of continuous operator from the previous lemma. Let {X_k} be the convex hull of {\{x_1 , \cdots, x_{N(k)} \}}, the center of covering balls. Set {M_k = X_k \cap M}, then by the definition of {T_k}, one can see {T_k} maps {M_k} into itself. Notice that {M_k} is finite-dimensional, convex and compact, we can apply Brouwer fixed point theorem to obtain a fixed point for {T_k}, say {y_k}.

Since {y_k \in T(M)} and {T(M)} is pre-compact, there exists a subsequence, still denote by {y_k}, that converges to {x_0 \in M}. Then

\displaystyle \|x_0 - T(x_0)\| = \lim_{k\rightarrow \infty} \|y_k - T(y_k) \| =\lim_{k\rightarrow \infty} \| T_k(y_k) - T(y_k)\| \le \lim_{k\rightarrow \infty} \frac{1}{k} = 0.

\Box

Remark 1 From the proof we can see the boundedness of {M} and compactness of {T} have only been used to deliver the pre-compactness of {T(M)}. So if we know {T(M)} is pre-compact, we can drop those 2 conditions, and of course, {T} still needs to be continuous.

Theorem 4 (Leray-Schauder fixed point theorem) Let {X} be a Banach space, {T} be a compact operator from {X} into itself, and suppose there is an a-priori bound

\displaystyle \|x\| < M

for all {x \in X} and {\sigma \in [0,1]} satisfying {x = \sigma T(x)}. Then {T} has a fixed point.

Proof: Define a mapping {T^* : \bar{B}_M \rightarrow \bar{B}_M} by

\displaystyle T^* (x) = \left\{ \begin{aligned} &T(x), &\text{if } \|T(x)\| \le M,\\ &\frac{M T(x)}{\|T(x)\|}, &\text{if } \|T(x)\| \ge M. \end{aligned} \right.

Then {T^*} is a continuous mapping. Since {T} is compact, so is {T^*} (one can test against a bounded sequence to see this). By Schauder fixed point theorem, there exists a fixed point {x} for {T^*}. It remains to show {x} is also a fixed point for {T}. Assume {\|T(x)\| \ge M}, then {x = T^*(x) = \sigma T(x)}, with {\sigma = \frac{M}{\|T(x)\|}}. And we will have {\|x\| = \|T^*(x)\| = M}, which contradicts the a-priori estimate. Therefore {\|T(x)\| < M} and {x = T^*(x) =T(x)}. \Box

Remark 2 None of the above fixed point theorems give uniqueness of fixed point, which is different from Banach (contracting) mapping theorem.

Remark 3 If one can reduce an equation to a fixed point problem of a compact operator (especially for integral equation), an a-priori estimate of the type above will give the existence of solution through Leray-Schauder fixed point theorem, but not the uniqueness!

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s