18.5. THE TYCHONOFF AND SCHAUDER FIXED POINT THEOREMS 499

Thusf (x)− fr (x) = ∑

{i: f (x)∈B(yi,r)}( f (x)− yi)ψ i ( f (x))

+ ∑{i: f (x)/∈B(yi,r)}

( f (x)− yi)ψ i ( f (x))

= ∑{i: f (x)−yi∈B(0,r)}

( f (x)− yi)ψ i ( f (x)) =

∑{i: f (x)−yi∈B(0,r)}

( f (x)− yi)ψ i ( f (x))+ ∑{i: f (x)/∈B(yi,r)}

0ψ i ( f (x)) ∈ B(0,r)

because 0 ∈ B(0,r), B(0,r) is convex, and 18.5.18. It is just a convex combination ofthings in B(0,r).

Note that we could have had the yi in f (K) in addition to being in f (K). This wouldmake it possible to eliminate the assumption that K is closed later on. All you really needis that K is convex.

We think of fr as an approximation to f . In fact it is uniformly within r of f on K. Thenext lemma shows that this fr has a fixed point. This is the main result and comes from theBrouwer fixed point theorem in Rn. It is an approximate fixed point.

Lemma 18.5.3 For each r > 0, there exists xr ∈ convex hull of f (K)⊆ K such that

fr (xr) = xr, ∥ fr (x)− f (x)∥< r for all x

Proof: If fr (xr) = xr and

xr =n

∑i=1

aiyi

for ∑ni=1 ai = 1 and the yi described in the above lemma, we need

fr (xr) =n

∑i=1

yiψ i ( f (xr)) =n

∑j=1

y jψ j

(f

(n

∑i=1

aiyi

))=

n

∑j=1

a jy j = xr.

Also, if this is satisfied, then we have the desired fixed point.This will be satisfied if for each j = 1, · · · ,n,

a j = ψ j

(f

(n

∑i=1

aiyi

)); (18.5.20)

so, let

Σn−1 ≡

{a ∈ Rn :

n

∑i=1

ai = 1, ai ≥ 0

}and let h : Σn−1→ Σn−1 be given by

h(a) j ≡ ψ j

(f

(n

∑i=1

aiyi

)).