Kenneth Kuttler

6.4. THE SCHAUDER FIXED POINT THEOREM 165

Lemma 6.4.2 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 = ∑ni=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

∑i=1

aiyi

))=

∑j=1

a jy j = xr.

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

a j = ψ j

∑i=1

aiyi

)); (6.8)

so, let

Σn−1 ≡

{a ∈ Rn :

∑i=1

ai = 1, ai ≥ 0

}and let h : Σn−1→ Σn−1 be given by

h(a) j ≡ ψ j

∑i=1

aiyi

)).

Since h is a continuous function of a, the Brouwer fixed point theorem applies and thereexists a fixed point for h which is a solution to 6.8. ■

The following is the Schauder fixed point theorem.

Theorem 6.4.3 Let K be a closed and convex subset of X, a normed linear space.Let f : K→ K be continuous and suppose f (K) is compact. Then f has a fixed point.

Proof: Recall that f (xr)− fr (xr) ∈ B(0,r) and fr (xr) = xr with xr ∈ convex hull off (K)⊆ K.

There is a subsequence, still denoted with subscript r with r→ 0 such that f (xr)→ x ∈f (K). Note that the fact that K is convex is what makes f defined at xr. xr is in theconvex hull of f (K) ⊆ K. This is where we use K convex. Then since fr is uniformlyclose to f , it follows that f (xr) = xr→ x also. Therefore,

f (x) = limr→0

f (xr) = limr→0

fr (xr) = limr→0

xr = x. ■

We usually have in mind the mapping defined on a Banach space. However, the com-pleteness was never used. Thus the result holds in a normed linear space.

There is a nice corollary of this major theorem which is called the Schaefer fixed pointtheorem or the Leray Schauder alterative principle [22].

Theorem 6.4.4 Let f : X → X be a compact map. Then either

1. There is a fixed point for f for all t ∈ [0,1] or

6.4. THE SCHAUDER FIXED POINT THEOREM 165Lemma 6.4.2 For each r > 0, there exists x, € convex hull of f (K) © K such thatfr (xr) = Xr, Il, (x) — f (x) < r for all xProof: If f, (x-) =x; and x, = Y_, aiy; for ¥"_, aj = 1 and the y; described in the abovelemma, we needfr) = yoy, (f(s) = YW) G (Zen) = Yay, =x,I= J= i= j=Also, if this is satisfied, then we have the desired approximate fixed point.This will be satisfied if for each j = 1,--- ,n,aj=W; G (z ew] ) ; (6.8)i=lnYn-1 ={ecr:Za-ta>0]i=1so, letand let h: X,_1; — X,_1 be given byh(a); =Y; U (Zen)Since / is a continuous function of a, the Brouwer fixed point theorem applies and thereexists a fixed point for h which is a solution to 6.8.The following is the Schauder fixed point theorem.Theorem 6.4.3 Let K be a closed and convex subset of X, a normed linear space.Let f : K — K be continuous and suppose f (K) is compact. Then f has a fixed point.Proof: Recall that f (x,) — f; (x) € B(0,r) and f, (x,) =x; with x, € convex hull ofF(K) CK.___ There is a subsequence, still denoted with subscript r with r + 0 such that f (x,) > x €f (K). Note that the fact that K is convex is what makes f defined at x,. x, is in theconvex hull of f(K) C K. This is where we use K convex. Then since f, is uniformlyclose to f, it follows that f (x,) =x, > x also. Therefore,f (x) = lim f (x,) = lim f, (x,) = limx, =x.r>0 r>0 r—>0We usually have in mind the mapping defined on a Banach space. However, the com-pleteness was never used. Thus the result holds in a normed linear space.There is a nice corollary of this major theorem which is called the Schaefer fixed pointtheorem or the Leray Schauder alterative principle [22].Theorem 6.4.4 Ler f :X — X be acompact map. Then either1. There is a fixed point for f for all t € [0,1] or