166 CHAPTER 6. FIXED POINT THEOREMS

2. For every r > 0, there exists a solution to x = t f (x) for t ∈ (0,1) such that ∥x∥> r.

Proof: Suppose there is t0 ∈ [0,1] such that t0 f has no fixed point. Then t0 ̸= 0. Ift0 = 0, then t0 f obviously has a fixed point. Thus t0 ∈ (0,1]. Then let rM be the radialretraction onto B(0,M).

rM f (x) = Mf (x)∥ f (x)∥

By Schauder’s theorem there exists x ∈ B(0,M) such that t0rM f (x) = x. Then if ∥ f (x)∥ ≤M, rM has no effect and so t0 f (x) = x which is assumed not to take place. Hence ∥ f (x)∥>M and so ∥rM f (x)∥ = M so ∥x∥ = t0M. Also t0rM f (x) = t0M f (x)

∥ f (x)∥ = x and so x =

t̂ f (x) , t̂ = t0 M∥ f (x)∥ < 1. Since M is arbitrary, it follows that the solutions to x = t f (x) for

t ∈ (0,1) are unbounded. It was just shown that there is a solution to x = t̂ f (x) , t̂ < 1 suchthat ∥x∥= t0M where M is arbitrary. Thus the second of the two alternatives holds. ■

As an example of the usefulness of the Schauder fixed point theorem, consider thefollowing application to the theory of ordinary differential equations. In the context of thistheorem, X =C ([0,T ] ;Rn), a Banach space with norm given by

∥x∥ ≡max{|x(t)| : t ∈ [0,T ]} .

I assume the reader knows about the Riemann integral in what follows and the elementaryfundamental theorem of calculus. More general versions of these things are presented laterin the book.

Theorem 6.4.5 Let f : [0,T ]×Rn → Rn be continuous and suppose there existsL > 0 such that for all λ ∈ (0,1), if

x′ = λf (t,x) , x(0) = x0 (6.9)

for all t ∈ [0,T ], then ∥x∥< L. Then there exists a solution to

x′ = f (t,x) , x(0) = x0 (6.10)

for t ∈ [0,T ].

Proof: Let F : X → X where X described above.

Fy (t)≡∫ t

0f (s,y (s)+x0)ds

Let B be a bounded set in X . Then |f (s,y (s)+x0)| is bounded for s ∈ [0,T ] if y ∈ B. Say|f (s,y (s)+x0)| ≤CB. Hence F (B) is bounded in X . Also, for y ∈ B,s < t,

|Fy (t)−Fy (s)| ≤∣∣∣∣∫ t

sf (s,y (s)+x0)ds

∣∣∣∣≤CB |t− s|

and so F (B) is pre-compact by the Ascoli Arzela theorem. By the Schaefer fixed pointtheorem, there are two alternatives. Either there are unbounded solutions y to

λF (y) = y