23.2. PROPERTIES OF THE DEGREE 763

so x→ gn (x, t) is in C∞(Ω;Rp

). Also,

∥gn (x, t)−hm (x,t)∥ ≤∫

B(0, 1n )∥h(x−u, t)−h(z, t)∥du < ε

provided n is large enough. This follows from uniform continuity of h on the compact setΩ× [a,b]. It follows that if n is large enough, one can replace hm in 23.2.5 with gn andobtain for large enough n

maxt∈[a,b]

∥gn (·, t)−h(·, t)∥∞,Ω < δ (23.2.7)

Now gn ∈C∞(Ω× [a,b] ;Rp

)because all partial derivatives with respect to either t or x are

continuous. Here

gn (x,t) =m

∑k=0

pk (t)h(·, tk)∗ψn (x)

Let τ ∈ (a,b]. Let gaτ (x, t) ≡ gn (x, t)−(

τ−tτ−a ya +yτ

t−aτ−a

)where ya is a regular value

of gn (·,a), yτ is a regular value of gn (x,τ) and both ya,yτ are so small that

maxt∈[a,b]

∥gaτ (·, t)−h(·, t)∥∞,Ω < δ . (23.2.8)

This uses Lemma 23.1.3. Thus if gaτ (x,τ) = 0, then gn (x,τ)− yτ = 0 so 0 is a regularvalue for gaτ (·,τ) . If gaτ (x,a) = 0, then gn (x, t) = ya so 0 is also a regular value forgaτ (·,a) . From 23.2.8, dist(gaτ (∂Ω× [a,b]) ,0) > 0 as in Lemma 23.1.5. Choosing ε <dist(gaτ (∂Ω× [a,b]) ,0), it follows from Lemma 23.1.12, the definition of the degree, andLemma 23.1.11 in the first and last equations that for ε small enough,

d (h(·,a) ,Ω,0) =∫

Ω

φ ε (gaτ (x,a))detD1gaτ (x,a)dx

=∫

Ω

φ ε (gaτ (x,τ))detD1gaτ (x,τ)dx = d (h(·,τ) ,Ω,0)

Since τ is arbitrary, this proves the theorem.Now the following theorem is a summary of the main result on properties of the degree.

Theorem 23.2.2 Definition 23.1.6 is well defined and the degree satisfies the followingproperties.

1. (homotopy invariance) If h ∈C(Ω× [0,1] ,Rp

)and y(t) /∈ h(∂Ω, t) for all t ∈ [0,1]

where y is continuous, then

t→ d (h(·, t) ,Ω,y(t))

is constant for t ∈ [0,1] .

2. If Ω⊇Ω1∪Ω2 where Ω1∩Ω2 = /0, for Ωi an open set, then if y /∈ f(Ω\ (Ω1∪Ω2)

),

thend (f,Ω1,y)+d (f,Ω2,y) = d (f,Ω,y)