600 CHAPTER 22. HILBERT SPACES

Thanks to the fundamental theorem of calculus, there exists a solution to 22.26 if and onlyif it is a solution to the integral equation

x(t) = x0 +∫ t

t0F (s,x(s))ds (22.28)

Then we have the following theorem.

Theorem 22.7.3 Let 22.27 hold. Then there exists a unique solution to 22.26 inBC ([a,b] ;X).

Proof: Use the norm of 22.25 where γ ̸= 0 is described later. Let T : BC ([a,b] ;X)→BC ([a,b] ;X) be defined by

T x(t)≡ x0 +∫ t

t0F (s,x(s))ds

Then

∥T x(t)−Ty(t)∥X =

∥∥∥∥∫ t

t0F (s,x(s))ds−

∫ t

t0F (s,y(s))ds

∥∥∥∥≤ K

∫ t

t0∥x(s)− y(s)∥ds = K

∫ t

t0

∥∥∥(x(s)− y(s))eγ(s−a)e−γ(s−a)∥∥∥ds

≤ K∫ t

t0e−γ(s−a)ds∥x− y∥

γ= K

(e−γ(t−a)

−γ+

e−γ(t0−a)

γ

)∥x− y∥

γ

Therefore, letting γ < 0

eγ(t−a) ∥T x(t)−Ty(t)∥X ≤ K

(1−γ

+eγ(t−t0)

γ

)∥x− y∥

γ< K

(1|γ|

)∥x− y∥

γ

∥T x−Ty∥γ≤ K

(1|γ|

)∥x− y∥

γ

Letting γ =−m2, this reduces to

∥T x−Ty∥−m2 ≤Km2 ∥x− y∥−m2

and so if K/m2 < 1/2, this shows the solution to the integral equation is the unique fixedpoint of a contraction mapping defined on BC ([a,b] ;X). This shows existence and unique-ness of the initial value problem 22.26. ■

Definition 22.7.4 Let S : [0,∞)→L (X ,X) be continuous and satisfy

1. S (t + s) = S (t)S (s) called the semigroup identity.

2. S (0) = I

3. limh→0+S(h)x−x

h = Ax for A a densely defined closed linear operator whenever x ∈D(A)⊆ X .