610 CHAPTER 22. HILBERT SPACES

Next suppose A is the generator of a semigroup S (t) having ∥S (t)∥ ≤ M. Then byProposition 22.8.5 for all λ > 0,(λ I−A) is onto and (λ I−A)−1 =

∫∞

0 e−λ tS (t)dt. Thus,∥∥∥((λ I−A)−1)n∥∥∥

=

∥∥∥∥∫ ∞

0· · ·∫

∞

0e−λ (t1+···+tn)S (t1 + · · ·+ tn)dt1 · · ·dtn

∥∥∥∥≤

∫∞

0· · ·∫

∞

0e−λ (t1+···+tn)Mdt1 · · ·dtn =

Mλ

n . ■

22.8.3 An Evolution EquationWhen Λ generates a continuous semigroup, one can consider a very interesting theoremabout evolution equations of the form y′−Λy = g(t) provided t→ g(t) is C1.

Theorem 22.8.11 Let Λ be the generator of S (t) , a continuous semigroup on X , aBanach space and let t→ g(t) be in C1 (0,∞;X). Then there exists a unique solution to theinitial value problem y′ = Λy+g, y(0) = y0 ∈ D(Λ) and it is given by

y(t) = S (t)y0 +∫ t

0S (t− s)g(s)ds. (22.44)

This solution is continuous having continuous derivative and has values in D(Λ).

Proof: First I show the following claim.Claim: For t > 0,

∫ t0 S (t− s)g(s)ds ∈ D(Λ) and

Λ

(∫ t

0S (t− s)g(s)ds

)= S (t)g(0)−g(t)+

∫ t

0S (t− s)g′ (s)ds

Proof of the claim:

1h

(S (h)

∫ t

0S (t− s)g(s)ds−

∫ t

0S (t− s)g(s)ds

)

=1h

(∫ t

0S (t− s+h)g(s)ds−

∫ t

0S (t− s)g(s)ds

)=

1h

(∫ t−h

−hS (t− s)g(s+h)ds−

∫ t

0S (t− s)g(s)ds

)

=1h

∫ 0

−hS (t− s)g(s+h)ds+

∫ t−h

0S (t− s)

g(s+h)−g(s)h

−1h

∫ t

t−hS (t− s)g(s)ds

Using the estimate in Theorem 22.8.3 on Page 603, the triangle inequality and the uniformconvergence of the integrands, the limit as h→ 0 of the above equals

S (t)g(0)−g(t)+∫ t

0S (t− s)g′ (s)ds