602 CHAPTER 22. HILBERT SPACES

where θ n (h) ∈ (0,h). Then taking a limit as h→ 0 and using the dominated convergencetheorem, the limit of the difference quotient is

∞

∑n=1

Antn−1

(n−1)!= A

∞

∑n=1

An−1tn−1

(n−1)!= A

∞

∑n=0

(At)n!

n

Thus ∑∞n=0

(At)n!

nsatisfies the differential equation. It clearly satisfies the initial condition.

Hence it equals S (t). ■Note that as a consequence of the above argument showing that T and S are the same, it

follows that T (t)A=AT (t) so one obtains that if the generator is a bounded linear operator,then the semigroup commutes with this operator.

When dealing with differential equations, one of the best tools is Gronwall’s inequality.This is presented next.

Theorem 22.7.6 Suppose u is nonnegative, continuous, and real valued and that

u(t)≤C+∫ t

0ku(s)ds, k ≥ 0

Then u(t)≤Cekt .

Proof: Let w(t)≡∫ t

0 ku(s)ds. Then

w′ (t) = ku(t)≤ kC+ kw(t)

and so w′ (t)− kw(t)≤ kC which implies ddt

(e−ktw(t)

)≤ kCe−kt . Therefore,

e−ktw(t)≤Ck∫ t

0e−ksds =Ck

(1k− 1

ke−kt

)so w(t)≤C

(ekt −1

). From the original inequality, u(t)≤C+w(t)≤C+Cekt−C =Cekt .

■

22.8 General Theory of Continuous SemigroupsMuch more on semigroups is available in Yosida [60]. This is just an introduction to thesubject.

22.8.1 Generators of Semigroups

Definition 22.8.1 A strongly continuous semigroup defined on X ,a Banach spaceis a function S : [0,∞)→ X which satisfies the following for all x0 ∈ X .

S (t) ∈ L (X ,X) ,S (t + s) = S (t)S (s) ,

t → S (t)x0 is continuous, limt→0+

S (t)x0 = x0

Sometimes such a semigroup is said to be C0. It is said to have the linear operator A as itsgenerator if

D(A)≡{

x : limh→0

S (h)x− xh

exists}