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}

602 CHAPTER 22. HILBERT SPACESwhere 9, (A) € (0,/). Then taking a limit as h > 0 and using the dominated convergencetheorem, the limit of the difference quotient isoo Alte! co An I yn— 1 (At)"Ln Ay Gopr n—1)! =ay\— 1!Thus )yr 9 {ai)” satisfies the differential equation. It clearly satisfies the initial condition.Hence it equals S(t). aNote that as a consequence of the above argument showing that T and S are the same, itfollows 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 thattu(t) <c+/ ku(s)ds,k>00Then u(t) < Ce.Proof: Let w(t) = fj ku(s)ds. Thenw(t) =ku(t) <kC+kw(t)and so w’ (t) — kw (t) < kC which implies 4 (e~"w(t)) < kCe~. Therefore,dtt . 1 1,e w(t) < ck | eds =Ck( ——-e™JO k okso w(t) <C (eM — 1) . From the original inequality, w(t) <C-+w(t) <C+Ce” —C=Ce™.a22.8 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 SemigroupsDefinition 22.8.1 A strongly continuous semigroup defined on X,a Banach spaceis a function S : [0,0c) — X which satisfies the following for all xy € X.S(t) € Y(X,X),S(t+s) =S(t)S(s),t > S(t)xo is continuous, lim S(t) xo = xot0+Sometimes such a semigroup is said to be Co. It is said to have the linear operator A as itsgenerator ifh-0D(A)= fe lim Wace exists}