22.8. GENERAL THEORY OF CONTINUOUS SEMIGROUPS 611

which proves the claim since the limit exists and is therefore, Λ(∫ t

0 S (t− s)g(s)ds).

Since y0 ∈ D(Λ) ,

S (t)Λy0 = S (t) limh→0

S (h)y0− y0

h= lim

h→0

S (t +h)−S (t)h

y0

= limh→0+

S (h)S (t)y0−S (t)y0

h≡ ΛS (t)y0 (22.45)

This is because the limit exists and so it is by definition the right side. So S (t)y0 ∈ D(Λ).Now consider 22.44.

y(t +h)− y(t)h

=S (t +h)−S (t)

hy0+

1h

(∫ t+h

0S (t− s+h)g(s)ds−

∫ t

0S (t− s)g(s)ds

)

=S (t +h)−S (t)

hy0 +

1h

∫ t+h

tS (t− s+h)g(s)ds

+1h

(S (h)

∫ t

0S (t− s)g(s)ds−

∫ t

0S (t− s)g(s)ds

)From the claim and 22.45, the limit of the right side is

ΛS (t)y0 +g(t)+Λ

(∫ t

0S (t− s)g(s)ds

)= Λ

(S (t)y0 +

∫ t

0S (t− s)g(s)ds

)+g(t)

Hence y′ (t) = Λy(t)+g(t) and from the formula, y′ is continuous since by the claim and22.45 it also equals

S (t)Λy0 +g(t)+S (t)g(0)−g(t)+∫ t

0S (t− s)g′ (s)ds

which is continuous. The claim and 22.45 also shows y(t) ∈ D(Λ). This proves the exis-tence part of the lemma.

It remains to prove the uniqueness part. It suffices to show that if

y′−Λy = 0, y(0) = 0

and y is C1 having values in D(Λ) , then y = 0. Suppose then that y is this way. Letting0 < s < t,

dds

(S (t− s)y(s))≡

limh→0

S (t− s−h)y(s+h)− y(s)

h− S (t− s)y(s)−S (t− s−h)y(s)

h

provided the limit exists. Since y′ exists and y(s) ∈ D(Λ) , this equals

S (t− s)y′ (s)−S (t− s)Λy(s) = 0.