19.14. GENERAL THEORY OF CONTINUOUS SEMIGROUPS 587

=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 19.14.94, 19.14.95 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)

Hencey′ (t) = Λy(t)+g(t)

and from the formula, y′ is continuous since by the claim and 19.14.95 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 19.14.95 also shows y(t) ∈ D(Λ). This proves theexistence 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.

Let y∗ ∈ X ′. This has shown that on the open interval (0, t) the function

s→ y∗ (S (t− s)y(s))

has a derivative equal to 0. Also from continuity of S and y, this function is continuouson [0, t]. Therefore, it is constant on [0, t] by the mean value theorem. At s = 0, thisfunction equals 0. Therefore, it equals 0 on [0, t]. Thus for fixed s > 0 and letting t >s,y∗ (S (t− s)y(s)) = 0. Now let t decrease toward s. Then y∗ (y(s)) = 0 and since y∗ wasarbitrary, it follows y(s) = 0. This proves uniqueness.