Kenneth Kuttler

45.3. INTRINSIC NORMS 1491

Since this limit exists, the last limit in the above exists and equals

ΛG(t)y0 (45.3.10)

and G(t)y0 ∈ D(Λ). Now consider 45.3.8.

y(t +h)− y(t)h

=G(t +h)−G(t)

hy0+

(∫ t+h

0G(t− s+h)g(s)ds−

∫ t

0G(t− s)g(s)ds

)

=G(t +h)−G(t)

hy0 +

∫ t+h

tG(t− s+h)g(s)ds

+1h

(G(h)

∫ t

0G(t− s)g(s)ds−

∫ t

0G(t− s)g(s)ds

)From the claim and 45.3.9, 45.3.10 the limit of the right side is

ΛG(t)y0 +g(t)+Λ

(∫ t

0G(t− s)g(s)ds

)= Λ

(G(t)y0 +

∫ t

0G(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 45.3.10 it also equals

G(t)Λy0 +g(t)+G(t)g(0)−g(t)+∫ t

0G(t− s)g′ (s)ds

which is continuous. The claim and 45.3.10 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

(G(t− s)y(s))

≡ limh→0

G(t− s−h)y(s+h)− y(s)

−G(t− s)y(s)−G(t− s−h)y(s)h

45.3. INTRINSIC NORMS 1491Since this limit exists, the last limit in the above exists and equalsAG (t) yo (45.3.10)and G(t) yo € D(A). Now consider 45.3.8.y(t+h)—yl(t) _ GU+h~COh; ( PGs h)e(o)ds~ ['G(r—s}e(o)as)= Ong [G0 s+ malo+7 (G10 [ Ge—sje(syas— ['G0-s)@6)4s)From the claim and 45.3.9, 45.3.10 the limit of the right side istAG(t)yo+g(t)+A (/ G(r—s)e(s)ds)t= A (Gio G(t -s)¢(s)ds) + g(t)Hencey (t) = Ay(t) +8 (t)and from the formula, y’ is continuous since by the claim and 45.3.10 it also equalsGWA +8] +G()20)-e()+ ['GU-s)e'(o)aswhich is continuous. The claim and 45.3.10 also shows y(t) € D(A). This proves theexistence part of the lemma.It remains to prove the uniqueness part. It suffices to show that ify’—Ay=0, y(0) =0and y is C! having values in D(A), then y = 0. Suppose then that y is this way. LettingO0<s<t,£(Gt-s)y(s))= tim G(r—s—n) 20+ HY)_ G(t—s)y(s)-—G(t—s—h)y(s)h