Kenneth Kuttler

566 CHAPTER 19. HILBERT SPACES

Theorem 19.12.3 Let 19.12.75 hold. Then there exists a unique solution to 19.12.74 inBC ([a,b] ;X).

Proof: Use the norm of 19.12.73 where γ ̸= 0 is described later. Let T : BC ([a,b] ;X)→BC ([a,b] ;X) be defined by

T x(t)≡ x0−∫ t

aF (s,x(s))ds

Then

∥T x(t)−Ty(t)∥X =

∥∥∥∥∫ t

aF (s,x(s))ds−

∫ t

aF (s,y(s))ds

∥∥∥∥≤ K

∫ t

a∥x(s)− y(s)∥ds = K

∫ t

∥∥∥(x(s)− y(s))eγ(s−a)e−γ(s−a)∥∥∥ds

≤ K∫ t

ae−γ(s−a)ds∥x− y∥

γ= K

(e−γ(t−a)

−γ+

1γ

)∥x− y∥

Therefore,

eγ(t−a) ∥T x(t)−Ty(t)∥X ≤ K

(eγ(t−a)

γ− 1

)∥x− y∥

∥T x−Ty∥γ≤ sup

t∈[a,b]K

(eγ(t−a)

γ− 1

)∥x− y∥

Letting γ =−m2, this reduces to

∥T x−Ty∥−m2 ≤Km2 ∥x− y∥−m2

and so if K/m2 < 1/2, this shows the solution to the integral equation is the unique fixedpoint of a contraction mapping defined on BC ([a,b] ;X). This shows existence and unique-ness of the initial value problem 19.12.74.

Definition 19.12.4 Let S : [0,∞)→L (X ,X) be continuous and satisfy

1. S (t + s) = S (t)S (s) called the semigroup identity.

2. S (0) = I

3. limh→0+S(h)x−x

h = Ax for A a densely defined closed linear operator whenever x ∈D(A)⊆ X .

Then S is called a continuous semigroup and A is said to generate S.

566 CHAPTER 19. HILBERT SPACESTheorem 19.12.3 Let 19.12.75 hold. Then there exists a unique solution to 19.12.74 inBC ([a,b];X).Proof: Use the norm of 19.12.73 where y £ 0 is described later. Let T : BC ([a, b];X) >BC ({a,b];X) be defined byTx(t) = x0 = [ F(s,x(s))dsThent t|Tx(t) Tye =|] fF s.x(9))as— [F(s.y(o))ast t< K[Ix(s)-y@)llas=K [ (a6) -y(s) ee | ast _vs—a eo Vt—a) 1< Kem slay 4 = +1) Ix—sll,Therefore,(t—a) ev(t—a) 1et STX() Ty Olle SK | 9 J ely¥(t—a)Tx—Ty|l,< sup K( £——— +} |Jx—yIY~ tela] yY Y yLetting y = —m7”, this reduces toK|x Ty] m2 S = |e vl neand so if K/m? < 1/2, this shows the solution to the integral equation is the unique fixedpoint of a contraction mapping defined on BC ([a, b] ;X). This shows existence and unique-ness of the initial value problem 19.12.74. JJDefinition 19.12.4 Let S : [0,0c) > & (X,X) be continuous and satisfy1. S(t+s) =S(t)S(s) called the semigroup identity.2. $(0)=1S(h)x3. limp+o4 — = Ax for A a densely defined closed linear operator whenever x €D(A) CX.Then S is called a continuous semigroup and A is said to generate S.