43.1. SOME STANDARD TECHNIQUES IN EVOLUTION EQUATIONS 1417
Since X ′ separates the points of X , it follows∫ b
a
∫ t
af ′ (s)φ
′ (t)dsdt =∫ b
a
∫ b
sf ′ (s)φ
′ (t)dtds.
Therefore, ∫ b
a
(f (t)−
∫ t
af ′ (s)ds
)φ′ (t)dt
=∫ b
af (t)φ
′ (t)dt−∫ b
a
∫ b
sf ′ (s)φ
′ (t)dtds
=∫ b
af (t)φ
′ (t)dt−∫ b
af ′ (s)
∫ b
sφ′ (t)dtds
=∫ b
af (t)φ
′ (t)dt +∫ b
af ′ (s)φ (s)ds = 0.
Therefore, by Lemma 43.1.8, there exists a constant, denoted as f (a) such that
f (t)−∫ t
af ′ (s)ds = f (a)
The integration by parts formula is also important.
Corollary 43.1.10 Suppose f , f ′ ∈ L1 (a,b;X) and suppose φ ∈ C1 ([a,b]) . Then the fol-lowing integration by parts formula holds.∫ b
af (t)φ
′ (t)dt = f (b)φ (b)− f (a)φ (a)−∫ b
af ′ (t)φ (t)dt.
Proof: From Theorem 43.1.9∫ b
af (t)φ
′ (t)dt
=∫ b
a
(f (a)+
∫ t
af ′ (s)ds
)φ′ (t)dt
= f (a)(φ (b)−φ (a))+∫ b
a
∫ t
af ′ (s)dsφ
′ (t)dt
= f (a)(φ (b)−φ (a))+∫ b
af ′ (s)
∫ b
sφ′ (t)dtds
= f (a)(φ (b)−φ (a))+∫ b
af ′ (s)(φ (b)−φ (s))ds
= f (a)(φ (b)−φ (a))−∫ b
af ′ (s)φ (s)ds+( f (b)− f (a))φ (b)
= f (b)φ (b)− f (a)φ (a)−∫ b
af ′ (s)φ (s)ds.
The interchange in order of integration is justified as in the proof of Theorem 43.1.9.There is an interesting theorem which is easy to present at this point.