34.2. STANDARD TECHNIQUES IN EVOLUTION EQUATIONS 1167
Corollary 34.2.11 Suppose fn→ f weakly in X where we assume also that X is reflexive.Then fn (t)→ f (t) weakly in X .
The integration by parts formula is also important.
Corollary 34.2.12 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 34.2.9∫ 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 34.2.9.With this integration by parts formula, the following interesting lemma is obtained.
This lemma shows why it was appropriate to define f as in Definition 34.2.2.
Lemma 34.2.13 Let f be given in Definition 34.2.2 and suppose f , f ′ ∈ L1 (a,b;X) . Thenf , f ′ ∈ L1 (2a−b,2b−a;X) also and
f ′ (t)≡
f ′ (t) if t ∈ [a,b]− f ′ (2a− t) if t ∈ [2a−b,a]− f ′ (2b− t) if t ∈ [b,2b−a]
(34.2.9)
Proof: It is clear from the definition of f that f ∈ L1 (2a−b,2b−a;X) and that in fact∣∣∣∣ f ∣∣∣∣L1(2a−b,2b−a;X)≤ 3 || f ||L1(a,b;X) . (34.2.10)