76.3. THE EXISTENCE OF APPROXIMATE SOLUTIONS 2549

Hence, since ζ = (Bu)′ ,0 =−⟨Bu(0) ,w⟩φ (0)

Then it follows that,⟨Bu(0) ,w⟩= 0. Since w was arbitrary, Bu(0) = 0 and ζ = (Bu)′.Thus, passing to a limit in 76.3.18,

(Bu)′+ξ = f in V ′ω , Bu(0) = 0

It is desired to identify ξ with Ā(ω)u. First let

Lh ≡I− τh

h

Then

Lh (Bu)(t) ={ 1

h∫ t

t−h (Bu)′ ds if t ≥ h1h∫ t

0 (Bu)′ ds if t < h

Then from standard considerations involving approximate identities,

limh→0

Lh (Bu) = (Bu)′ strongly in V ′ω (76.3.19)

Thus ⟨Lh (Buh)− (Bu)′ ,uh−u

⟩=

⟨Lh (Buh)−Lh (Bu) ,uh−u⟩+⟨Lh (Bu)− (Bu)′ ,uh−u

⟩≥

⟨Lh (Bu)− (Bu)′ ,uh−u

⟩and the above strong convergence implies that this converges to 0. Therefore, from 76.3.18,

LhBuh + Āuh = f in V ′ω

and so⟨LhBuh,uh−u⟩+

⟨Āuh,uh−u

⟩= ⟨ f ,uh−u⟩

From the above,⟨(Bu)′ ,uh−u

⟩+⟨Lh (Bu)− (Bu)′ ,uh−u

⟩+⟨Āuh,uh−u

⟩≤ ⟨ f ,uh−u⟩

and so, taking limsuph→0 of both sides, it follows from 76.3.19 that

lim suph→0

⟨Āuh,uh−u

⟩≤ 0, lim sup

h→0

⟨Āuh,uh

⟩≤ ⟨ξ ,u⟩

Since A is monotone and hemicontinuous, the same is true of Ā and so

Āu = ξ

Thus ((Bu)′ (·,ω)

)+ Ā(ω)u(·,ω) = f (·,ω) in V ′ω , Bu(0,ω) = 0 (76.3.20)