48.7. LIMIT CONDITIONS FOR NEMYTSKII OPERATORS 1583

the last term being bounded independent of t,n by assumption. Thus there exists c ∈L1 (0,T ) and a positive constant C such that

⟨zn (t) ,un (t)− y(t)⟩ ≥ −c(t)−C ||y(t)||pV . (48.7.26)

Letting y = u, we use Fatou’s lemma to write

lim infn→∞

∫ T

0

(⟨zn (t) ,un (t)−u(t)⟩+ c(t)+C ||u(t)||pV

)dt ≥

∫ T

0lim inf

n→∞⟨zn (t) ,un (t)−u(t)⟩+

(c(t)+C ||u(t)||pV

)dt

≥∫ T

0

(c(t)+C ||u(t)||pV

)dt.

Here, we added the term c(t)+C ||u(t)||pV to make the integrand nonnegative in order toapply Fatou’s lemma. Thus,

lim infn→∞

∫ T

0⟨zn (t) ,un (t)−u(t)⟩dt ≥ 0.

Consequently, using the claim in the last inequality,

0 ≥ lim supn→∞

⟨zn,un−u⟩V ′,V

≥ lim infn→∞

∫ T

0⟨zn (t) ,un (t)−u(t)⟩dt

= lim infn→∞⟨zn,un−u⟩V ′,V

≥∫ T

0lim inf

n→∞⟨zn (t) ,un (t)−u(t)⟩dt ≥ 0,

hence, we find thatlimn→∞⟨zn,un−u⟩V ′,V = 0. (48.7.27)

We need to show that if y is given in V then

lim infn→∞⟨zn,un− y⟩V ′,V ≥ ⟨z(y) ,u− y⟩ V ′,V , z(y) ∈ Âu

Suppose to the contrary that there exists y such that

η = lim infn→∞⟨zn,un− y⟩V ′,V < ⟨z,u− y⟩V ′,V , (48.7.28)

for all z ∈ Âu. Take a subsequence, denoted still with subscript n such that

η = limn→∞⟨zn,un− y⟩V ′,V

Thuslimn→∞⟨zn,un− y⟩V ′,V < ⟨z,u− y⟩V ′,V (48.7.29)