48.7. LIMIT CONDITIONS FOR NEMYTSKII OPERATORS 1591

The estimate 48.7.40 implies,

0≤ ⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩− ≤ c(t,ω)+C ||u(t,ω)||pV , (48.7.44)

where c is a function in L1 (0,T ). Thanks to (48.7.39),

lim infn→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩ ≥ 0, a.e.

and, therefore, the following pointwise limit exists,

limn→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩− = 0, a.e.

and so we may apply the dominated convergence theorem using (48.7.44) and conclude

limn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩−dtdP

=∫

Ω

∫ T

0limn→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩−dtdP = 0

Now, it follows from (48.7.41) and the above equation, that

limn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩+dtdP

= limn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩

+⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩−dtdP

= limn→∞⟨zn,un−u⟩V ′,V = 0.

Therefore, both ∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩+dtdP

and ∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩−dtdP

converge to 0, thus,

limn→∞

∫Ω

∫ T

0|⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩|dtdP = 0 (48.7.45)

From the above, it follows that there exists a further subsequence {nk} not depending ont,ω such that ∣∣⟨znk (t,ω) ,unk (t,ω)−u(t,ω)⟩

∣∣→ 0 a.e. (t,ω) . (48.7.46)

Therefore, by the pseudomonotone limit condition for A there exists

wt,ω ∈ A(u(t,ω) , t,ω)