25.1. SOME NONLINEAR SINGLE VALUED OPERATORS 827

for all v ∈V . Consequently, for all v ∈V,

⟨Au,u− v⟩ ≤ lim infn→∞⟨Aun,un− v⟩

= lim infn→∞

(⟨Aun,u− v⟩+ ⟨Aun,un−u⟩)

= ⟨ξ ,u− v⟩+ lim infn→∞⟨Aun,un−u⟩ ≤ ⟨ξ ,u− v⟩

and so Au = ξ .An interesting result is the following which states that a monotone linear function added

to a type M is also type M.

Proposition 25.1.9 Suppose A : V → V ′ is type M and suppose L : V → V ′ is monotone,bounded and linear. Then L+A is type M. Let V be separable or reflexive so that the weakconvergences in the following argument are valid.

Proof: Suppose un ⇀ u and Aun +Lun ⇀ ξ and also that

lim supn→∞

⟨Aun +Lun,un⟩ ≤ ⟨ξ ,u⟩

Does it follow that ξ = Au+ Lu? Suppose not. There exists a further subsequence, stillcalled n such that Lun ⇀ Lu. This follows because L is linear and bounded. Then frommonotonicity,

⟨Lun,un⟩ ≥ ⟨Lun,u⟩+ ⟨L(u) ,un−u⟩

Hence with this further subsequence, the limsup is no larger and so

lim supn→∞

⟨Aun,un⟩+ limn→∞

(⟨Lun,u⟩+ ⟨L(u) ,un−u⟩)≤ ⟨ξ ,u⟩

and solim sup

n→∞

⟨Aun,un⟩ ≤ ⟨ξ −Lu,u⟩

It follows since A is type M that Au = ξ −Lu, which contradicts the assumption that ξ ̸=Au+Lu.

There is also the following useful generalization of the above proposition.

Corollary 25.1.10 Suppose A : V → V ′ is type M and suppose L : W →W ′ is monotone,bounded and linear where V ⊆W and V is dense in W so that W ′ ⊆ V ′. Then for u0 ∈Wdefine M (u) ≡ L(u−u0) . Then M +A is type M. Let V be separable or reflexive so thatthe weak convergences in the following argument are valid.

Proof: Suppose un ⇀ u and Aun +Mun ⇀ ξ and also that

lim supn→∞

⟨Aun +Mun,un⟩ ≤ ⟨ξ ,u⟩

Does it follow that ξ = Au+Mu? Suppose not. By assumption, un ⇀ u and so,

un−u0 ⇀ u−u0 weak convergence in W