Chapter 75

Some Nonlinear OperatorsIn this chapter is a description and properties of some standard nonlinear maps.

75.1 An Assortment Of Nonlinear OperatorsDefinition 75.1.1 For V a real Banach space, A : V → V ′ is a pseudomonotone map ifwhenever

un ⇀ u (75.1.1)

andlim sup

n→∞

⟨Aun,un−u⟩ ≤ 0 (75.1.2)

it follows that for all v ∈V,

lim infn→∞⟨Aun,un− v⟩ ≥ ⟨Au,u− v⟩. (75.1.3)

The half arrows denote weak convergence.

Definition 75.1.2 A : V →V ′ is monotone if for all v,u ∈V,

⟨Au−Av,u− v⟩ ≥ 0,

and A is Hemicontinuous if for all v,u ∈V,

limt→0+⟨A(u+ t (v−u)) ,u− v⟩= ⟨Au,u− v⟩.

Theorem 75.1.3 Let V be a Banach space and let A : V →V ′ be monotone and hemicon-tinuous. Then A is pseudomonotone.

Proof: Let A be monotone and Hemicontinuous. First here is a claim.Claim: If 75.1.1 and 75.1.2 hold, then limn→∞⟨Aun,un−u⟩= 0.Proof of the claim: Since A is monotone,

⟨Aun−Au,un−u⟩ ≥ 0

so⟨Aun,un−u⟩ ≥ ⟨Au,un−u⟩.

Therefore,

0 = lim infn→∞⟨Au,un−u⟩ ≤ lim inf

n→∞⟨Aun,un−u⟩ ≤ lim sup

n→∞

⟨Aun,un−u⟩ ≤ 0.

Now using that A is monotone again, then letting t > 0,

⟨Aun−A(u+ t (v−u)) ,un−u+ t (u− v)⟩ ≥ 0

and so⟨Aun,un−u+ t (u− v)⟩ ≥ ⟨A(u+ t (v−u)) ,un−u+ t (u− v)⟩.

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