Kenneth Kuttler

832 CHAPTER 25. NONLINEAR OPERATORS

The right side is bounded and so it follows that the left side is also bounded. Therefore,∥un∥ must be bounded. Taking a subsequence and using the assumption that V is reflexive,we can obtain

un→ u weakly in V

By the fact that convex closed sets are weakly closed also, it follows that u∈K. Also, givenM, eventually all ∥un∥ and ∥u∥ are less than M. Now from the inequality,

⟨Aun,un− v⟩ ≤ ⟨ f ,un− v⟩

Thus⟨Aun,un−u⟩+ ⟨Aun,u− v⟩ ≤ ⟨ f ,un−u⟩+ ⟨ f ,u− v⟩

Then taking limsupn→∞ one gets

lim supn→∞

⟨Aun,un−u⟩+ ⟨ξ ,u− v⟩ ≤ ⟨ f ,u− v⟩

This holds for v ∈ Km where m is arbitrary. Hence one could let vm→ u. Thus eventually∥vm∥< M and so for large m,vm ∈ Km. Then it follows that

lim supn→∞

⟨Aun,un−u⟩ ≤ 0.

Consequently, by the assumption that A is pseudomonotone on K, for every v ∈ K,

⟨Au,u− v⟩ ≤ lim infn→∞⟨Aun,un− v⟩ (*)

for all v ∈ K. Then from the inequality obtained from Browder’s lemma,

⟨Aun,un− v⟩V ′,V ≤ ⟨ f ,un− v⟩V ′,V

and so * implies on taking liminf that for all v ∈ K,

⟨Au,u− v⟩V ′,V ≤ ⟨ f ,u− v⟩V ′,V

25.2 Duality MapsThe duality map is an attempt to duplicate some of the features of the Riesz map in Hilbertspace which is discussed in the chapter on Hilbert space.

Definition 25.2.1 A Banach space is said to be strictly convex if whenever ||x||= ||y|| andx ̸= y, then ∣∣∣∣∣∣∣∣x+ y

∣∣∣∣∣∣∣∣< ||x||.F : X→ X ′ is said to be a duality map if it satisfies the following: a.) ||F(x)||= ||x||p−1. b.)F(x)(x) = ||x||p, where p > 1.

832 CHAPTER 25. NONLINEAR OPERATORSThe right side is bounded and so it follows that the left side is also bounded. Therefore,||u,|| must be bounded. Taking a subsequence and using the assumption that V is reflexive,we can obtainUn — u weakly in VBy the fact that convex closed sets are weakly closed also, it follows that u € K. Also, givenM, eventually all ||u,|| and ||u|| are less than M. Now from the inequality,(Aun, Un —v) < (fun — v)Thus(Aun, Un — U) + (Aun, U— v) < (f Un —Uu) +(f,u—v)Then taking limsup,,_,., one getslim sup (Auy,Un —u) + (€,u—v) < (f,u—v)nooThis holds for v € K,, where m is arbitrary. Hence one could let v,, — u. Thus eventually||¥m|| <M and so for large m, vm € Kn. Then it follows thatlim sup (Auy, uy, —u) <0.Nn—ooConsequently, by the assumption that A is pseudomonotone on K, for every v € K,(Au,u—v) <lim inf (Auy,u, — Vv) (*)Nn—yoofor all v € K. Then from the inequality obtained from Browder’s lemma,(Aun, Un — V)yry < (f,un— V)yryand so * implies on taking liminf that for all v € K,(Au,u-V)yry <(f,u-V)yry Oe25.2 Duality MapsThe duality map is an attempt to duplicate some of the features of the Riesz map in Hilbertspace which is discussed in the chapter on Hilbert space.Definition 25.2.1 A Banach space is said to be strictly convex if whenever ||x|| = ||y|| andxy, thenx+y—|1< .|| <iF :X — X' is said to be a duality map if it satisfies the following: a.) ||F(x)|| =||x||?~!. B.)F (x)(x) = ||x||?, where p > 1.