Kenneth Kuttler

926 CHAPTER 25. NONLINEAR OPERATORS

It follows that we can pass to a limit in 25.8.92 and obtain

Lu+w∗ = f (25.8.93)

Now by assumption on A, it is L modified bounded pseudomonotone and so there is asubsequence, still denoted as uε such that the liminf pseudomonotone limit condition holds.This will be what is referred to in what follows. Then⟨

εL∗(F−1 (Luε)

),uε −u

⟩+ ⟨Luε ,uε −u⟩+ ⟨w∗ε ,uε −u⟩= ⟨ f ,uε −u⟩

and so,

ε⟨F−1 (Luε) ,Luε −Lu

⟩+ ⟨Luε ,uε −u⟩+ ⟨w∗ε ,uε −u⟩= ⟨ f ,uε −u⟩

using the monotonicity of L,

ε⟨Luε −Lu,F−1 (Luε)−F−1 (Lu)

⟩+ ε⟨Luε −Lu,F−1 (Lu)

⟩+⟨Lu,uε −u⟩+ ⟨w∗ε ,uε −u⟩ ≤ ⟨ f ,uε −u⟩

Now using monotonicity of F−1,

ε⟨Luε −Lu,F−1 (Lu)

⟩+ ⟨Lu,uε −u⟩+ ⟨w∗ε ,uε −u⟩ ≤ ⟨ f ,uε −u⟩

and so, passing to a limit as ε → 0,

lim supε→0⟨w∗ε ,uε −u⟩ ≤ 0

It follows that for all v ∈ X = D(L) there exists w∗ (v) ∈ Au

lim infε→0⟨w∗ε ,uε − v⟩ ≥ ⟨w∗ (v) ,u− v⟩

But the left side equals

lim infε→0

[⟨w∗ε ,uε −u⟩+ ⟨w∗ε ,u− v⟩]

≤ lim supε→0⟨w∗ε ,uε −u⟩+ ⟨w∗,u− v⟩ ≤ ⟨w∗,u− v⟩

and so⟨w∗,u− v⟩ ≥ ⟨w∗ (v) ,u− v⟩

for all v.Is w∗ ∈ Au? Suppose not. Then Au is a closed convex set and w∗ is not in it. Hence,

since V is reflexive, there exists z ∈ V such that whenever y∗ ∈ Au,⟨w∗,z⟩ < ⟨y∗,z⟩ . Nowsimply choose v such that u− v = z and it follows that

⟨w∗ (v) ,u− v⟩> ⟨w∗,u− v⟩ ≥ ⟨w∗ (v) ,u− v⟩

which is clearly a contradiction. Hence w∗ ∈ Au. Thus from 25.8.93, this has shown thatL+A is onto.

926 CHAPTER 25. NONLINEAR OPERATORSIt follows that we can pass to a limit in 25.8.92 and obtainLu+w* =f (25.8.93)Now by assumption on A, it is L modified bounded pseudomonotone and so there is asubsequence, still denoted as ug such that the liminf pseudomonotone limit condition holds.This will be what is referred to in what follows. Then(eL* (F~! (Lute)) ,ue —u) + (Lite, Ue — u) + (We,Ue — UU) = (f Ue — U)and so,e(F! (Lute) ,Lug — Lu) + (Lite, Ug — u) + (We, Ue —U) = (fe —U)using the monotonicity of L,€ (Lute — Lu, F | (Lug) — F~' (Lu)) + € (Lite — Lu, F~' (Lu))+ (Lu, Ug —u) + (We, Ue —U) < (f,Ue — U)Now using monotonicity of F~!,€ (Lue —Lu,F7! (Lu)) + (Lu,ug —u) + (We,Ue —U) < (f Ue —u)and so, passing to a limit as € > 0,lim sup (we,Ue —u) <0€0It follows that for all v € X = D(L) there exists w* (v) € Aulim inf (wg,ue —v) > (w* (v) ,u—v)e0But the left side equalslim inf [(w3,ue —u) + (we,u—v)]€>0<_ limsup (w3,Ue —u) + (w*,u—v) < (w*,u—v)€>0and so(w*,u—v) > (w* (v) ,u—v)for all v.Is w* € Au? Suppose not. Then Au is a closed convex set and w* is not in it. Hence,since V is reflexive, there exists z € V such that whenever y* € Au, (w*,z) < (y*,z). Nowsimply choose v such that u — v = z and it follows that(w* (v),u—v) > (w*,u—v) > (w* (v) ,u—v)which is clearly a contradiction. Hence w* € Au. Thus from 25.8.93, this has shown thatL-+A is onto.