898 CHAPTER 25. NONLINEAR OPERATORS

Note also that for a given x there is only one solution Jλ x to 0 ∈ F (Jλ x− x)+λp−1A(Jλ x).

By monotonicity of F,

0≤ ⟨F (Jλ xn− xn)−F (Jλ xm− xm) ,xm− xn + Jλ xn− Jλ xm⟩

Then from the above,

⟨F (Jλ xn− xn)−F (Jλ xm− xm) ,xn− xm⟩≤ ⟨F (Jλ xn− xn)−F (Jλ xm− xm) ,Jλ xn− Jλ xm⟩

Now from the boundedness of these operators, the left side of the above inequality con-verges to 0 as n,m→ ∞. Thus

lim infm,n→∞

⟨F (Jλ xn− xn)−F (Jλ xm− xm) ,Jλ xn− Jλ xm⟩ ≥ 0 (25.7.55)

lim infm,n→∞

⟨−λ

p−1Aλ (xn)−(−λ

p−1Aλ (xm)),Jλ xn− Jλ xm

⟩≥ 0

lim infm,n→∞

⟨λ

p−1

∈A(Jλ xm)︷ ︸︸ ︷Aλ (xm)−λ

p−1

A(Jλ xn)︷ ︸︸ ︷Aλ (xn),Jλ xn− Jλ xm

⟩≥ 0

The expression on the left in the above is non positive. Multiplying by −1,

0 ≥ lim supm,n→∞

⟨Aλ (xn)−Aλ (xm) ,Jλ xn− Jλ xm⟩

≥ lim infm,n→∞

⟨Aλ (xn)−Aλ (xm) ,Jλ xn− Jλ xm⟩ ≥ 0 (25.7.56)

Thus, in fact,the expression in 25.7.55 converges to 0. By boundedness considerations andthe strong convergence given,

limm,n→∞

⟨F (Jλ xn− xn)−F (Jλ xm− xm) ,Jλ xn− xn− (Jλ xm− xm)⟩= 0 (25.7.57)

From boundedness again, there is a subsequence still denoted with the subscript n suchthat

Jλ xn− xn→ a− x, F (Jλ xn− xn)→ b both weakly.

Since F is maximal monotone, (Theorem 25.7.9) it follows from Lemma 25.7.34 that[a− x,b]∈G (F) and so in fact F (a− x)= b. Thus this has just shown that F (Jλ xn− xn)→F (a− x). Next consider 25.7.56. We have Jλ xn→ a weakly and

Aλ (xn) =−λ−(p−1)F (Jλ xn− xn)→−λ

−(p−1)b

weakly. Then from Lemma 25.7.34 again,[a,−λ

−(p−1)b]∈ G (A) so −λ

−(p−1)b ∈ A(a)

so b ∈ −λp−1A(a) . But it was just shown that b = F (a− x) and so

F (a− x) ∈ −λp−1A(a) so 0 ∈ F (a− x)+λ

p−1A(a) , so a = Jλ x.