838 CHAPTER 25. NONLINEAR OPERATORS
Hence
⟨Fu−Fv,u− v⟩ = ∥u∥p +∥v∥p−⟨Fu,v⟩−⟨Fv,u⟩≥ ∥u∥p +∥v∥p−∥u∥p−1 ∥v∥−∥u∥∥v∥p−1
≥ ∥u∥p +∥v∥p−(∥u∥p
p′+∥v∥p
p
)−(∥u∥p
p+∥v∥p
p′
)= 0
Now suppose ∥x∥= ∥y∥= 1 but x ̸= y. Then⟨Fx,
x+ y2
⟩≤ ∥x∥p−1
∥∥∥∥x+ y2
∥∥∥∥< ∥x∥+∥y∥2= 1
It follows that12⟨Fx,x⟩+ 1
2⟨Fx,y⟩= 1
2+
12⟨Fx,y⟩< 1
and so⟨Fx,y⟩< 1
It is easy to check that for nonzero α, F (αx) = |α|p−2αF (x) . This is because∥∥∥|α|p−2
αF (x)∥∥∥= |α|p−1 ∥x∥p−1 = ∥αx∥p−1
⟨|α|p−2
αF (x) ,αx⟩= |α|p ∥x∥p = ∥αx∥p
and so, since |α|p−2αF (x) acts like F (αx) , it is F (αx). It follows that for arbitrary x,y,
such that x/∥x∥ ̸= y/∥y∥
⟨Fx,y⟩= ∥x∥p−1 ∥y∥⟨
F(
x∥x∥
),
(y∥y∥
)⟩Therefore,
⟨Fx,y⟩= ∥x∥p−1 ∥y∥⟨
F(
x∥x∥
),
(y∥y∥
)⟩< ∥x∥p−1 ∥y∥ (25.2.10)
Now say that x ̸= y and consider
⟨Fx−Fy,x− y⟩
First suppose x = αy. This is the case where x is a multiple of y. Then the above is
⟨F (αy)−Fy,(α−1)y⟩= (α−1)(⟨F (αy) ,y⟩−∥y∥p)
= (α−1)(|α|p−2
α ∥y∥p−∥y∥p)= (α−1)
(|α|p−2
α−1)∥y∥p > 0
by the above observation that x→ |x|p−2 x is strictly monotone. Similarly,
⟨Fx−Fy,x− y⟩> 0