 Solution

ID: past-exam-of-the-methods-of-applied-functional-analysis-course-of-the-university-of-oxford/2010/b/6/a/solution

Using the dual pairing between

X^{*}

and

X

⟨ A (u) - A (v), u - v ⟩ \geq 0 (u, v \in M) .

u_{0}

satisfies

\frac{⟨ A ( u ) , u - u _{0} ⟩}{∥ u - u _{0} ∥} ⟶ + \infty as u \in M, ∥ u - u_{0} ∥ \to \infty.

\partial_{v} F (u) = lim_{t \to 0} [F (u + t v) - F (u)] / t

, when this finite limit exists through admissible points. At a boundary point of

M

only an inward direction may be admissible; then one uses the corresponding one-sided limit.

A : X \to X^{*}

is monotone and

s < t

, apply its defining inequality at

u + t v

and

u + s v

. It becomes

(t - s) [φ (t) - φ (s)] \geq 0

, so

φ

is nondecreasing. Conversely choose

u = y

v = x - y

. Nondecrease from zero to one gives

⟨ A (x) - A (y), x - y ⟩ \geq 0

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