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 $X^{\ast }$ and $X$, a monotone operator satisfiesA nonlinear coercive operator with respect to $u_0$ satisfiesThe directional derivative is ${\partial }_{v}F(u)={\lim }_{t\to 0}[F(u+tv)-F(u)]/t$, 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.

If $A:X\to X^{\ast }$ is monotone and $s<t$, apply its defining inequality at $u+tv$ and $u+sv$. It becomes $(t-s)[\phi (t)-\phi (s)]\ge 0$, so $\phi$ is nondecreasing. Conversely choose $u=y$, $v=x-y$. Nondecrease from zero to one gives $⟨A(x)-A(y),x-y⟩\ge 0$. This proves the line criterion for a monotone operator.