I'm reading Barbus & Precupanu's 'Convexity and optimization in Banach spaces'. The authors define what I think is the resolvent for an operator $A: X \to X^*$ (or subset $A \subset X \times X^*$) where $X^*$ is the dual space:
Let $ X,X^*$ be reflexive and strictly convex and $A$ maximal monotone and $F$ the duality mapping ($F = \{ x^*: (x,x^*) = \lVert x \rVert^2 = \lVert x^* \rVert^2$) . Then the equation $$ 0 \in F(y-x) + \lambda Ay$$ has a unique solution $x_\lambda$.
The maximal monotonicity follows since $F$ is also a maximal monotone operator and demicontinuous (since $X^*$ is strictly convex ) I think.
The authors then claim that the uniqueness follows from the fact that $X$ and $X^*$ are strictly convex and the inequality:
$$ (F(x) -F(y), x-y) \geq \left( \lVert x \rVert - \lVert y \rVert \right)^2$$.
I wonder if anyone can help me and explain how the uniqueness follows from this?
Kind regards and thanks in advance,