Let $f\colon H\to\mathbb{R}\cup \{+\infty\}$ is convex, lower semicontinuous, proper and coercive function. $H$ is a Hilbert space.
$f_{\lambda}:H\to\mathbb{R}\cup \{+\infty\}$ is the Moreau-Yosida approximation with $\lambda>0$: $$f_\lambda(x)=\inf_{y\in X} \left\{ f(y)+\frac{1}{2\lambda}||x-y||^2\right\}$$
Define $J_{\lambda}(x)=y$, $y$ is the point where the infimum is attained.
Determine $\partial f_{\lambda}(x)$ (the subdifferential of a convex function). I don't know how to do this. Any help would be greatly appreciated!