Differentiability of Moreau-Yosida approximation. I want to show that if $X$ is a reflexive Banach space with norm of class $\mathcal{C}^1$ and $f\colon X\to\mathbb{R}\cup \{+\infty\}$ is convex and lower semicontinuous, then $f_{\lambda}$ is differentiable of class $\mathcal{C}^1$.
(where $f_{\lambda}:X\to\mathbb{R}\cup \{+\infty\}$ is the Moreau-Yosida approximation: $$f_\lambda(x)=\inf_{y\in X} \left\{ f(y)+\frac{1}{2\lambda}|x-y|^2\right\})$$ 
Maybe, this result could be useful:
If $g\colon X\to\mathbb{R}$ is convex and differentiable in every point then $g\in\mathcal{C}^1(X)$.
Many thanks in advance.

Observe that the subdifferential of the function $y\to \frac{\|x-y\|^2}{2\lambda}+f(y)$ is the operator
$$y\to F(y-x)+\partial f(y),$$
where $F:X\to X^*$ is a duality mapping ($Fx=\{f^*\in X^*\,|\,\langle f,x\rangle=\|x\|^2=\|f\|^2\}$). Now, recall that a point $y$ is a minimazer of a convex function $g$ iff $0\in \partial g(y)$. Since $F$ is a duality mapping, $A=\partial f$ is maximal monotone and $X$ is reflexive, invoking Rockafellar (or Minty - don't remember) theorem, we have that the equation 
$$F(y-x)+Ay\ni0$$
has a unigue solution, which gives that the infimum is attained. Contrary to user53153 answer, the argement which realizes the infimum is not only a weak limit of a minimazing sequance, but also a solution to a certain equation. This has a direct impact on Gâteaux differentiability of $f_\lambda$ if we don't assume that the norm is $\mathcal{C}^1$.
