Let $f,g$ be two convex functions and $g$ has Lipchitz continuous gradient. The proximal - gradient mapping, which defined as $$ x \mapsto \textrm{prox}_{\lambda f} \left( x - \lambda \nabla g \left( x \right) \right) . $$
How can I prove that the proximal - gradient mapping is nonexpansive?
I only able to prove it when $g \equiv 0$. In particular, for any $$ x^{+} = \textrm{prox}_{\lambda f} \left( x - \lambda \nabla g \left( x \right) \right) \textrm{ and } y^{+} = \textrm{prox}_{\lambda f} \left( y - \lambda \nabla g \left( y \right) \right) $$ we have that $$ x - \lambda \nabla g \left( x \right) - x^{+} \in \lambda \partial f \left( x^{+} \right) \textrm{ and } y - \lambda \nabla g \left( y \right) - y^{+} \in \lambda \partial f \left( y^{+} \right) $$ which, by the monotonicity of $\partial f$ implies further $$ \left\langle x - \lambda \nabla g \left( x \right) - x^{+} - \left( y - \lambda \nabla g \left( y \right) - y^{+} \right) , x^{+} - y^{+} \right\rangle \geq 0 $$ or equivalently $$ \left\langle x - y - \lambda \left( \nabla g \left( x \right) - \nabla g \left( y \right) \right) , x^{+} - y^{+} \right\rangle \geq \left\lVert x^{+} - y^{+} \right\Vert ^{2} . $$ Infact, when $g \equiv 0$ we have that $$ \left\langle x - y , x^{+} - y^{+} \right\rangle \geq \left\lVert x^{+} - y^{+} \right\Vert ^{2} $$ which means the $\textrm{prox}$ mapping is not only nonexpansive but also firmly nonexpansive. However, the appearance of $g$ make me confuse. I tried to use the Cauchy - Schwarz inequality then Lipchitz continuity of $\nabla g$ but it leads to some constant bigger than $1$: \begin{align*} \left\lVert x^{+} - y^{+} \right\Vert ^{2} & \leq \left\langle x - y , x^{+} - y^{+} \right\rangle - \lambda \left\langle \nabla g \left( x \right) - \nabla g \left( y \right) , x^{+} - y^{+} \right\rangle \\ & \leq \left\lVert x - y \right\Vert \left\lVert x^{+} - y^{+} \right\Vert + \lambda \left\lVert\nabla g \left( x \right) - \nabla g \left( y \right) \right\Vert \left\lVert x^{+} - y^{+} \right\Vert \\ & \leq \left( 1 + \lambda L \right) \left\lVert x - y \right\Vert \left\lVert x^{+} - y^{+} \right\Vert . \end{align*} Can anyone tell me where did I miss?