Is this mapping contractive?

Question

Given a sufficiently smooth nonlinear function $g(x): \mathbb{R}^{n}\rightarrow\mathbb{R}$, we define a mapping $\Phi(x)=x_0-\lambda(x)\nabla g(x)^T$, where $\lambda(x)\in\mathbb{R}$ is chosen to make $g(x_0-\lambda(x)\nabla g(x)^T)=0$. Can we say that $\Phi$ is a contractive mapping? I'm stuck with this for a month.