# Property of Fixed Point Function

Given an operator $$\mathcal{T}$$ that maps from a function $$f: \mathbb{R}^d\rightarrow \mathbb{R}$$ to another function $$f': \mathbb{R}^d\rightarrow \mathbb{R}$$, we are interested in the fixed point $$f^*$$ (assuming that $$T$$ is strictly contractive so that the fixed point exists and is unique): $$f^* = \mathcal{T}f^*.$$ One motivating example is Q-iteration in Markov decision process, where $$f(s, a) \leftarrow \mathcal{T} f(s,a): =r(s) + \gamma \mathbf{E}_{s'\sim p(\cdot | s, a)} \left\{\max_{a'\in \mathcal{A}}f(s', a')\right\}\quad (\gamma<1).$$ Here $$s\in \mathbb{R}^d$$ and $$a \in \mathcal{A}$$ denote the state and action, where $$\mathcal{A}$$ is a discrete set. Meanwhile, $$r(s): \mathbb{R}^d \rightarrow \mathbb{R}$$ is the reward function, $$p(\cdot | s, a)$$ is the transition kernel, that is, a conditional distribution of the next state $$s'$$ given the current state and action, and $$a'$$ is the action taken at the next state $$a'$$.

Without the max operator, the above fixed-point iteration corresponds to a linear operator. In such a simplified case, what can we say about the properties of the fixed point $$f^*$$, assuming properties of the operator, which depends on the transition kernel and reward function? (For example, the smoothness of $$f^*$$, or more generally, its Sobolev- or Hölder-norm.)

What about the case with the max operator? More generally, what about general strictly contractive nonlinear operators?