Show that operator T is a contraction mapping I want to check whether the operator T defined as:
$Tf(x) = \beta \left\lbrace \sum_{\theta} \mu_\theta \left[ h_\theta(x) + \int f(x') Q_\theta(x,dx') \right]^\alpha \right\rbrace^{1/\alpha} $
is a contraction mapping. It is known that:


*

*$\beta \in (0,1) $ and $ \alpha >1 $.

*The function $Q_\theta(a,A) = Pr_\theta(x' \in A | x = a ) $ is a transition function. The transition function differs for each value of $\theta$. 

*$\sum_\theta \mu_\theta = 1 $ and $ \mu_\theta \geq 0 , \forall \theta$. Furhtermore $ \theta \in \left\lbrace 0,1,...,N \right\rbrace $. 

*The functions $h,f$ are continous and bounded. Denote the set of continuous and bouded functions by $C(X)$. 


The integral can be represented by the operator M: $ M_\theta f(x) = \int f(x') Q_\theta (x,dx') $. This operator preserves boundedness and continuity. Accordingly, $ T: C(X) \rightarrow C(X) $. 
Usually, I use Blackwell's sufficient conditions to show that the operator $T$ is a contraction mapping or check the definition of a contraction mapping directly. Unfortuantely, with the operator defined above, I am not able to conclude whether $T$ is a contraction mapping. Can somebody tell me other conditions than Blackwell's to look at for determining whether $T$ is a contaction mapping, or does somebody know outright whether $T$ is a contaction mapping?   
 A: As I was not able to show with Blackwell's sufficient conditions or directly that the operator T constitutes an contraction mapping, I rewrote the problem, to attack it with different methods. Define the function $g= h + f$ and add $h$ to each side of the equation above. Then, define the operator $\tilde{T}$ as:
$\tilde{T}g(x) = h(x) + \beta \left\lbrace  \sum_{\theta} \mu_{\theta} \left[ \int g(x') Q_\theta (x, dx') \right]^{\alpha} \right\rbrace^{1/\alpha}$
Now, Marinacci and Montrucchio in their paper titled ''Unique solutions for stochastic recursive utilities'' operators of the folllwing form: $\hat{T} g = W \left( h,\mathcal{M}(g) \right)  $  with with $W: \mathbf{R}^{2}_{+} \rightarrow \mathbf{R}_{+}$ and $\mathcal{M}: C(X) \rightarrow C(X) $. In particular, they lay out sufficient conditions for the operator $\hat{T}$ to be a contraction mapping. I then showed that my problem satifies these conditions and that I can formulate my problem through the function $W$ and operator $\mathcal{M}$. Having found a solution $g^{\star}$, I can retrieve the function $f^{\star}$. 
