I have a compact convex domain $\mathcal{D} \subset R_{0+}^{n}$ (nonnegative vectors in $R^n$), and a mapping $f:\mathcal{D} \to \mathcal{D}$.
The function $f$ is continuous, but nonsmooth ($f$ is a piecewise linear nondecreasing concave function over $\mathcal{D}$)
Because of continuity of $f$ and a compact-convex $\mathcal{D}$, we know by Brouwer fixed-point theorem that there exists atleast one $\bar{x} \in \mathcal{D}$ such that $\bar{x}=f(\bar{x})$ (Existence of fixed-point).
Now, I am trying to show that the iterations $x_{k+1}=f(x_{k})$ converge to some fixed-point $\tilde{x} \in \mathcal{D}$, i.e., to some point satisfying $\tilde{x}=f(\tilde{x})$, for any initial-condition $x_0 \in \mathcal{D}$.
One fact I know is that there exists a matrix $M \in R^{n \times n}$, such that for all $x \in \mathcal{D}$, the component-wise inequality $f(x) \leq Mx$ holds.
The properties of matrix $M$ are as follows:
- Each component $M_{ij} \geq 0, i=1,..,n, j=1,..n$.
- $\rho(M)<1$ (Largest absolute value of the eigen values of $M$ is less than 1)
Is there any way to show that the iterations $x_{k+1}=f(x_k)$ converge to some fixed-point in $\mathcal{D}$ from any initial-condition $x_0 \in \mathcal{D}$?
I intuitively see that the iterations $x_+=f(x)$ are always upper-bounded by the iterations of the decaying linear system $x_+=Mx$ (since $\rho(M)<1$). However, I am unable to use this matrix to establish convergence.
Very grateful for your time.