Solving iterative functional equations for a class of piecewise linear -functions

We consider the problem of solving functional equations in K ∞ , the space of homeomorphisms of [ 0 , ∞ ) . The problem is treated for the subclass of continuous functions that are defined as affine functions on a countable partition of [ 0 , ∞ ) and for which the partition has no accumulation of intervals from the right. Within this class it can be shown that the problem has a unique solution. The proof is constructive and can be implemented numerically. Some properties of solutions are discussed.