Non-monotonic solutions and continuously differentiable solutions of conjugacy equations

In this paper solutions of conjugacy equation phi(f(x)) = g(phi(x)) for a strictly decreasing continuous given function f and a continuous given function g (maybe non-monotonic) are constructed by piecewise de. ning. We determine the conditions for piecewise continuously differentiable solutions of conjugacy equations with a strictly decreasing continuously differentiable given function f and a continuously differentiable given function g. Finally, the recursive algorithm is implemented in MATLAB software and two examples are respectively presented for a non-monotonic solution and a continuously differentiable one. (C) 2009 Elsevier Inc. All rights reserved.