Some Properties of the Continuous Single-Valley Expansion Solution to the Feigenbaum's Functional Equation

In this paper, some fundamental properties of the continuous single-valley expansion solution to Feigenbaum's functional equation were obtained. For A G ( 0,1) and p 2, we will discuss the completeness of the function space which consists of unique continuous single-valley expansion solution (resp. unique continuous non-single-valley expansion solution) to P order Feigenbaum's functional equation. Let p,q >= 2. It was proved that the system of equations {f (x) = 1/lambda f(p)(lambda x), f(0) = 1,(lambda is an element of (0,1) for decision) f (x), x is an element of [0,1]; f (x) = 1/lambda f(q) (lambda x), f (0)=1, (lambda is an element of (0,1) for decision) f (x), x is an element of [0,1]. does not have continuous single-valley expansion solution.