Existence, uniqueness and stability of Cm solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x, f(x). ..., f(n)(x)) = 0 (for all x is an element ofJ), where J is a connected closed subset of the real number axis R, G is an element ofC(m)(J(n+1), R), and n greater than or equal to 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability of Cm solutions of the above equation for any integer m greater than or equal to0 under relatively weak conditions, and generalize related results in reference in different aspects.