Existence, uniqueness and stability of C m 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∈J ), where J is a connected closed subset of the real number axis R , G ∈ C m ( J n+1 , R )，and n ≥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 C m solutions of the above equation for any integer m ≥0 under relatively weak conditions, and generalize related results in reference in different aspects.