Research Of Iterative Functional Equations

By the iterative functional equation we usually mean an equation containing at least an n-th iterate (n ≥ 2) of an unknown function. A relatively general form of it isG(x, f(x),…, fⁿ(x)) = 0, for any x ∈ J, (1)where n ≥ 2 is a given integer, J is a connected closed subset of the real number axis R, G is a given C^m function from Jⁿ⁺¹ to R, and f is an unknown function to be solved. Each C^m function f which makes (1) hold is called a C^m solution of the equation.The existence, uniqueness and stability of C⁰, C¹ and C² solutions of some particular kinds of iterative functional equations have been discussed by means of structural operations. It was pointed out that even if for m ≥ 3, the existence of C^m solutions of the linear iterative functional equation F(x) - = 0 is a difficult and very interesting problem.In Chapter 2, we discuss C^m solutions of equation (1) for any integer m ≥ 0. 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 obtain a series of theorems on the existence, uniqueness and stability of C^m solutions of equation (1) under some relatively weak conditions, and generalize related results in a number of papers in different aspects.In Chapter 3, by means of the ideas, methods and skills used in Chapter 2, we go further into the system of iterative functional equations(where n ≥ 2, G, H are given C^m functions from J²ⁿ⁺¹ to R), and give some relatively weak conditions of the existence and uniqueness of C^m solutions and C^∞ solutions of the system of equations.In Chapter 4, we study the analytic solutions of the following iterative functional equationG(z, f(z), …, fⁿ(z)) = 0, for any z ∈ D, (3)where D is a closed disc in the complex plane C whose center is the origin, n ≥ 2, G is a given continuous function from Dⁿ⁺¹ to C, and G|(Dⁿ⁺¹)° is analytic. We present some conditions for the equation to have an analytic solution and a unique analytic solution.In Chapter 5, we discuss the system of equations consisted of equations in the form (3), and give some weak conditions of the existence and uniqueness of analytic solutions of the system of equations.Chapter 6 is based on Chapter 2, in which we discuss the following functional iterated equationG(x, f(x), f(H2(x, f(x)), â– â– â– , r(Hn(x, fix), â– â– ., fn^\x)))) = 0, for any x E J, (4)where n > 2, G is a given Cm function from Jn+1 to R, and H^ is a given Cm function from Jk to R, k = 2, ? ? ? ,n. For any integer ra > 0, we obtain some conditions for the equation to have a Cm solution and a unique Cm solution.