Hyers-Ulam Stability of Iterative Equation in the Class of Lipschitz Functions

It is pointed out that the theory of Ulam’s type stability (also quite often connected, e.g., with the names of Bourgin, Găvruţa, Aoki, Hyers, and Rassias) is a very popular subject of investigations at the moment in [1]. For more details of Ulam’s type stability, we refer the audience to [1–3]. In [1], the authors present a survey of some selected recent developments (results and methods) in the theory of Ulam’s type stability, such as the Forti method and the methods of fixed points and stability in non-Archimedean spaces. These results and methods have not been treated at all or have been treated only marginally. The very book of Jung [2] covers and offers almost all classic results on theHyers-UlamRassias stability in an integrated and self-contained fashion. And the authors of [3] discussed Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equations, nonlinear functional equations, and iterative equations. And they also clarified the relation between Hyers-Ulam stability and other senses of stability which are used for functional equations. Let C(X,X) be the set of all continuous self-mappings on a topological space X. For any f ∈ C(X,X), let fm denote the mth iteration of f; that is, fm = f ∘ f,


Introduction
It is pointed out that the theory of Ulam's type stability (also quite often connected, e.g., with the names of Bourgin, G ȃ vrut ¸a, Aoki, Hyers, and Rassias) is a very popular subject of investigations at the moment in [1].For more details of Ulam's type stability, we refer the audience to [1][2][3].In [1], the authors present a survey of some selected recent developments (results and methods) in the theory of Ulam's type stability, such as the Forti method and the methods of fixed points and stability in non-Archimedean spaces.These results and methods have not been treated at all or have been treated only marginally.The very book of Jung [2] covers and offers almost all classic results on the Hyers-Ulam-Rassias stability in an integrated and self-contained fashion.And the authors of [3] discussed Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equations, nonlinear functional equations, and iterative equations.And they also clarified the relation between Hyers-Ulam stability and other senses of stability which are used for functional equations.
Let (, ) be the set of all continuous self-mappings on a topological space .For any  ∈ (,), let   denote the th iteration of ; that is,   =  ∘  −1 ,  0 = id,  = 1, 2, . ... Equations having iterations as their main operation, that is, including iterations of the unknown mapping, are called iterative equations.It is one of the most interesting classes of functional equations [4][5][6][7].As a natural generalization of the problem of iterative roots, iterative equations of the following form: are known as polynomial-like iterative equations.Here  ≥ 2 is an integer,   ∈ R ( = 1, 2, . . ., ),  :  → R is a given mapping, and  :  →  is unknown.As mentioned in [8,9], polynomial-like iterative equations are important not only in the study of functional equations but also in the study of dynamical systems.
In 1986, Zhang [10] constructed an interesting operator called "structural operator" for (1) and used the fixed point theory in Banach space to get the solutions of (1).From then on (1) and other types of equations were discussed extensively by employing this idea (see [8,9,[11][12][13][14][15][16] and references therein).In 2002, by means of a modification of Zhang's method applied in [10], Kulczycki and Tabor [11] investigated the existence of Lipschitzian solutions of the iterative equation where B is a compact convex subset of R  and  : B → B is a given Lipschitz function.They generalized the results of Zhang [10] to a more general case.
In 2002, Xu and Zhang [17] discussed the Hyers-Ulam stability for a nonlinear iterative equation which includes the polynomial-like iterative equation.In 2003, Agarwal et al. [3] investigated the Hyers-Ulam stability for a general form of iterative equations which include the polynomial-like iterative equation with variable coefficients.Motivated by the above results, in this paper, we will discuss the Hyers-Ulam stability of (2).As in [11], we give a result on the Hyers-Ulam stability of a functional equation with a more general form firstly. And, by means of this result, the Hyers-Ulam stability of (2) is investigated.In fact, we want to generalize the result of Xu and Zhang [17] to high-dimensional case.

Basic Assumptions, Definitions, and Lemmas
Let B be a compact convex subset of R  ,  ∈ Z + , with nonempty interior.Let In (B, R  ), we use the supremum norm where ‖ ⋅ ‖ denotes the usual metric of R  .Obviously, (B, R  ) is a complete metric space.
Definition 1 (see [11]).Let B be a convex compact subset of R  with nonempty interior.For  ∈ [0, 1], and  ∈ [ where B denotes the boundary of B.
And let It is easy to see that Lip B (0, ∞) ∈ A.
Let  ∈ A and let P :  →  be a map.For any  ∈  we will consider a functional equation of the following form: where  ∈  is unknown.
Definition 8. If, for every  ∈ Lip B (0, ∞) such that ‖ − P() ∘ ‖ B ≤ , there exists a solution  of ( 17) such that ‖ − ‖ B ≤ , where  > 0 is a constant and  does not depend on the choice of , then ( 17) is said to have Hyers-Ulam stability.Then the following two facts hold:

Main Results
(c1): for any  ∈  ∩ Lip B (0, ), P() is a homeomorphism with lip((P()) −1 ) ≤ 1/(); (c2): for any given  0 ∈  ∩ Lip B (0, ), the sequence is continuous.By Lemma 4 and assumptions of , the mapping is well defined and continuous.Now, by the assumption (P1) and the fact P :  → , it is easy to see that, for every  ∈ Lip B (0, ) ∩ , Since (A2) holds and by Lemma 2, we know that all elements of Lip B ((), ∞) are invertible.Moreover we can get that and by means of (A1) we have Finally, by inequality (P2), we obtain that the mapping is well defined.Take any  0 ∈  ∩ Lip B (0, ).By means of the mapping (25), it is easy to see that the sequence is well defined and each   ∈  ∩ Lip B (0, ).By virtue of Lemma 5, it is easy to see that all   are surjective. where Proof.Since all the assumptions of Lemma 9 hold, so we can construct a sequence {  } of functions as follows.We take first and then by means of equality (18) repeatedly, define By the conclusions (c1) and (c2) of Lemma hold for  = 1, 2, . ... We prove these two inequalities inductively.
For  = 1, by virtue of the definition of  1 , we have has Hyers-Ulam stability.