Series-Like Iterative Functional Equation for PM Functions

Given a non-empty subset X of the real line and a self map G on X, the functional equation representing G as an infinite linear combination of iterations of a self map g on X is known as the series-like functional equation. The solutions of the series-like functional equation have been studied only for the class of continuous strictly monotone functions. In this paper, we prove the existence of solutions of series-like functional equations for the class of continuous non-monotone functions using characteristic interval.


Introduction
Given a set X ⊆ R, a function G : X → X and a sequence of real numbers {λ n }, the functional equation of the form ∞ n=1 λ n g n (t) = G(t) for all t ∈ X is called a series-like iterated functional equation. Any function f : X → X satisfies (1) is called a solution to the series-like iterated functional equation (1). If λ n = 0 for all n > m for some m ∈ N then the functional equation (1) reduces to m n=1 λ n g n (t) = G(t) for all t ∈ X.
The functional equation (2) is known as polynomial-like iterative functional equation [12]. If λ 1 = 1 and λ n = 0 for all n ≥ 2 then the functional equation (1) further reduces to g n (t) = G(t) for all t ∈ X.
The functional equation (3) is known as an iterative functional equation [14]. The equation (3) is one of the simplest form of iterative functional equation, and it is examined in the classical works of Babbage [1]. The solution of (3) has been studied extensively for the class of continuous monotone functions in [2] and [4]. More precisely, for a given strictly increasing continuous function G on an interval I the equation (3) has solution for all n ∈ N,  (3) has solution for all odd n ∈ N and has no solution for all even n ∈ N (see Theorem 11.2.2 and 11.2.4 [5]). Zhang [14], introduced the concept of height and characteristic interval for a continuous non-monotone functions having finitely many non-monotone points, known as Piecewise Monotone(PM) functions, and studied the solutions of (3). Further developments and some open problems on the existence of solutions of (3) for PM functions and non-PM functions can be found in [7], [3], [6], [8].
The polynomial-like iterative functional equation (2), which is the generalization of (3), whose continuous and differentiable solutions has been studied in [11], [12], [13] for the class of continuous strictly monotone functions. The existence of solutions of (2) for the class of PM functions has been studied in [15] by using the method of characteristic interval.
The existence of continuous solutions of series-like iterative functional equation (1) has been discussed in [9], [10], but only for the class of strictly monotone functions. The existence of solutions of series-like iterative functional equation (1) for the class of PM functions remains unsolved. In this paper, we prove the existence of solutions of series-like iterative functional equation (1) for the class of PM functions by using the method of characteristic interval. At first, we prove the existence of solution of (1) on the characteristic interval and then extend that solution to the whole domain of the PM function. We also provide an example to illustrate our main theorem.

Preliminaries
Throughout this paper, we fix the domain of all functions to be the closed and bounded interval I = [a, b], and all functions are assumed to be continuous on I. We say that a point t ∈ I is a fort of the function g, if g is non-monotone in any neighborhood of t. Further, g is said to be piecewise monotone(PM) if it has finite number of forts. We denote the set of all continuous functions from I into itself by C(I) and the set of all PM functions from I into itself by P M (I).
For any PM function g, if we denote the number of forts by N (g) then it is easy to observe that {N (g n )} ∞ n=1 is a non-decreasing sequence of positive integers. We say a positive integer k as height of g, if k is the least positive integer such that N (g k ) = N (g k+1 ). If there is no such positive integer then we say the height of g is infinity [14]. Height of the function g is denoted by H(g).
The characteristic interval of g is the smallest closed interval containing the range of g whose endpoints are either forts of g or the endpoints of I and it is denoted by Ch g .
We observe that, for any PM function g, H(g) ≤ 1 if and only if g is strictly monotone in Ch g . A detailed study of functions having different height can be found in [3,6,7,14]. Let a , b ∈ I such that a < b and m, M be positive real numbers. We define a class of functions as follows: 3 existence of solution of (1) in the characteristic interval of G and then we extend that solution to the whole interval. The following lemma will lead to our main result.
Then for any Proof of the Lemma 2.2 follows from Corollary 3.5 in [10]. For each G ∈ S([a , b ], m, M ), Lemma 2.2 guarantees the solution of the series-like iterative functional equation (1) on the characteristic interval of G. Hence, to study existence of solutions of equation (1) for PM functions, in addition to the hypothesis of Lemma 2.2, it is enough to discuss the extension of such solution to the whole interval.

Extension of Solutions from the Characteristic Interval
In this section we prove our main theorem of extending solutions of (1) from the characteristic interval of G to the whole interval I. The following lemma will be useful in extending the above solution to the whole interval I.
Then for each g ∈ F ([a , b ], K 0 , K 1 ), the function L g : is invertible and Proof. Proof of Lemma 3.1 follows from Lemma 3.2 in [10].  Proof.
We now extend this g 0 to the whole interval I. For this, define Then, by Lemma 3.1, . Now, we define g : I → I as follows: The function g is well defined by condition (iii). Consequently, for each t ∈ [a , b ] we have ∞ n=1 λ n g n (t) = ∞ n=1 λ n g n 0 (t) = G 0 (t) = G(t).
Hence g satisfies the functional equation (1) for all t ∈ I. To prove g is continuous on I, by definition of g, it is suffices to prove g is continuous at the points a and b . If {t n } be a sequence in [a, a ) such that t n → a as n → ∞ then lim n→∞ g(t n ) = lim n→∞ L −1 g 0 (G(t n )) = L −1 g 0 (G(a )) = L −1 g 0 (a ) = g(a ).