Difference and functional equations Definition

Iterative functional equation

Builds on Functional equation Iterated function Function

Definition

Let $X$ and $Y$ be sets. An iterative functional equation for an unknown function $\varphi \colon X \to Y$ is a functional equation in one variable $x \in X$ of the following kind. In it, $\varphi$ is evaluated at $x$ and at images of $x$ under given maps $f_1, \dots, f_k \colon X \to X$ or under iterates of $\varphi$ itself. The basic linear form, for $Y = \R$ and given $f \colon X \to X$ and $g, h \colon X \to \R$ , is $\varphi(f(x)) = g(x)\, \varphi(x) + h(x) \quad \text{for all } x \in X.$

Remarks

Example: $\varphi(2x) = \varphi(x) + 1$ on $X = (0, \infty)$ is of the basic linear form with $f(x) = 2x$ , $g = 1$ , $h = 1$ , and $\varphi = \log_2$ solves it. Non-example: Cauchy's equation $\varphi(x + y) = \varphi(x) + \varphi(y)$ is not iterative, since it involves two independent variables. The case $g = 1$ , $h = 1$ is the Abel equation, and the case $g = s$ constant, $h = 0$ is Schröder's equation. Equations with iterates of the unknown, such as $\varphi(\varphi(x)) = f(x)$ , ask for iterative roots of $f$ .

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Function concept
Functional equation concept
Iterated function concept
Iterative functional equation definition

Tag	`GGDJN5KXN4`
URI	`m2://GGDJN5KXN4`
Kind	definition
Writing	W0 stub
MSC	39B12, 26A18
Defines	Iterative functional equation
Uses	Functional equation, Iterated function, Function

Iterative functional equation

Definition

Remarks

Reading path

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