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  "body": "\n## Definition\n\nLet $X$ and $Y$ be sets.\nAn *iterative functional equation* for an unknown [[function|0EPAT51DXQ]] $\\varphi \\colon X \\to Y$ is a [[functional equation|SRAN4EKWS0]] in one variable $x \\in X$ of the following kind.\nIn 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|AYVTR2ES5B]] of $\\varphi$ itself.\nThe basic linear form, for $Y = \\R$ and given $f \\colon X \\to X$ and $g, h \\colon X \\to \\R$, is\n$$\\varphi(f(x)) = g(x)\\, \\varphi(x) + h(x) \\quad \\text{for all } x \\in X.$$\n\n## Remarks\n\nExample: $\\varphi(2x) = \\varphi(x) + 1$ on $X = (0, \\infty)$ is of the basic linear form with $f(x) = 2x$, $g = 1$, $h = 1$, and $\\varphi = \\log_2$ solves it.\nNon-example: Cauchy's equation $\\varphi(x + y) = \\varphi(x) + \\varphi(y)$ is not iterative, since it involves two independent variables.\nThe case $g = 1$, $h = 1$ is the [[Abel equation|68SHB7Z721]], and the case $g = s$ constant, $h = 0$ is [[Schröder's equation|VTJV7JVJEA]].\nEquations with iterates of the unknown, such as $\\varphi(\\varphi(x)) = f(x)$, ask for [[iterative roots|27C6J3YNZY]] of $f$.\n",
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