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  "body": "\n## Definition\n\nLet $X$ be a nonempty set, let $k \\ge 1$, and let $(Y, d)$ be a [[metric space|0VKD1TP7F3]].\nLet $\\mathcal{F}$ be a set of maps $X \\to Y$, and let $E_1, E_2$ assign to each $f \\in \\mathcal{F}$ a map $X^k \\to Y$.\nConsider the [[functional equation|SRAN4EKWS0]] $E_1(f)(u) = E_2(f)(u)$ for all $u \\in X^k$.\nThe equation is *Hyers–Ulam stable* in $\\mathcal{F}$ if there is a constant $K \u003e 0$ such that the following holds for every $\\delta \\ge 0$.\nIf $f \\in \\mathcal{F}$ satisfies $d(E_1(f)(u), E_2(f)(u)) \\le \\delta$ for all $u \\in X^k$, then there is a solution $g \\in \\mathcal{F}$ of the equation with $d(f(x), g(x)) \\le K \\delta$ for all $x \\in X$.\n\n## Remarks\n\nExample: the Cauchy equation $f(x + y) = f(x) + f(y)$ for maps from a normed space to a Banach space is Hyers–Ulam stable with $K = 1$; here $k = 2$, $E_1(f)(x, y) = f(x + y)$ and $E_2(f)(x, y) = f(x) + f(y)$.\nNon-example: the periodicity equation $f(x + 1) = f(x)$ for maps $\\R \\to \\R$ is not stable.\nFor $\\delta \u003e 0$ the map $f(x) = \\delta x$ satisfies $\\abs{f(x + 1) - f(x)} = \\delta$.\nIf $g$ is $1$-periodic, then $f(x + n) - g(x + n) = f(x) - g(x) + n\\delta$ for every integer $n$, so $f - g$ is unbounded and no $K$ works.\nSome authors use a weaker form: for every $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that a $\\delta$-approximate solution lies within $\\varepsilon$ of a solution.\nIn the Hyers–Ulam–Rassias form, the constant bound $\\delta$ is replaced by a control function such as $\\theta(\\norm{x}^p + \\norm{y}^p)$.\n",
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    "name": "Hyers–Ulam stability",
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  "msc": [
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  "name": "Hyers–Ulam stability",
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