Difference and functional equations Stability of functional equations Theorem

Hyers' theorem

Builds on Approximately additive function Additive function Normed vector space Banach space Linear map

Statement

Let $E_1$ be a real normed vector space, let $E_2$ be a real Banach space, and let $\delta \ge 0$ . Let $f \colon E_1 \to E_2$ be $\delta$ -additive, that is, $\norm{f(x + y) - f(x) - f(y)} \le \delta$ for all $x, y \in E_1$ . Then:

For each $x \in E_1$ the limit $A(x) = \lim_{n \to \infty} 2^{-n} f(2^n x)$ exists in $E_2$ .
The map $A \colon E_1 \to E_2$ is additive and satisfies $\norm{f(x) - A(x)} \le \delta$ for all $x \in E_1$ .
$A$ is the only additive map $E_1 \to E_2$ such that $f - A$ is bounded.
If moreover, for each $x \in E_1$ , the map $t \mapsto f(tx)$ from $\R$ to $E_2$ is continuous, then $A$ is linear.

Remarks

Setting $x = y$ gives $\norm{f(2x) - 2f(x)} \le \delta$ , so consecutive terms of the sequence $2^{-n} f(2^n x)$ differ by at most $2^{-n-1}\delta$ . The sequence is therefore Cauchy, it converges by completeness of $E_2$ , and summing the geometric bounds gives $\norm{f(x) - A(x)} \le \delta$ . Additivity follows from $\norm{2^{-n}(f(2^n(x + y)) - f(2^n x) - f(2^n y))} \le 2^{-n}\delta$ . For uniqueness, the difference $D$ of two such additive maps is additive and bounded, and $D(x) = D(nx)/n \to 0$ . Example: $f(x) = x + \sin x$ on $\R$ is $3$ -additive, and the theorem gives $A(x) = x$ with $\abs{f(x) - x} \le 1$ . Completeness of $E_2$ is used to make the limit exist. D. H. Hyers proved the theorem in 1941, answering Ulam's stability problem for the Cauchy equation.

Going deeper

For a student

The theorem shows that the Cauchy equation $f(x + y) = f(x) + f(y)$ is Hyers–Ulam stable with constant $1$ . The method, now called the direct method, works for many other equations, such as the Jensen and quadratic equations, after a suitable rescaling. Replacing the constant $\delta$ by a bound that grows with $\norm{x}$ and $\norm{y}$ leads to the Aoki–Rassias theorem.

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Aoki–Rassias theorem

Tag	`VFRJ0P2P15`
URI	`m2://VFRJ0P2P15`
Kind	statement, theorem
Writing	W0 stub
MSC	39B82, 39B52
Status	This copy of the site has no build data. Run `m2 status VFRJ0P2P15`.
About	Hyers–Ulam stability, Approximately additive function
Uses	Approximately additive function, Additive function, Normed vector space, Banach space, Linear map

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Aoki–Rassias theorem