Hyers-Ulam-Rassias stability

Hyers–Ulam stability

In almost-all areas of mathematical analysis one can ask the following question: "When is it true that a mathematical object satisfying a certain property approximately must be close to an object satisfying the property exactly?"

If one applies this question to the case of functional equations, one can particularly ask when the solutions of an equation differing slightly from a given one must be close to a solution of the given equation. The stability problem of functional equations originates from such a fundamental question.

In 1940, S.M. Ulam [a29] raised a question concerning the stability of homomorphisms: Let $G_1$ be a group and let $G_2$ be a metric group with a metric $d ( \cdot , \cdot )$. Given $\varepsilon > 0$, does there exist a $\delta > 0$ such that if a function $h : G _ { 1 } \rightarrow G _ { 2 }$ satisfies the inequality $d ( h ( x y ) , h ( x ) h ( y ) ) < \delta$ for all $x , y \in G _ { 1 }$, then there is a homomorphism $H : G _ { 1 } \rightarrow G _ { 2 }$ with $d ( h ( x ) , H ( x ) ) < \varepsilon$ for all $x \in G_1$?

In the following year 1941, D.H. Hyers [a11] gave a partial solution to Ulam's question. Hyers' theorem says that if a function $f : E _ { 1 } \rightarrow E _ { 2 }$ defined between Banach spaces (cf. also Banach space) satisfies the inequality

\begin{equation*} \| f ( x + y ) - f ( x ) - f ( y ) \| \leq \varepsilon \end{equation*}

for all $x , y \in E _ { 1 }$, then there exists a unique additive function $a : E _ { 1 } \rightarrow E _ { 2 }$ with

\begin{equation*} \| f ( x ) - a ( x ) \| \leq \varepsilon \end{equation*}

for all $x \in E _ { 1 }$.

For the above case one says that the additive Cauchy equation $f ( x + y ) = f ( x ) + f ( y )$ has the Hyers–Ulam stability on $( E _ { 1 } , E _ { 2 } )$.

Hyers explicitly constructed the additive function $a$ directly from the given function $f$ by the formula

\begin{equation*} a ( x ) = \operatorname { lim } _ { n \rightarrow \infty } 2 ^ { - n } f ( 2 ^ { n } x ). \end{equation*}

This method is called a direct method and is a powerful tool for studying the stability of several functional equations. It is often used to construct a certain function which is a solution of a given functional equation.

Following its appearance, Hyers' theorem was further extended in various directions (see [a3], [a5], [a7], [a8], [a14], [a15], [a20], [a21], [a25], [a26], [a27]). In particular, Th.M. Rassias [a20] considered a generalized version of it in which the Cauchy difference is allowed to become unbounded. He assumed that a function $f : E _ { 1 } \rightarrow E _ { 2 }$ between Banach spaces satisfies the inequality

\begin{equation*} \| f ( x + y ) - f ( x ) - f ( y ) \| \leq \theta ( \| x \| ^ { p } + \| y \| ^ { p } ) \end{equation*}

for some $\theta \geq 0$, $0 \leq p < 1$ and for all $x , y \in E _ { 1 }$. Functions such as $f$ are called approximately additive functions. Using a direct method, he proved that in this case too there exists a unique additive function $a : E _ { 1 } \rightarrow E _ { 2 }$ such that

\begin{equation*} \| f ( x ) - a ( x ) \| \leq K \| x \| ^ { p } \end{equation*}

for all $x \in E _ { 1 }$, where $K > 0$ depends on $p$ as well as on $\theta$. This phenomenon of Hyers–Ulam–Rassias stability was later extended to all $p \neq 1$ and generalizations of it were given (see [a7], [a8], [a14], [a15], [a25], [a16]). In general, Hyers–Ulam stability is a special case of Hyers–Ulam–Rassias stability.

Superstability.

An equation involving homomorphisms is called superstable if each approximate homomorphism is actually a true homomorphism. For example, in [a2] it is proved that if a real-valued function $f$ defined on a vector space $V$ satisfies the inequality

\begin{equation*} | f ( x + y ) - f ( x ) f ( y ) | \leq \varepsilon \end{equation*}

for some $\varepsilon > 0$ and for all $x , y \in V$, then either $f$ is a bounded function or $f ( x + y ) = f ( x ) f ( y )$ for all $x , y \in V$. This result was further generalized in [a1] and [a28] (cf. also [a9], [a10]). Superstability phenomena can also be regarded as special cases of Hyers–Ulam–Rassias stability.

For results concerning the stability of other equations, see [a4], [a17], [a18], [a19], [a22], or the references listed in [a12], [a30]. The survey papers [a13], [a6], [a16], [a23], [a24] contain general information on stability. See [a12], [a30] for a comprehensive introduction to the general theory of Hyers–Ulam–Rassias stability of functional equations.

In the same vein there is a theory of almost isomorphisms of Banch algebras, -perturbations of the multiplication and -isometries of Banach algebras. See [a31] for a selection of results.

References

[a1]	J. Baker, "The stability of the cosine equation" Proc. Amer. Math. Soc. , 80 (1980) pp. 411–416
[a2]	J. Baker, J. Lawrence, F. Zorzitto, "The stability of the equation $f ( x + y ) = f ( x ) f ( y )$" Proc. Amer. Math. Soc. , 74 (1979) pp. 242–246
[a3]	C. Borelli, G.L. Forti, "On a general Hyers–Ulam stability result" Internat. J. Math. Math. Sci. , 18 (1995) pp. 229–236
[a4]	S. Czerwik, "On the stability of the quadratic mapping in normed spaces" Abh. Math. Sem. Univ. Hamburg , 62 (1992) pp. 59–64
[a5]	G.L. Forti, "The stability of homomorphisms and amenability with applications to functional equations" Abh. Math. Sem. Univ. Hamburg , 57 (1987) pp. 215–226
[a6]	G.L. Forti, "Hyers–Ulam stability of functional equations in several variables" Aequat. Math. , 50 (1995) pp. 143–190
[a7]	Z. Gajda, "On stability of additive mappings" Internat. J. Math. Math. Sci. , 14 (1991) pp. 431–434
[a8]	P. Găvruta, "A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings" J. Math. Anal. Appl. , 184 (1994) pp. 431–436
[a9]	R. Ger, "Superstability is not natural" Rocznik Naukowo–Dydaktyczny WSP w Krakowie, Prace Mat. , 159 (1993) pp. 109–123
[a10]	R. Ger, P. Šemrl, "The stability of the exponential equation" Proc. Amer. Math. Soc. , 124 (1996) pp. 779–787
[a11]	D.H. Hyers, "On the stability of the linear functional equation" Proc. Nat. Acad. Sci. USA , 27 (1941) pp. 222–224
[a12]	D.H. Hyers, G. Isac, Th.M. Rassias, "Stability of functional equations in several variables" , Birkhäuser (1998)
[a13]	D.H. Hyers, Th.M. Rassias, "Approximate homomorphisms" Aequat. Math. , 44 (1992) pp. 125–153
[a14]	G. Isac, Th.M. Rassias, "On the Hyers–Ulam stability of $\psi$-additive mappings" J. Approx. Th. , 72 (1993) pp. 131–137
[a15]	S.-M. Jung, "On the Hyers–Ulam–Rassias stability of approximately additive mappings" J. Math. Anal. Appl. , 204 (1996) pp. 221–226
[a16]	S.-M. Jung, "Hyers–Ulam–Rassias stability of functional equations" Dynamic Syst. Appl. , 6 (1997) pp. 541–566
[a17]	S.-M. Jung, "Hyers–Ulam–Rassias stability of Jensen's equation and its application" Proc. Amer. Math. Soc. , 126 (1998) pp. 3137–3143
[a18]	S.-M. Jung, "On the Hyers–Ulam stability of the functional equations that have the quadratic property" J. Math. Anal. Appl. , 222 (1998) pp. 126–137
[a19]	Z. Kominek, "On a local stability of the Jensen functional equation" Demonstratio Math. , 22 (1989) pp. 499–507
[a20]	Th.M. Rassias, "On the stability of the linear mapping in Banach spaces" Proc. Amer. Math. Soc. , 72 (1978) pp. 297–300
[a21]	Th.M. Rassias, "On a modified Hyers–Ulam sequence" J. Math. Anal. Appl. , 158 (1991) pp. 106–113
[a22]	Th.M. Rassias, "On the stability of the quadratic functional equation" Mathematica (to appear)
[a23]	Th.M. Rassias, "On the stability of functional equations originated by a problem of Ulam" Studia Univ. Babes–Bolyai (to appear)
[a24]	Th.M. Rassias, "On the stability of functional equations and a problem of Ulam" Acta Applic. Math. , 62 (2000) pp. 23–130
[a25]	Th.M. Rassias, P. Šemrl, "On the Hyers–Ulam stability of linear mappings" J. Math. Anal. Appl. , 173 (1993) pp. 325–338
[a26]	Th.M. Rassias, J. Tabor, "What is left of Hyers–Ulam stability?" J. Natural Geometry , 1 (1992) pp. 65–69
[a27]	F. Skof, "Sull'approssimazione delle applicazioni localmente $\delta$-additive" Atti Accad. Sci. Torino , 117 (1983) pp. 377–389
[a28]	L. Székelyhidi, "On a theorem of Baker, Lawrence and Zorzitto" Proc. Amer. Math. Soc. , 84 (1982) pp. 95–96
[a29]	S.M. Ulam, "A collection of mathematical problems" , Interscience (1960)
[a30]	S.-M. Jung, "Hyers–Ulam–Rassias stability of functional equations in mathematical analysis" , Hadronic Press (2001)
[a31]	K. Jarosz, "Perturbations of Banach algebras" , Springer (1985)