View source for 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 [[#References|[a29]]] raised a question concerning the stability of homomorphisms: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h1301201.png" /> be a [[Group|group]] and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h1301202.png" /> be a metric group with a [[Metric|metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h1301203.png" />. Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h1301204.png" />, does there exist a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h1301205.png" /> such that if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h1301206.png" /> satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h1301207.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h1301208.png" />, then there is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h1301209.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012011.png" />?

In the following year 1941, D.H. Hyers [[#References|[a11]]] gave a partial solution to Ulam's question. Hyers' theorem says that if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012012.png" /> defined between Banach spaces (cf. also [[Banach space|Banach space]]) satisfies the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012013.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012014.png" />, then there exists a unique additive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012015.png" /> with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012016.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012017.png" />.

For the above case one says that the additive Cauchy equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012018.png" /> has the Hyers–Ulam stability on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012019.png" />.

Hyers explicitly constructed the additive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012020.png" /> directly from the given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012021.png" /> by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012022.png" /></td> </tr></table>

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 [[#References|[a3]]], [[#References|[a5]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a14]]], [[#References|[a15]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a25]]], [[#References|[a26]]], [[#References|[a27]]]). In particular, Th.M. Rassias [[#References|[a20]]] considered a generalized version of it in which the Cauchy difference is allowed to become unbounded. He assumed that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012023.png" /> between Banach spaces satisfies the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012024.png" /></td> </tr></table>

for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012026.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012027.png" />. Functions such as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012028.png" /> are called approximately additive functions. Using a direct method, he proved that in this case too there exists a unique additive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012029.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012030.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012032.png" /> depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012033.png" /> as well as on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012034.png" />. This phenomenon of Hyers–Ulam–Rassias stability was later extended to all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012035.png" /> and generalizations of it were given (see [[#References|[a7]]], [[#References|[a8]]], [[#References|[a14]]], [[#References|[a15]]], [[#References|[a25]]], [[#References|[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 [[#References|[a2]]] it is proved that if a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012036.png" /> defined on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012037.png" /> satisfies the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012038.png" /></td> </tr></table>

for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012039.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012040.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012041.png" /> is a bounded function or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012042.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012043.png" />. This result was further generalized in [[#References|[a1]]] and [[#References|[a28]]] (cf. also [[#References|[a9]]], [[#References|[a10]]]). Superstability phenomena can also be regarded as special cases of Hyers–Ulam–Rassias stability.

For results concerning the stability of other equations, see [[#References|[a4]]], [[#References|[a17]]], [[#References|[a18]]], [[#References|[a19]]], [[#References|[a22]]], or the references listed in [[#References|[a12]]], [[#References|[a30]]]. The survey papers [[#References|[a13]]], [[#References|[a6]]], [[#References|[a16]]], [[#References|[a23]]], [[#References|[a24]]] contain general information on stability. See [[#References|[a12]]], [[#References|[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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012046.png" />-perturbations of the multiplication and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012048.png" />-isometries of Banach algebras. See [[#References|[a31]]] for a selection of results.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Baker,   "The stability of the cosine equation"  ''Proc. Amer. Math. Soc.'' , '''80'''  (1980)  pp. 411–416</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Baker,   J. Lawrence,   F. Zorzitto,   "The stability of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012049.png" />"  ''Proc. Amer. Math. Soc.'' , '''74'''  (1979)  pp. 242–246</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C. Borelli,   G.L. Forti,   "On a general Hyers–Ulam stability result"  ''Internat. J. Math. Math. Sci.'' , '''18'''  (1995)  pp. 229–236</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Czerwik,   "On the stability of the quadratic mapping in normed spaces"  ''Abh. Math. Sem. Univ. Hamburg'' , '''62'''  (1992)  pp. 59–64</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  G.L. Forti,   "The stability of homomorphisms and amenability with applications to functional equations"  ''Abh. Math. Sem. Univ. Hamburg'' , '''57'''  (1987)  pp. 215–226</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G.L. Forti,   "Hyers–Ulam stability of functional equations in several variables"  ''Aequat. Math.'' , '''50'''  (1995)  pp. 143–190</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  Z. Gajda,   "On stability of additive mappings"  ''Internat. J. Math. Math. Sci.'' , '''14'''  (1991)  pp. 431–434</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  P. Găvruta,   "A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings"  ''J. Math. Anal. Appl.'' , '''184'''  (1994)  pp. 431–436</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R. Ger,   "Superstability is not natural"  ''Rocznik Naukowo–Dydaktyczny WSP w Krakowie, Prace Mat.'' , '''159'''  (1993)  pp. 109–123</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  R. Ger,   P. Šemrl,   "The stability of the exponential equation"  ''Proc. Amer. Math. Soc.'' , '''124'''  (1996)  pp. 779–787</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  D.H. Hyers,   "On the stability of the linear functional equation"  ''Proc. Nat. Acad. Sci. USA'' , '''27'''  (1941)  pp. 222–224</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  D.H. Hyers,   G. Isac,   Th.M. Rassias,   "Stability of functional equations in several variables" , Birkhäuser  (1998)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  D.H. Hyers,   Th.M. Rassias,   "Approximate homomorphisms"  ''Aequat. Math.'' , '''44'''  (1992)  pp. 125–153</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  G. Isac,   Th.M. Rassias,   "On the Hyers–Ulam stability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012050.png" />-additive mappings"  ''J. Approx. Th.'' , '''72'''  (1993)  pp. 131–137</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  S.-M. Jung,   "On the Hyers–Ulam–Rassias stability of approximately additive mappings"  ''J. Math. Anal. Appl.'' , '''204'''  (1996)  pp. 221–226</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  S.-M. Jung,   "Hyers–Ulam–Rassias stability of functional equations"  ''Dynamic Syst. Appl.'' , '''6'''  (1997)  pp. 541–566</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  S.-M. Jung,   "Hyers–Ulam–Rassias stability of Jensen's equation and its application"  ''Proc. Amer. Math. Soc.'' , '''126'''  (1998)  pp. 3137–3143</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  Z. Kominek,   "On a local stability of the Jensen functional equation"  ''Demonstratio Math.'' , '''22'''  (1989)  pp. 499–507</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  Th.M. Rassias,   "On the stability of the linear mapping in Banach spaces"  ''Proc. Amer. Math. Soc.'' , '''72'''  (1978)  pp. 297–300</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  Th.M. Rassias,   "On a modified Hyers–Ulam sequence"  ''J. Math. Anal. Appl.'' , '''158'''  (1991)  pp. 106–113</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  Th.M. Rassias,   "On the stability of the quadratic functional equation"  ''Mathematica''  (to appear)</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top">  Th.M. Rassias,   "On the stability of functional equations originated by a problem of Ulam"  ''Studia Univ. Babes–Bolyai''  (to appear)</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  Th.M. Rassias,   "On the stability of functional equations and a problem of Ulam"  ''Acta Applic. Math.'' , '''62'''  (2000)  pp. 23–130</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top">  Th.M. Rassias,   P. Šemrl,   "On the Hyers–Ulam stability of linear mappings"  ''J. Math. Anal. Appl.'' , '''173'''  (1993)  pp. 325–338</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top">  Th.M. Rassias,   J. Tabor,   "What is left of Hyers–Ulam stability?"  ''J. Natural Geometry'' , '''1'''  (1992)  pp. 65–69</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top">  F. Skof,   "Sull'approssimazione delle applicazioni localmente <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h130/h130120/h13012051.png" />-additive"  ''Atti Accad. Sci. Torino'' , '''117'''  (1983)  pp. 377–389</TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top">  L. Székelyhidi,   "On a theorem of Baker, Lawrence and Zorzitto"  ''Proc. Amer. Math. Soc.'' , '''84'''  (1982)  pp. 95–96</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top">  S.M. Ulam,   "A collection of mathematical problems" , Interscience  (1960)</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top">  S.-M. Jung,   "Hyers–Ulam–Rassias stability of functional equations in mathematical analysis" , Hadronic Press  (2001)</TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top">  K. Jarosz,   "Perturbations of Banach algebras" , Springer  (1985)</TD></TR></table>