On approximate homomorphisms: a fixed point approach
Consider the functional equation ℑ 1 ( f ) = ℑ 2 ( f ) ( ℑ )in a certain general setting. A function g is an approximate solution of ( ℑ )if ℑ 1 ( g )and ℑ 2 ( g )are close in some sense. The Ulam stability problem asks whether or not there is a true solution of ( ℑ )near g . A functional equation i...
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| Published in | Mathematical sciences (Karaj, Iran) Vol. 6; no. 1; pp. 59 - 8 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2012
Springer Nature B.V BioMed Central Ltd |
| Subjects | |
| Online Access | Get full text |
| ISSN | 2008-1359 2251-7456 2251-7456 |
| DOI | 10.1186/2251-7456-6-59 |
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| Abstract | Consider the functional equation
ℑ
1
(
f
) =
ℑ
2
(
f
) (
ℑ
)in a certain general setting. A function
g
is an approximate solution of (
ℑ
)if
ℑ
1
(
g
)and
ℑ
2
(
g
)are close in some sense. The Ulam stability problem asks whether or not there is a true solution of (
ℑ
)near
g
. A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each
m
= 1,2,3,4, we will find out the general solution of the functional equation
f
(
ax
+
y
)
+
f
(
ax
-
y
)
=
a
m
-
2
[
f
(
x
+
y
)
+
f
(
x
-
y
)
]
+
2
(
a
2
-
1
)
[
a
m
-
2
f
(
x
)
+
(
m
-
2
)
(
1
-
(
m
-
2
)
2
)
6
f
(
y
)
]
for any fixed integer
a
with
a
≠ 0, ± 1.
Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions.
2010 Mathematics Subject Classification
39B52, 47H10, 39B82 |
|---|---|
| AbstractList | Consider the functional equation
ℑ
1
(
f
) =
ℑ
2
(
f
) (
ℑ
)in a certain general setting. A function
g
is an approximate solution of (
ℑ
)if
ℑ
1
(
g
)and
ℑ
2
(
g
)are close in some sense. The Ulam stability problem asks whether or not there is a true solution of (
ℑ
)near
g
. A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each
m
= 1,2,3,4, we will find out the general solution of the functional equation
f
(
ax
+
y
)
+
f
(
ax
-
y
)
=
a
m
-
2
[
f
(
x
+
y
)
+
f
(
x
-
y
)
]
+
2
(
a
2
-
1
)
[
a
m
-
2
f
(
x
)
+
(
m
-
2
)
(
1
-
(
m
-
2
)
2
)
6
f
(
y
)
]
for any fixed integer
a
with
a
≠ 0, ± 1.
Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions.
2010 Mathematics Subject Classification
39B52, 47H10, 39B82 Consider the functional equation sub(1)(f) = sub(2)(f) ()in a certain general setting. A function g is an approximate solution of ()if sub(1)(g)and sub(2)(g)are close in some sense. The Ulam stability problem asks whether or not there is a true solution of ()near g. A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each m = 1,2,3,4, we will find out the general solution of the functional equation [Equation not available: see fulltext.] for any fixed integer a with a [ne] 0, plus or minus 1. Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions. 2010 Mathematics Subject Classification: 39B52, 47H10, 39B82 Consider the functional equation ^sub 1^(f) = ^sub 2^(f) ()in a certain general setting. A function g is an approximate solution of ()if ^sub 1^(g)and ^sub 2^(g)are close in some sense. The Ulam stability problem asks whether or not there is a true solution of ()near g. A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each m = 1,2,3,4, we will find out the general solution of the functional equation [Equation not available: see fulltext.] for any fixed integer a with a [not =] 0, ± 1. Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions. 39B52, 47H10, 39B82 : Consider the functional equation ℑ1(f) = ℑ2(f) (ℑ)in a certain general setting. A function g is an approximate solution of (ℑ)if ℑ1(g)and ℑ2(g)are close in some sense. The Ulam stability problem asks whether or not there is a true solution of (ℑ)near g. A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each m = 1,2,3,4, we will find out the general solution of the functional equationf(ax+y)+f(ax−y)=am−2[f(x+y)+f(x−y)]+2(a2−1)[am−2f(x)+(m−2)(1−(m−2)2)6f(y)]for any fixed integer a with a≠0,±1.Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions. 2010 MATHEMATICS SUBJECT CLASSIFICATION: 39B52, 47H10, 39B82 |
| Author | Khodaei, Hamid Gordji, Madjid Eshaghi Alizadeh, Zahra Park, Choonkil |
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| Copyright | Gordji et al.; licensee Springer. 2012. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Author(s) 2012 |
| Copyright_xml | – notice: Gordji et al.; licensee Springer. 2012. This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. – notice: The Author(s) 2012 |
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| Keywords | Fixed point approach Additive Approximate homomorphism Banach algebra Quadratic Cubic and quartic functional equation |
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| References | Eshaghi GordjiMKhodaeiHOn the generalized Hyers-Ulam-Rassias stability of quadratic functional equationsAbstract and Appl. Anal20092009112516012 GordgiMEGhobadipourNHyers-Ulam-Aoki-Rassias Stability and Ulam-Gǎvruta-Rassias Stability of the Quadratic Homomorphisms and Quadratic Derivations on Banach Algebras2010Nova Science Publishers Inc IsacGRassiasThMOn the Hyers-Ulam stability of ψ-additive mappingsJ. Approx. Theory19937213113710.1006/jath.1993.101012041350770.41018 FortiGLAn existence and stability theorem for a class of functional equationsStochastica19804233010.1080/174425080088331555737230442.39005 CădariuLRaduVFixed points and the stability of Jensen functional equationJ. Ineq. Pure Appl. Math200347 RassiasJMSolution of a problem of UlamJ. Approx. Theory19895726827310.1016/0021-9045(89)90041-59998610672.41027 ParkCLie ∗-homomorphisms between Lie C∗-algebras and Lie ∗-derivations on Lie C∗-algebrasJ. Math. Anal. Appl200429341943410.1016/j.jmaa.2003.10.05120538881051.46052 BakerJLawrenceJZorzittoFThe stability of the equation f(x+y) = f(x)f(y)Proc. Amer. Math. Soc1979742422465242940397.39010 Eshaghi GordjiMKarimiTKaboli GharetapehSApproximately n-Jordan homomorphisms on Banach algebrasJ. Ineq. Appl. Appl2009 MargolisBDiazJBA fixed point theorem of the alternative for contractions on the generalized complete metric spaceBull. Amer. Math. Soc1968126305309220267 UlamSMProblems in Modern Mathematics, Chapter VI, science Editions1964New YorkWiley Eshaghi GordjiMNajatiAApproximately J∗-homomorphisms: A fixed point approachJ. Geometry and Phys20106080981410.1016/j.geomphys.2010.01.01226085301192.39020 GordjiMESavadkouhiMBApproximation of generalized homomorphisms in quasi–Banach algebrasAnalele Univ. Ovidius Constata, Math Ser20091722032141199.39061 NajatiAThe generalized Hyers-Ulam-Rassias stability of a cubic functional equationTurk. J. Math20073139540823649091140.39014 RassiasJMOn Approximation of approximately linear mappings by linear mappingsJ. Funct. Anal19824612613010.1016/0022-1236(82)90048-96544690482.47033 LeeYChungSStability of quartic functional equation in the spaces of generalized functionsAdv. Difference Equations200920091610.1155/2009/8383472491087 BourginDGClasses of transformations and bordering transformationsBull. Amer. Math. Soc19515722323710.1090/S0002-9904-1951-09511-7426130043.32902 RassiasThMOn the stability of the linear mapping in Banach spacesProc. Amer. Math. Soc19787229730010.1090/S0002-9939-1978-0507327-15073270398.47040 LeeYJunKOn the stability of approximately additive mappingsProc. Amer. Math. Soc20001281361136910.1090/S0002-9939-99-05156-416411280961.47039 RassiasThMOn the stability of functional equations and a problem of UlamActa Appl. Math2000622313010.1023/A:100649922357217780160981.39014 RaduVThe fixed point alternative and the stability of functional equationsFixed Point Theory20034919620318241051.39031 CzerwikSOn the stability of the quadratic mapping in normed spacesAbh. Math. Sem. Univ. Hamburg199262596410.1007/BF0294161811828410779.39003 RassiasThMNew characterization of inner product spacesBull. Sci. Math198410895997516630544.46016 GajdaZOn stability of additive mappingsInt. J. Math. Math. Sci19911443143410.1155/S016117129100056X11100360739.39013 AczelJDhombresJFunctional Equations in Several Variables1989CambridgeCambridge University Press10.1017/CBO97811390865780685.39006 KannappanPQuadratic functional equation and inner product spacesResults Math19952736837210.1007/BF0332284113311100836.39006 Gordji EshaghiMKhodaeiHSolution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spacesNonlinear Analysis.–TMA2009715629564310.1016/j.na.2009.04.0521179.39034 KhodaeiHRassiasThMApproximately generalized additive functions in several variablesInt. J. Nonlinear Anal. Appl2010122411281.39041 AokiTOn the stability of the linear transformation in Banach spacesJ. Math. Soc. Japan19502646610.2969/jmsj/00210064405800040.35501 LeeSImSHawngIQuartic functional equationJ. Math. Anal. Appl200530738739410.1016/j.jmaa.2004.12.06221424321072.39024 JungSHyers-Ulam-Rassias stability of Jensen’s equation and its applicationProc. Amer. Math. Soc19981263137314310.1090/S0002-9939-98-04680-214761420909.39014 BaeJParkWA functional equation having monomials as solutionsAppl. Mathematics and Comput2010216879410.1016/j.amc.2010.01.00625961351191.39026 BadoraROn approximate ring homomorphismsJ. Math. Anal. Appl200227658959710.1016/S0022-247X(02)00293-719447771014.39020 JungSHyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis2001Palm Harbor, FloridaHadronic Press Inc.0980.39024 ParkCHomomorphisms between Lie JC∗-algebras and Cauchy-Rassias stability of Lie JC∗-algebra derivationsJ. Lie Theory20051539341421474351091.39006 GávrutaPA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappingsJ. Math. Anal. Appl199418443143610.1006/jmaa.1994.121112815180818.46043 JunKKimHThe generalized Hyers-Ulam-Rassias stability of a cubic functional equationJ. Math. Anal. Appl200227486787810.1016/S0022-247X(02)00415-819367351021.39014 Că dariuLRaduVFixed point methods for the generalized stability of functional equations in a single variableFixed Point Theory and App2008200815 LeeYChungSStability for quadratic functional equation in the spaces of generalized functionsJ. Math. Anal. Appl200733610111010.1016/j.jmaa.2007.02.05323484941125.39026 RassiasThMSemrlPOn the behavior of mappings which do not satisfy Hyers-Ulam stabilityProc. Amer. Math. Soc199211498999310.1090/S0002-9939-1992-1059634-110596340761.47004 FortiGLElementary remarks on Ulam-Hyers stability of linear functional equationsJ. Math. Anal. Appl200732810911810.1016/j.jmaa.2006.04.07922855361111.39026 BourginDGApproximately isometric and multiplicative transformations on continuous function ringsDuke Math. J19491638539710.1215/S0012-7094-49-01639-7311940033.37702 CădariuLRaduVOn the stability of the Cauchy functional equation: a fixed point approachGrazer Math. Ber200434643521060.39028 HyersDHOn the stability of the linear functional equationProc. Nat. Acad. Sci. USA19412722222410.1073/pnas.27.4.2224076 GordgiMEBavand SavadkouhiMOn approximate cubic homomorphismsAdv. Difference Equations2009200911 HyersDHIsacGRassiasThMStability of Functional Equations in Several Variables199810.1007/978-1-4612-1790-9 ParkCOn an approximate automorphism on a C∗-algebraProc. Amer. Math. Soc20041321739174510.1090/S0002-9939-03-07252-620511351055.47032 ParkCJangS-YCauchy–Rassias stability of sesquilinear n-quadratic mappings in Banach modulesRocky Mountain J. Math20093962015202710.1216/RMJ-2009-39-6-201525758911195.39009 |
| References_xml | – reference: Eshaghi GordjiMKhodaeiHOn the generalized Hyers-Ulam-Rassias stability of quadratic functional equationsAbstract and Appl. Anal20092009112516012 – reference: LeeSImSHawngIQuartic functional equationJ. Math. Anal. Appl200530738739410.1016/j.jmaa.2004.12.06221424321072.39024 – reference: RaduVThe fixed point alternative and the stability of functional equationsFixed Point Theory20034919620318241051.39031 – reference: CădariuLRaduVOn the stability of the Cauchy functional equation: a fixed point approachGrazer Math. Ber200434643521060.39028 – reference: NajatiAThe generalized Hyers-Ulam-Rassias stability of a cubic functional equationTurk. J. Math20073139540823649091140.39014 – reference: JunKKimHThe generalized Hyers-Ulam-Rassias stability of a cubic functional equationJ. Math. Anal. Appl200227486787810.1016/S0022-247X(02)00415-819367351021.39014 – reference: MargolisBDiazJBA fixed point theorem of the alternative for contractions on the generalized complete metric spaceBull. Amer. Math. Soc1968126305309220267 – reference: GajdaZOn stability of additive mappingsInt. J. Math. Math. Sci19911443143410.1155/S016117129100056X11100360739.39013 – reference: Eshaghi GordjiMNajatiAApproximately J∗-homomorphisms: A fixed point approachJ. Geometry and Phys20106080981410.1016/j.geomphys.2010.01.01226085301192.39020 – reference: AokiTOn the stability of the linear transformation in Banach spacesJ. Math. Soc. Japan19502646610.2969/jmsj/00210064405800040.35501 – reference: GordgiMEGhobadipourNHyers-Ulam-Aoki-Rassias Stability and Ulam-Gǎvruta-Rassias Stability of the Quadratic Homomorphisms and Quadratic Derivations on Banach Algebras2010Nova Science Publishers Inc – reference: BourginDGClasses of transformations and bordering transformationsBull. Amer. Math. Soc19515722323710.1090/S0002-9904-1951-09511-7426130043.32902 – reference: KannappanPQuadratic functional equation and inner product spacesResults Math19952736837210.1007/BF0332284113311100836.39006 – reference: Eshaghi GordjiMKarimiTKaboli GharetapehSApproximately n-Jordan homomorphisms on Banach algebrasJ. Ineq. Appl. Appl2009 – reference: CădariuLRaduVFixed points and the stability of Jensen functional equationJ. Ineq. Pure Appl. Math200347 – reference: JungSHyers-Ulam-Rassias stability of Jensen’s equation and its applicationProc. Amer. Math. Soc19981263137314310.1090/S0002-9939-98-04680-214761420909.39014 – reference: Că dariuLRaduVFixed point methods for the generalized stability of functional equations in a single variableFixed Point Theory and App2008200815 – reference: JungSHyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis2001Palm Harbor, FloridaHadronic Press Inc.0980.39024 – reference: Gordji EshaghiMKhodaeiHSolution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spacesNonlinear Analysis.–TMA2009715629564310.1016/j.na.2009.04.0521179.39034 – reference: HyersDHIsacGRassiasThMStability of Functional Equations in Several Variables199810.1007/978-1-4612-1790-9 – reference: RassiasThMOn the stability of functional equations and a problem of UlamActa Appl. Math2000622313010.1023/A:100649922357217780160981.39014 – reference: UlamSMProblems in Modern Mathematics, Chapter VI, science Editions1964New YorkWiley – reference: BakerJLawrenceJZorzittoFThe stability of the equation f(x+y) = f(x)f(y)Proc. Amer. Math. Soc1979742422465242940397.39010 – reference: GávrutaPA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappingsJ. Math. Anal. Appl199418443143610.1006/jmaa.1994.121112815180818.46043 – reference: FortiGLElementary remarks on Ulam-Hyers stability of linear functional equationsJ. Math. Anal. Appl200732810911810.1016/j.jmaa.2006.04.07922855361111.39026 – reference: LeeYChungSStability for quadratic functional equation in the spaces of generalized functionsJ. Math. Anal. Appl200733610111010.1016/j.jmaa.2007.02.05323484941125.39026 – reference: LeeYChungSStability of quartic functional equation in the spaces of generalized functionsAdv. 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| Snippet | Consider the functional equation
ℑ
1
(
f
) =
ℑ
2
(
f
) (
ℑ
)in a certain general setting. A function
g
is an approximate solution of (
ℑ
)if
ℑ
1
(
g
)and
ℑ
2
(... Consider the functional equation ^sub 1^(f) = ^sub 2^(f) ()in a certain general setting. A function g is an approximate solution of ()if ^sub 1^(g)and ^sub... Consider the functional equation sub(1)(f) = sub(2)(f) ()in a certain general setting. A function g is an approximate solution of ()if sub(1)(g)and... : Consider the functional equation ℑ1(f) = ℑ2(f) (ℑ)in a certain general setting. A function g is an approximate solution of (ℑ)if ℑ1(g)and ℑ2(g)are close in... |
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| SubjectTerms | Algebra Applications of Mathematics Approximation Classification Exact solutions Fixed points (mathematics) Functions (mathematics) Homomorphisms Mathematical analysis Mathematics Mathematics and Statistics Original Research Stability |
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| Title | On approximate homomorphisms: a fixed point approach |
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