A fixed point approach to the stability of an AQ-functional equation on β-Banach modules
Using the fixed point method, we prove the Hyers-Ulam stability of the following mixed additive and quadratic functional equation f ( kx + y ) + f ( kx - y ) = f ( x + y ) + f ( x - y ) + ( k - 1) [( k + 2) f ( x ) + kf (- x )] ( k ∈ ℕ, k ≠ 1) in β -Banach modules on a Banach algebra. MR(2000) Subje...
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| Published in | Fixed point theory and applications (Hindawi Publishing Corporation) Vol. 2012; no. 1; pp. 1 - 14 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.03.2012
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1687-1812 1687-1820 1687-1812 |
| DOI | 10.1186/1687-1812-2012-32 |
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| Abstract | Using the fixed point method, we prove the Hyers-Ulam stability of the following mixed additive and quadratic functional equation
f
(
kx
+
y
) +
f
(
kx
-
y
) =
f
(
x
+
y
) +
f
(
x
-
y
) + (
k
- 1) [(
k
+ 2)
f
(
x
) +
kf
(-
x
)] (
k
∈ ℕ,
k
≠ 1) in
β
-Banach modules on a Banach algebra.
MR(2000) Subject Classification
. 39B82; 39B52; 46H25. |
|---|---|
| AbstractList | Using the fixed point method, we prove the Hyers-Ulam stability of the following mixed additive and quadratic functional equation f (kx + y) + f(kx - y) = f(x + y) + f(x - y) + (k - 1) [(k + 2) f(x) + kf(-x)] (k , k [ne] 1) in beta -Banach modules on a Banach algebra. MR(2000) Subject Classification. 39B82; 39B52; 46H25. Using the fixed point method, we prove the Hyers-Ulam stability of the following mixed additive and quadratic functional equation f ( kx + y ) + f ( kx - y ) = f ( x + y ) + f ( x - y ) + ( k - 1) [( k + 2) f ( x ) + kf (- x )] ( k ∈ ℕ, k ≠ 1) in β -Banach modules on a Banach algebra. MR(2000) Subject Classification . 39B82; 39B52; 46H25. |
| ArticleNumber | 32 |
| Author | Rassias, John Michael Xu, Tian Zhou |
| Author_xml | – sequence: 1 givenname: Tian Zhou surname: Xu fullname: Xu, Tian Zhou email: xutianzhou@bit.edu.cn organization: School of Mathematics, Beijing Institute of Technology – sequence: 2 givenname: John Michael surname: Rassias fullname: Rassias, John Michael organization: Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens |
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| Cites_doi | 10.1007/s00010-008-2945-7 10.1007/978-0-387-89492-8 10.1016/j.jmaa.2009.08.021 10.1090/S0002-9904-1968-11933-0 10.1142/4875 10.1007/BF03322841 10.5486/PMD.1996.1559 10.1016/j.na.2009.02.123 10.1155/S016117129100056X 10.1016/j.jmaa.2003.09.032 10.1007/BF02941618 10.1016/j.jmaa.2005.09.027 10.1016/0022-1236(82)90048-9 10.1007/BF02192660 10.1155/S0161171296000324 10.1016/j.jmaa.2011.02.048 10.1016/j.aml.2011.03.002 10.1007/978-1-4419-9637-4 10.1007/BF02924890 10.1017/CBO9781139086578 10.1007/BF01831117 10.1155/2008/672618 10.2969/jmsj/00210064 10.1090/S0002-9939-1978-0507327-1 10.1073/pnas.27.4.222 10.1006/jmaa.1994.1211 10.1007/BF01830975 10.1007/978-1-4612-1790-9 10.1007/s00009-010-0082-8 |
| ContentType | Journal Article |
| Copyright | Xu and Rassias; 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. |
| Copyright_xml | – notice: Xu and Rassias; 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. |
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| Keywords | Banach module fixed point method AQ-functional equation Hyers-Ulam stability generalized metric space unital Banach algebra |
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Acta Math Sci Ser B, in press. AokiTOn the stability of the linear transformation in Banach spacesJ Math Soc Japan1950264660040.355014058010.2969/jmsj/00210064 AgarwalRPXuBZhangWStability of functional equations in single variableJ Math Anal Appl20032888528691053.39042202020110.1016/j.jmaa.2003.09.032 EskandaniGZGăvruţaPRassiasJMZarghamiRGeneralized Hyers-Ulam stability for a general mixed functional equation in quasi-β-normed spacesMediterr J Math201183313481236.39026282458510.1007/s00009-010-0082-8 UlamSMA Collection of the Mathematical Problems1960New YorkInterscience ParkCRassiasThMHyers-Ulam stability of a generalized quadratic Apollonius type mappingJ Math Anal Appl20063223713811101.39020223924510.1016/j.jmaa.2005.09.027 JungSMHyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis2011New YorkSpringer10.1007/978-1-4419-9637-4 SkofFLocal properties and approximations of operatorsRend Sem Mat Fis Milano1983531131290599.3900785854110.1007/BF02924890 HyersDHRassiasThMApproximate homomorphismsAequationes Math1992441251530806.47056118126410.1007/BF01830975 RassiasJMOn approximation of approximately linear mappings by linear mappingsJ Funct Anal1982461261300482.4703365446910.1016/0022-1236(82)90048-9 RassiasJMOn approximation of approximately linear mappings by linear mappingsBull des Sci Math19841084454460599.47106 GrabiecAThe generalized Hyers-Ulam stability of a class of functional equationsPubl Math Debrecen1996482172351274.390581394843 XuTZRassiasJMRassiasMJXuWXA fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spacesJ Inequal Appl2010201023Article ID 423231 CădariuLRaduVOn the stability of the Cauchy functional equation: A fixed point approachGrazer Math Ber200434643521060.39028 IsacGRassiasThMStability of ψ-additive mappings: applications to non-linear analysisInt J Math Math Sci1996192192280843.47036137598310.1155/S0161171296000324 GajdaZOn stability of additive mappingsInt J Math Math Sci1991144314340739.39013111003610.1155/S016117129100056X MosznerZOn the stability of functional equationsAequationes Math20097733881207.39044249571810.1007/s00010-008-2945-7 CzerwikSFunctional Equations and Inequalities in Several Variables2002New Jersey, London, Singapore, Hong KongWorld Scientific Publishing Company GăvruţaPA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappingsJ Math Anal Appl19941844314360818.46043128151810.1006/jmaa.1994.1211 Brzd¸ekJOn the quotient stability of a family of functional equationsNonlinear Anal20097143964404254866910.1016/j.na.2009.02.123 AczélJDhombresJFunctional Equations in Several Variables1989CambridgeCambridge Univ. Press10.1017/CBO9781139086578 JungSMHyers-Ulam stability of zeros of polynomialsAppl Math Lett201124132213251217.26023279362510.1016/j.aml.2011.03.002 CholewaPWRemarks on the stability of functional equationsAequationes Math19842776860549.3900675886010.1007/BF02192660 SaadatiRVaezpourSMParkCThe stability of the cubic functional equation in various spacesMath Commun2011161311451225.390292835961 CzerwikSOn the stability of the quadratic mapping in normed spacesAbh Math Sem Univ Hamburg19926259640779.39003118284110.1007/BF02941618 GordjiMEKhodaeiHNajatiAFixed points and quartic functional equations in β-Banach modulesResults Math2011 CieplińskiKStability of the multi-Jensen equationJ Math Anal Appl20103632492541211.39017255906010.1016/j.jmaa.2009.08.021 BrzdękJOn approximately additive functionsJ Math Anal Appl20113812993071235.39017279621010.1016/j.jmaa.2011.02.048 RaduVThe fixed point alternative and the stability of functional equationsFixed Point Theory2003491961051.390312031824 MoradlouFVaeziHParkCFixed points and stability of an additive functional equation of n-Apollonius type in C*-algebrasAbstract Appl Anal2008200813243925510.1155/2008/672618Article ID 672618 BalachandranVKTopological Algebras1999New DelhiNarosa Publishing House KannappanPlFunctional Equations and Inequalities with Applications2009New YorkSpringer10.1007/978-0-387-89492-8 HyersDHIsacGRassiasThMStability of Functional Equations in Several Variables1998Boston, Basel, BerlinBirkhauser10.1007/978-1-4612-1790-9 DiazJBMargolisBA fixed point theorem of the alternative for the contractions on generalized complete metric spaceBull Am Math Soc1968743053090157.2990422026710.1090/S0002-9904-1968-11933-0 J Brzd¸ek (147_CR23) 2009; 71 SM Jung (147_CR28) 2011 SM Ulam (147_CR1) 1960 TZ Xu (147_CR20) 2010; 2010 VK Balachandran (147_CR39) 1999 L Cădariu (147_CR10) 2004; 346 HA Kenary (147_CR30) 2011 RP Agarwal (147_CR9) 2003; 288 A Grabiec (147_CR37) 1996; 48 Z Gajda (147_CR41) 1991; 14 147_CR21 Pl Kannappan (147_CR29) 2009 P Găvruţa (147_CR5) 1994; 184 Z Moszner (147_CR17) 2009; 77 PW Cholewa (147_CR36) 1984; 27 F Moradlou (147_CR16) 2008; 2008 DH Hyers (147_CR2) 1941; 27 R Saadati (147_CR32) 2011; 16 F Skof (147_CR35) 1983; 53 SM Jung (147_CR27) 2011; 24 T Aoki (147_CR3) 1950; 2 Pl Kannappan (147_CR15) 1995; 27 ThM Rassias (147_CR4) 1978; 72 DH Hyers (147_CR24) 1998 JB Diaz (147_CR40) 1968; 74 JM Rassias (147_CR7) 1984; 108 GL Forti (147_CR14) 1995; 50 K Ciepliński (147_CR31) 2010; 363 ME Gordji (147_CR12) 2011 GZ Eskandani (147_CR11) 2011; 8 J Brzdęk (147_CR22) 2011; 381 V Radu (147_CR19) 2003; 4 L Cădariu (147_CR33) 2008 DH Hyers (147_CR25) 1992; 44 JM Rassias (147_CR6) 1982; 46 J Aczél (147_CR8) 1989 G Isac (147_CR26) 1996; 19 S Czerwik (147_CR38) 2002 ME Gordji (147_CR13) 2011 C Park (147_CR18) 2006; 322 S Czerwik (147_CR34) 1992; 62 |
| References_xml | – reference: GajdaZOn stability of additive mappingsInt J Math Math Sci1991144314340739.39013111003610.1155/S016117129100056X – reference: HyersDHOn the stability of the linear functional equationProc Nat Acad Sci USA194127222224407610.1073/pnas.27.4.222 – reference: FortiGLHyers-Ulam stability of functional equations in several variablesAequationes Math1995501431900836.39007133686610.1007/BF01831117 – reference: JungSMHyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis2011New YorkSpringer10.1007/978-1-4419-9637-4 – reference: SaadatiRVaezpourSMParkCThe stability of the cubic functional equation in various spacesMath Commun2011161311451225.390292835961 – reference: CzerwikSOn the stability of the quadratic mapping in normed spacesAbh Math Sem Univ Hamburg19926259640779.39003118284110.1007/BF02941618 – reference: CădariuLRaduVOn the stability of the Cauchy functional equation: A fixed point approachGrazer Math Ber200434643521060.39028 – reference: KannappanPlQuadratic functional equation and inner product spacesResults Math1995273683720836.39006133111010.1007/BF03322841 – reference: HyersDHIsacGRassiasThMStability of Functional Equations in Several Variables1998Boston, Basel, BerlinBirkhauser10.1007/978-1-4612-1790-9 – reference: GordjiMEKhodaeiHRassiasThMFixed points and stability for quadratic mappings in β-normed left Banach modules on Banach algebrasResults Math2011 – reference: RassiasJMOn approximation of approximately linear mappings by linear mappingsJ Funct Anal1982461261300482.4703365446910.1016/0022-1236(82)90048-9 – reference: JungSMHyers-Ulam stability of zeros of polynomialsAppl Math Lett201124132213251217.26023279362510.1016/j.aml.2011.03.002 – reference: AgarwalRPXuBZhangWStability of functional equations in single variableJ Math Anal Appl20032888528691053.39042202020110.1016/j.jmaa.2003.09.032 – reference: RaduVThe fixed point alternative and the stability of functional equationsFixed Point Theory2003491961051.390312031824 – reference: GăvruţaPA generalization of the Hyers-Ulam-Rassias stability of approximately additive mappingsJ Math Anal Appl19941844314360818.46043128151810.1006/jmaa.1994.1211 – reference: Brzd¸ekJOn the quotient stability of a family of functional equationsNonlinear Anal20097143964404254866910.1016/j.na.2009.02.123 – reference: RassiasJMOn approximation of approximately linear mappings by linear mappingsBull des Sci Math19841084454460599.47106 – reference: DiazJBMargolisBA fixed point theorem of the alternative for the contractions on generalized complete metric spaceBull Am Math Soc1968743053090157.2990422026710.1090/S0002-9904-1968-11933-0 – reference: XuTZRassiasJMRassiasMJXuWXA fixed point approach to the stability of quintic and sextic functional equations in quasi-β-normed spacesJ Inequal Appl2010201023Article ID 423231 – reference: BrzdękJOn approximately additive functionsJ Math Anal Appl20113812993071235.39017279621010.1016/j.jmaa.2011.02.048 – reference: CădariuLRaduVFixed point methods for the generalized stability of functional equations in a single variableFixed Point Theory Appl2008Art ID 749392 – reference: EskandaniGZGăvruţaPRassiasJMZarghamiRGeneralized Hyers-Ulam stability for a general mixed functional equation in quasi-β-normed spacesMediterr J Math201183313481236.39026282458510.1007/s00009-010-0082-8 – reference: KenaryHAJangSYParkCA fixed point approach to the Hyers-Ulam stability of a functional equation in various normed spacesFixed Point Theory Appl2011 – reference: CzerwikSFunctional Equations and Inequalities in Several Variables2002New Jersey, London, Singapore, Hong KongWorld Scientific Publishing Company – reference: AokiTOn the stability of the linear transformation in Banach spacesJ Math Soc Japan1950264660040.355014058010.2969/jmsj/00210064 – reference: Xu TZ, Rassias JM, Xu WX: A fixed point approach to the stability of a general mixed additive-cubic equation on Banach modules. Acta Math Sci Ser B, in press. – reference: GrabiecAThe generalized Hyers-Ulam stability of a class of functional equationsPubl Math Debrecen1996482172351274.390581394843 – reference: AczélJDhombresJFunctional Equations in Several Variables1989CambridgeCambridge Univ. Press10.1017/CBO9781139086578 – reference: CieplińskiKStability of the multi-Jensen equationJ Math Anal Appl20103632492541211.39017255906010.1016/j.jmaa.2009.08.021 – reference: SkofFLocal properties and approximations of operatorsRend Sem Mat Fis Milano1983531131290599.3900785854110.1007/BF02924890 – reference: RassiasThMOn the stability of the linear mapping in Banach spacesProc Am Math Soc1978722973000398.4704010.1090/S0002-9939-1978-0507327-1 – reference: ParkCRassiasThMHyers-Ulam stability of a generalized quadratic Apollonius type mappingJ Math Anal Appl20063223713811101.39020223924510.1016/j.jmaa.2005.09.027 – reference: UlamSMA Collection of the Mathematical Problems1960New YorkInterscience – reference: HyersDHRassiasThMApproximate homomorphismsAequationes Math1992441251530806.47056118126410.1007/BF01830975 – reference: KannappanPlFunctional Equations and Inequalities with Applications2009New YorkSpringer10.1007/978-0-387-89492-8 – reference: CholewaPWRemarks on the stability of functional equationsAequationes Math19842776860549.3900675886010.1007/BF02192660 – reference: MosznerZOn the stability of functional equationsAequationes Math20097733881207.39044249571810.1007/s00010-008-2945-7 – reference: BalachandranVKTopological Algebras1999New DelhiNarosa Publishing House – reference: GordjiMEKhodaeiHNajatiAFixed points and quartic functional equations in β-Banach modulesResults Math2011 – reference: MoradlouFVaeziHParkCFixed points and stability of an additive functional equation of n-Apollonius type in C*-algebrasAbstract Appl Anal2008200813243925510.1155/2008/672618Article ID 672618 – reference: IsacGRassiasThMStability of ψ-additive mappings: applications to non-linear analysisInt J Math Math Sci1996192192280843.47036137598310.1155/S0161171296000324 – volume: 77 start-page: 33 year: 2009 ident: 147_CR17 publication-title: Aequationes Math doi: 10.1007/s00010-008-2945-7 – volume-title: Functional Equations and Inequalities with Applications year: 2009 ident: 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| Snippet | Using the fixed point method, we prove the Hyers-Ulam stability of the following mixed additive and quadratic functional equation
f
(
kx
+
y
) +
f
(
kx
-
y
) =... Using the fixed point method, we prove the Hyers-Ulam stability of the following mixed additive and quadratic functional equation f (kx + y) + f(kx - y) = f(x... |
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| SubjectTerms | Additives Algebra Analysis Applications of Mathematics Classification Differential Geometry Fixed points (mathematics) Mathematical analysis Mathematical and Computational Biology Mathematics Mathematics and Statistics Modules Stability Topology |
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| Title | A fixed point approach to the stability of an AQ-functional equation on β-Banach modules |
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