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Journal of Inequalities and Applications

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Open Access 01.12.2009 | Research Article

Fixed Points and Stability of a Generalized Quadratic Functional Equation

verfasst von: Abbas Najati, Choonkil Park

Erschienen in: Journal of Inequalities and Applications | Ausgabe 1/2009

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Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the generalized quadratic functional equation

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_IEq1_HTML.gif

in Banach modules, where

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_IEq2_HTML.gif

are nonzero rational numbers with

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_IEq3_HTML.gif

Vorheriger Artikel On a Multiple Hilbert-Type Integral Operator and Applications

Nächster Artikel Existence and Analytic Approximation of Solutions of Duffing Type Nonlinear Integro-Differential Equation with Integral Boundary Conditions

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Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.

Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941,27(4):222–224. 10.1073/pnas.27.4.222MathSciNetCrossRefMATH

Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064MathSciNetCrossRefMATH

Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetCrossRefMATH

Rassias JM: On approximation of approximately linear mappings by linear mappings. Journal of Functional Analysis 1982,46(1):126–130. 10.1016/0022-1236(82)90048-9MathSciNetCrossRefMATH

Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques 1984,108(4):445–446.MathSciNetMATH

Rassias JM: Solution of a problem of Ulam. Journal of Approximation Theory 1989,57(3):268–273. 10.1016/0021-9045(89)90041-5MathSciNetCrossRefMATH

Găvruţa P: On the stability of some functional equations. In Stability of Mappings of Hyers-Ulam Type, Hadronic Press Collection of Original Articles. Hadronic Press, Palm Harbor, Fla, USA; 1994:93–98.

Aczél J, Dhombres J: Functional Equations in Several Variables with Applications to Mathematics, Information Theory and to the Natural and Social Sciences, Encyclopedia of Mathematics and Its Applications. Volume 31. Cambridge University Press, Cambridge, UK; 1989:xiv+462.MATH

10.

Amir D: Characterizations of Inner Product Spaces, Operator Theory: Advances and Applications. Volume 20. Birkhäuser, Basel, Switzerland; 1986:vi+200.CrossRef

11.

Jordan P, von Neumann J: On inner products in linear, metric spaces. Annals of Mathematics 1935,36(3):719–723. 10.2307/1968653MathSciNetCrossRefMATH

12.

Kannappan P: Quadratic functional equation and inner product spaces. Results in Mathematics 1995,27(3–4):368–372.MathSciNetCrossRefMATH

13.

Skof F: Local properties and approximation of operators. Rendiconti del Seminario Matematico e Fisico di Milano 1983,53(1):113–129. 10.1007/BF02924890MathSciNetCrossRefMATH

14.

Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.MathSciNetCrossRefMATH

15.

Rassias JM: On the stability of the Euler-Lagrange functional equation. Chinese Journal of Mathematics 1992,20(2):185–190.MathSciNetMATH

16.

Czerwik S: On the stability of the quadratic mapping in normed spaces. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1992,62(1):59–64. 10.1007/BF02941618MathSciNetCrossRefMATH

17.

Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations. Publicationes Mathematicae Debrecen 1996,48(3–4):217–235.MathSciNetMATH

18.

Rassias JM: On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces. Journal of Mathematical and Physical Sciences 1994,28(5):231–235.MathSciNetMATH

19.

Rassias JM: On the stability of the general Euler-Lagrange functional equation. Demonstratio Mathematica 1996,29(4):755–766.MathSciNetMATH

20.

Jun K-W, Lee Y-H: On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality. Mathematical Inequalities & Applications 2001,4(1):93–118.MathSciNetCrossRefMATH

21.

Bourgin DG: Classes of transformations and bordering transformations. Bulletin of the American Mathematical Society 1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7MathSciNetCrossRefMATH

22.

Cădariu L, Radu V: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory and Applications 2008, 2008:-15.

23.

Cădariu L, Radu V: Remarks on the stability of monomial functional equations. Fixed Point Theory 2007,8(2):201–218.MathSciNetMATH

24.

Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory (ECIT '02), Grazer Mathematische Berichte. Volume 346. Karl-Franzens-Universität Graz, Graz, Austria; 2004:43–52.

25.

Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003,4(1, article 4):1–7.MATH

26.

Forti GL: On an alternative functional equation related to the Cauchy equation. Aequationes Mathematicae 1982,24(2–3):195–206.MathSciNetCrossRefMATH

27.

Forti GL: The stability of homomorphisms and amenability, with applications to functional equations. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1987,57(1):215–226. 10.1007/BF02941612MathSciNetCrossRefMATH

28.

Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae 1995,50(1–2):143–190. 10.1007/BF01831117MathSciNetCrossRefMATH

29.

Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211MathSciNetCrossRefMATH

30.

Găvruţa P: On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings. Journal of Mathematical Analysis and Applications 2001,261(2):543–553. 10.1006/jmaa.2001.7539MathSciNetCrossRefMATH

31.

Găvruţa P: On the Hyers-Ulam-Rassias stability of the quadratic mappings. Nonlinear Functional Analysis and Applications 2004,9(3):415–428.MathSciNetMATH

32.

Jun K-W, Kim H-M: On the Hyers-Ulam stability of a difference equation. Journal of Computational Analysis and Applications 2005,7(4):397–407.MathSciNetMATH

33.

Jun K-W, Kim H-M: Stability problem of Ulam for generalized forms of Cauchy functional equation. Journal of Mathematical Analysis and Applications 2005,312(2):535–547. 10.1016/j.jmaa.2005.03.052MathSciNetCrossRefMATH

34.

Jun K-W, Kim H-M: Stability problem for Jensen-type functional equations of cubic mappings. Acta Mathematica Sinica 2006,22(6):1781–1788. 10.1007/s10114-005-0736-9MathSciNetCrossRefMATH

35.

Jun K-W, Kim H-M: Ulam stability problem for a mixed type of cubic and additive functional equation. Bulletin of the Belgian Mathematical Society. Simon Stevin 2006,13(2):271–285.MathSciNetMATH

36.

Jun K-W, Kim H-M, Rassias JM: Extended Hyers-Ulam stability for Cauchy-Jensen mappings. Journal of Difference Equations and Applications 2007,13(12):1139–1153. 10.1080/10236190701464590MathSciNetCrossRefMATH

37.

Jung S-M, Kim T-S: A fixed point approach to the stability of the cubic functional equation. Boletín de la Sociedad Matemática Mexicana 2006,12(1):51–57.MathSciNetMATH

38.

Jung S-M, Rassias JM: A fixed point approach to the stability of a functional equation of the spiral of Theodorus. Fixed Point Theory and Applications 2008, 2008:-7.

39.

Mirzavaziri M, Moslehian MS: A fixed point approach to stability of a quadratic equation. Bulletin of the Brazilian Mathematical Society 2006,37(3):361–376. 10.1007/s00574-006-0016-zMathSciNetCrossRefMATH

40.

Moslehian MS, Rassias ThM: Stability of functional equations in non-Archimedean spaces. Applicable Analysis and Discrete Mathematics 2007,1(2):325–334. 10.2298/AADM0702325MMathSciNetCrossRefMATH

41.

Najati A: Hyers-Ulam stability of an -Apollonius type quadratic mapping. Bulletin of the Belgian Mathematical Society. Simon Stevin 2007,14(4):755–774.MathSciNetMATH

42.

Najati A: On the stability of a quartic functional equation. Journal of Mathematical Analysis and Applications 2008,340(1):569–574. 10.1016/j.jmaa.2007.08.048MathSciNetCrossRefMATH

43.

Najati A, Moghimi MB: Stability of a functional equation deriving from quadratic and additive functions in quasi-Banach spaces. Journal of Mathematical Analysis and Applications 2008,337(1):399–415. 10.1016/j.jmaa.2007.03.104MathSciNetCrossRefMATH

44.

Najati A, Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. Journal of Mathematical Analysis and Applications 2007,335(2):763–778. 10.1016/j.jmaa.2007.02.009MathSciNetCrossRefMATH

45.

Najati A, Park C: The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms between -algebras. Journal of Difference Equations and Applications 2008,14(5):459–479. 10.1080/10236190701466546MathSciNetCrossRefMATH

46.

Park C: On the stability of the linear mapping in Banach modules. Journal of Mathematical Analysis and Applications 2002,275(2):711–720. 10.1016/S0022-247X(02)00386-4MathSciNetCrossRefMATH

47.

Ravi K, Arunkumar M, Rassias JM: Ulam stability for the orthogonally general Euler-Lagrange type functional equation. International Journal of Mathematics and Statistics 2008,3(A08):36–46.MathSciNetMATH

48.

Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.CrossRefMATH

49.

Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.MATH

50.

Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATH

51.

Rassias ThM (Ed): Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.MATH

52.

Diaz JB, Margolis B: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bulletin of the American Mathematical Society 1968, 74: 305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetCrossRefMATH

53.

Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003,4(1):91–96.MathSciNetMATH

54.

Rassias JM: Alternative contraction principle and Ulam stability problem. Mathematical Sciences Research Journal 2005,9(7):190–199.MathSciNet

55.

Rassias JM: Alternative contraction principle and alternative Jensen and Jensen type mappings. International Journal of Applied Mathematics & Statistics 2006,4(M06):1–10.MathSciNet

56.

Isac G, Rassias ThM: Stability of -additive mappings: applications to nonlinear analysis. International Journal of Mathematics and Mathematical Sciences 1996,19(2):219–228. 10.1155/S0161171296000324MathSciNetCrossRefMATH

57.

Kurepa S: On the quadratic functional. Publications de l'Institut Mathématique de l'Académie Serbe des Sciences 1961, 13: 57–72.MathSciNetMATH

Titel: Fixed Points and Stability of a Generalized Quadratic Functional Equation
verfasst von: Abbas Najati
Choonkil Park
Publikationsdatum: 01.12.2009
Verlag: Springer International Publishing
Erschienen in: Journal of Inequalities and Applications / Ausgabe 1/2009
Elektronische ISSN: 1029-242X
DOI: https://doi.org/10.1155/2009/193035

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