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Hyers–Ulam Stability of the Quadratic Functional Equation

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Functional Equations in Mathematical Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 52))

Abstract

We prove a stability theorem for the quadratic functional equation

$$f(x + y) + f(x + \sigma (y)) = 2f(x) + 2f(y),\quad x,y \in G,$$

where G is an abelian group and σ is an involution of G. We also prove that for functions f from G to an inner product space E, the inequality

$$\|2f(x) + 2f(y) - f(x + \sigma (y))\| \leq \| f(x + y)\|,\quad x,y \in G.$$

implies that f is a solution to the equation.

Mathematics Subject Classification (2000): Primary 39B52, 39B82

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References

  1. Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bouikhalene, B., Elqorachi, E., Rassias, Th.M.: On the generalized Hyers-Ulam stability of the quadratic functional equation with a general involution. Nonlinear Funct. Anal. Appl. 12, 247–262 (2007)

    MathSciNet  Google Scholar 

  3. Cholewa, P.W.: Remarks on the stability of functional equations. Aequationes Math. 27, 76–86 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  4. Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg. 62, 59–64 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gǎvruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    Article  MathSciNet  Google Scholar 

  6. Gajda, Z.: On stability of additive mappings. Internat. J. Math. Math. Sci. 14, 431–434 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gilanyi, A.: Eine zur Parallelogrammgleichung aquivalente ungleichung. Aequationes Math. 62, 303–309 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U. S. A. 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

  9. Hyers, D.H., Rassias, Th.M.: Approximate homomorphisms. Aequationes Math. 44, 125–153 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hyers, D.H., Isac, G. Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhäuser, Basel (1998)

    Book  MATH  Google Scholar 

  11. Jung, S.-M.: Stability of the quadratic equation of Pexider type. Abh. Math. Sem. Univ. Hamburg 70, 175–190 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jung, S.-M., Sahoo, P.K.: Hyers–Ulam–Rassias stability of the quadratic equation of Pexider type. J. Korean Math. Soc. 38, 645–656 (2001)

    MathSciNet  MATH  Google Scholar 

  13. Lee, J.-R., An, J.-S., Park, C.: On the stability of quadratic functional equations. Abstr. Appl. Anal., doi: 10.1155/2008/628178

    Google Scholar 

  14. Maksa, Gy., Volkmann, P.: Characterization of group homomorphisms having values in an inner product space. Publ. Math. (Debrecen) 56, 197–200 (2000)

    Google Scholar 

  15. Rätz, J.: On inequality associated with the Jordan-von Neumann functional equation. Aequationes Math. 54, 191–200 (2003)

    Article  Google Scholar 

  16. Rassias, Th.M.: On the stability of linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72, 297–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rassias, Th.M.: On a modified Hyers-Ulam sequence. J. Math. Anal. Appl. 158, 106–113 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rassias, Th.M.: Functional Equations and Inequalities. Kluwer Academic Publishers, Dordrecht–Boston–London (2001)

    Google Scholar 

  19. Rassias, Th.M.: Functional Equations, Inequalities and Applications. Kluwer Academic Publishers, Dordrecht–Boston–London (2003)

    Google Scholar 

  20. Rassias, Th.M.: The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246, 352–378 (2000)

    Google Scholar 

  21. Rassias, Th.M.: On the stability of minimum points. Mathematica 45, 93–104 (2003)

    MathSciNet  Google Scholar 

  22. Rassias, Th.M.: On the stability of the functional equations and a problem of Ulam. Acta Appl. Math. 62, 23–130 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rassias, Th.M., S̆emrl, P.: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proc. Amer. Math. Soc. 114, 989–993 (1992)

    Google Scholar 

  24. Rassias, Th.M., S̆emrl, P.: On the Hyers-Ulam stability of linear mappings. J. Math. Anal. Appl. 173, 325–338 (1993)

    Google Scholar 

  25. Rassias, Th.M., Tabor, J. (eds.): Stability of Mappings of Hyers-Ulam Type. Hadronic Press, Inc., Palm Harbor (1994)

    MATH  Google Scholar 

  26. Skof, F.: Approssimazione di funzioni δ-quadratic su dominio restretto. Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 118, 58–70 (1984)

    MathSciNet  MATH  Google Scholar 

  27. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publ., New York (1961) (reprinted as: Problems in Modern Mathematics. Wiley, New York (1964))

    Google Scholar 

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Acknowledgements

Special thanks to Professor Janusz Brzdȩk for reading a preliminary version of the paper.

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Authors and Affiliations

  1. Faculty of Sciences, Department of Mathematics, University Ibn Zohr, Agadir, Morocco

    Elhoucien Elqorachi & Youssef Manar

  2. Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece

    Themistocles M. Rassias

Authors
  1. Elhoucien Elqorachi
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  2. Youssef Manar
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  3. Themistocles M. Rassias
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Corresponding author

Correspondence to Elhoucien Elqorachi .

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Editors and Affiliations

  1. , Department of Mathematics, National Technical University of Athens, Iroon Polytechneiou 9, Athens, 15780, Greece

    Themistocles M. Rassias

  2. Institute of Mathematics, Pedagogical University, ul. Podchorazych 2, Krakow, 30-084, Poland

    Janusz Brzdek

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Elqorachi, E., Manar, Y., Rassias, T.M. (2011). Hyers–Ulam Stability of the Quadratic Functional Equation. In: Rassias, T., Brzdek, J. (eds) Functional Equations in Mathematical Analysis. Springer Optimization and Its Applications(), vol 52. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0055-4_8

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