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Fixed Points and Stability for Quadratic Mappings in β–Normed Left Banach Modules on Banach Algebras

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Abstract

The goal of the present paper is to investigate some new stability results by applying the alternative fixed point of generalized quadratic functional equation

$$\begin{array}{ll}f\left(\sum\limits_{i=1}^{n}a_ix_i\right)+{\sum\limits_{i=1}^{n-1}}{\sum\limits_{j=i+1}^{n}}\left[f(a_ix_i+a_jx_j)+2f(a_ix_i-a_jx_j)\right]\\ \qquad \quad = (3n-2){\sum\limits_{i=1}^{n}}a^2_{i}f(x_{i})\end{array}$$

in β–Banach modules on Banach algebras, where \({a_{1},\dots,a_{n}\in \mathbb{Z}{\setminus}\{0\}}\) and some \({\ell\in\{1 , 2 ,\dots, n-1\},}\) a ℓ  ≠ ±1 and a n  = 1, where n is a positive integer greater or at least equal to two.

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References

  1. Aczel J., Dhombres J.: Functional Equations in Several Variables. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  3. Balachandran V.K.: Topological Algebras. Narosa Publishing House, New Delhi (1999)

    MATH  Google Scholar 

  4. Cădariu L., Radu V.: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber 346, 43–52 (2004)

    MATH  Google Scholar 

  5. 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 

  6. Eshaghi Gordji M., Khodaei H.: Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi—Banach spaces. Nonlinear Anal. -TMA 71, 5629–5643 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

  9. Gǎvruta P., Gǎvruta L.: A new method for the generalized Hyers–Ulam– Rassias stability. Int. J. Nonlinear Anal. Appl 1(2), 11–18 (2010)

    Google Scholar 

  10. Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  12. Isac G., Rassias Th.M.: On the Hyers–Ulam stability of ψ-additive mappings. J. Approx. Theory 72, 131–137 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jung S.-M.: On the Hyers–Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl 222, 126–137 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jung S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press lnc, Palm Harbor (2001)

    MATH  Google Scholar 

  15. Jung S.-M., Sahoo P.K.: Hyers–Ulam stability of the quadratic equation of Pexider type. J. Korean Math. Soc 38(3), 645–656 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Kannappan Pl.: Quadratic functional equation and inner product spaces. Results Math 27, 368–372 (1995)

    MathSciNet  MATH  Google Scholar 

  17. Khodaei H., Rassias Th.M.: Approximately generalized additive functions in several variables. Int. J. Nonlinear Anal. Appl 1, 22–41 (2010)

    Google Scholar 

  18. Margolis B., Diaz J.B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc 74, 305–309 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  19. Moslehian, M.S.: On the orthogonal stability of the pexiderized quadratic equation. J. Differ. Equ. Appl. 999–1004 (2005)

  20. Najati A., Moghimi M.B.: Stability of a functional equation deriving from quadratic and additive function in quasi-Banach spaces. J. Math. Anal. Appl 337, 399–415 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Park C.: On the stability of the quadratic mapping in Banach modules. J. Math. Anal. Appl 27, 135–144 (2002)

    Article  Google Scholar 

  22. Park C.: On the Hyers Ulam Rassias stability of generalized quadratic mappings in Banach modules. J. Math. Anal. Appl 291, 214–223 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Ulam S.M.: Problems in Modern Mathematics, Chapter VI, Science Editions. Wiley, New York (1964)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Semnan University, P. O. Box 35195–363, Semnan, Iran

    M. Eshaghi. Gordji & H. Khodaei

  2. Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Semnan, Iran

    M. Eshaghi. Gordji & H. Khodaei

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

    Th. M. Rassias

Authors
  1. M. Eshaghi. Gordji
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  2. H. Khodaei
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  3. Th. M. Rassias
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Correspondence to M. Eshaghi. Gordji.

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Gordji, M.E., Khodaei, H. & Rassias, T.M. Fixed Points and Stability for Quadratic Mappings in β–Normed Left Banach Modules on Banach Algebras. Results. Math. 61, 393–400 (2012). https://doi.org/10.1007/s00025-011-0123-z

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