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Stability of a pexiderial functional equation in random normed spaces

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Abstract

The concept of Hyers-Ulam-Rassias stability originated from Th.M. Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72:297–300, 1978. Recently, the generalized Hyers-Ulam-Rassias stability of the following quadratic functional equation

$$f(x+y)+f(x-y)=2f(x)+2f(y)$$

proved in the earlier work. In this paper, using direct method we prove the generalized Hyers-Ulam stability of the following Pexiderial functional equation

$$f(x+y)+f(x-y)=2g(x)+2g(y)$$

in random normed space.

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References

  1. Azadi Kenary, H., Cho, Y.J.: Stability of mixed additive-quadratic Jensen type functional equation in various spaces. Comput. Math. Appl. (2011). doi:10.1016/j.camwa.2011.03.024

    MATH  Google Scholar 

  2. Cholewa, P.W.: Remarks on the stability of functional equations. Aequ. Math. 27(1–2), 76–86 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Gavruta, P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184(3), 431–436 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of functional equations in several variables. In: Progress in Nonlinear Differential Equations and their Applications, vol. 34. Birkhäuser, Basel (1998)

    Google Scholar 

  7. Hyers, D.H., Isac, G., Rassias, Th.M.: On the asymptoticity aspect of Hyers-Ulam stability of mappings. Proc. Am. Math. Soc. 126(2), 425–430 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hyers, D.H., Rassias, Th.M.: Approximate homomorphisms. Aequ. Math. 44(2–3), 125–153 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jordan, P., von Neumann, J.: On inner products in linear metric spaces. Ann. Math. 36(3), 719–723 (1935)

    Article  MATH  Google Scholar 

  10. Jung, S.-M.: On the Hyers-Ulam stability of the functional equations that have the quadratic property. J. Math. Anal. Appl. 222(1), 126–137 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jung, S.-M.: On the Hyers-Ulam-Rassias stability of a quadratic functional equation. J. Math. Anal. Appl. 232(2), 384–393 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  14. Najati, A., Park, C.: On the stability of a cubic functional equation. Acta Math. Sin. Engl. Ser. (2011, to appear)

  15. Park, C.: Universal Jensen’s equations in Banach modules over a C ∗-algebra and its unitary group. Acta Math. Sin. Engl. Ser. 20(6), 1047–1056 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Park, C., Hou, J., Oh, S.: Homomorphisms between JC ∗-algebras and Lie C ∗-algebras. Acta Math. Sin. Engl. Ser. 21(6), 1391–1398 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Park, C., Rassias, Th.M.: The N-isometric isomorphisms in linear N-normed C ∗-algebras. Acta Math. Sin. Engl. Ser. 22(6), 1863–1890 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72(2), 297–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251(1), 264–284 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Saadati, R., Vaezpour, M., Cho, Y.J.: A note to paper “On the stability of cubic mappings and quartic mappings in random normed spaces”. J. Inequal. Appl. 2009, 214530 (2009)

    Article  MathSciNet  Google Scholar 

  21. Schewizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland, New York (1983)

    Google Scholar 

  22. Skof, F.: Local properties and approximation of operators. Rend. Semin. Mat. Fis. Milano 53, 113–129 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ulam, S.M.: A collection of mathematical problems. Intersci. Tracts Pure Appl. Math. (8) (1960)

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Yasouj University, Yasouj, 75914-353, Iran

    Hassan Azadi Kenary

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  1. Hassan Azadi Kenary
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Correspondence to Hassan Azadi Kenary.

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Azadi Kenary, H. Stability of a pexiderial functional equation in random normed spaces. Rend. Circ. Mat. Palermo 60, 59–68 (2011). https://doi.org/10.1007/s12215-011-0027-5

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