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Ulam Type Stability pp 167–198Cite as

Solutions and Stability of Some Functional Equations on Semigroups

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Abstract

In this paper we investigate the solutions and the Hyers-Ulam stability of the μ-Jensen functional equation

$$\displaystyle f(xy)+\mu (y)f(x\sigma (y))=2f(x),\;x,y \in S, $$

a variant of the μ-Jensen functional equation

$$\displaystyle f(xy)+\mu (y)f(\sigma (y)x)=2f(x),\;x,y \in S, $$

and the μ-quadratic functional equation

$$\displaystyle f(xy)+\mu (y)f(x\sigma (y))=2f(x)+2f(y),\;x,y \in S, $$

where S is a semigroup, σ is a morphism of S and μ: \(S\longrightarrow \mathbb {C}\) is a multiplicative function such that μ(xσ(x)) = 1 for all x ∈ S.

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References

  1. 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, vol. 31. Cambridge University Press, Cambridge (1989)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. 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(2), 247–262 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Bouikhalene, B., Elqorachi, E., Redouani, A.: Hyers-Ulam stability of the generalierd quadratic functional equation in Amenables semigroups. J. Inequal. Pure Appl. Math. (JIPAM) 8(2), 18 pp. (2007). Article 56

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Dilian, Y.: Contributions to the theory of functional equations. PhD Thesis, University of Waterloo, Waterloo, Ontario, Canada (2006)

    Google Scholar 

  8. Elqorachi, E., Rassias M.Th.: Generalized Hyers-Ulam stability of trigonometric functional equations. Mathematics 6(5), 11 pp. (2018)

    Google Scholar 

  9. Elqorachi, E., Manar, Y., Rassias, Th.M.: Hyers-Ulam stability of Wilson’s functional equation. In: Pardalos, P.M., Rassias, Th.M. (eds.) Contributions in Mathematics and Engineering: Honor of Constantin Carathéodory. Springer, New York (2016)

    MATH  Google Scholar 

  10. Elqorachi, E., Redouani, A.: Solutions and stability of a variant of Wilson’s functional equation. Proyecciones J. Math. 37(2), 317–344 (2018)

    Article  MathSciNet  Google Scholar 

  11. Faiziev, V.A., Sahoo. P.K.: On the stability of Jensen’s functional equation on groups. Proc. Indian Acad. Sci. (Math. Sci.) 117, 31–48 (2007)

    Google Scholar 

  12. Forti, G.L.: Hyers-Ulam stability of functional equations in several variables. Aequationes Math. 50, 143–190 (1995)

    Article  MathSciNet  Google Scholar 

  13. Gadja, Z.: On stability of additive mapping. Int. J. Math. Math. Sci. 14, 431–434 (1991)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  15. Greenleaf, F.P.: Invariant Means on topological Groups and their Applications. Van Nostrand, New York (1969)

    MATH  Google Scholar 

  16. 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 

  17. Hyers, D.H., Isac, G.I., Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkh\(\ddot {a}\)user, Basel (1998)

    Google Scholar 

  18. Jung, S-M.: Hyers-Ulam-Rassias stability of Jensen’s equation and its application. Proc. Amer. Math. Soc. 126, 3137–3134 (1998)

    Article  MathSciNet  Google Scholar 

  19. Jung, S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48. Springer, New York (2011)

    Book  Google Scholar 

  20. Kim, J.H., The stability of d’Alembert and Jensen type functional equations. J. Math. Anal. Appl. 325, 237–248 (2007)

    Article  MathSciNet  Google Scholar 

  21. Ng, C.T.: Jensen’s functional equation on groups, III. Aequationes Math. 62(1–2), 143–159 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Rassias, J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–120 (1982)

    Article  MathSciNet  Google Scholar 

  24. Rassias, M.Th.: Solution of a functional equation problem of Steven Butler. Octogon Math. Mag. 12, 152–153 (2004)

    Google Scholar 

  25. Stetkær, H.: Functional equations on abelian groups with involution. Aequationes Math. 54, 144–172 (1997)

    Article  MathSciNet  Google Scholar 

  26. Stetkær, H.: Functional Equations on Groups. World Scientific Publishing, New Jersey (2013)

    Book  Google Scholar 

  27. Stetkær, H.: D’Alember’s functional equations on groups. Banach Cent. Publ. 99, 173–191 (2013)

    Article  MathSciNet  Google Scholar 

  28. Stetkær, H.: A variant of d’Alembert’s functional equation. Aequationes Math. 89(3), 657–662 (2015)

    Article  MathSciNet  Google Scholar 

  29. Stetkær, H.: A note on Wilson’s functional equation. Aequationes Math. 91(5), 945–947 (2017)

    Article  MathSciNet  Google Scholar 

  30. Szekelyhidi, L.: Note on a stability theorem. Can. Math. Bull. 25(4), 500–501 (1982)

    Article  MathSciNet  Google Scholar 

  31. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publications, New york (1961). Problems in Modern Mathematics (Wiley, New York, 1964)

    Google Scholar 

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Authors and Affiliations

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

    Keltouma Belfakih & Elhoucien Elqorachi

  2. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

Authors
  1. Keltouma Belfakih
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  2. Elhoucien Elqorachi
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  3. Themistocles M. Rassias
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Corresponding author

Correspondence to Themistocles M. Rassias .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, AGH University of Science and Technology, Krakow, Poland

    Janusz Brzdęk

  2. Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

    Dorian Popa

  3. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Belfakih, K., Elqorachi, E., Rassias, T.M. (2019). Solutions and Stability of Some Functional Equations on Semigroups. In: Brzdęk, J., Popa, D., Rassias, T. (eds) Ulam Type Stability . Springer, Cham. https://doi.org/10.1007/978-3-030-28972-0_9

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