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A Functional Equation Having Monomials and Its Stability

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Handbook of Functional Equations

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 96))

Abstract

We use some results about the Fréchet functional equation to consider the following functional equation:

$$\begin{aligned} f\left(\left(\sum_{i=1}^{m}a_ix_i^p\right)^\frac{1}{p}\right)=\sum_{i=1}^{m}a_if(x_i).\end{aligned}$$

We also apply a fixed point method and homogeneous functions of degree α to investigate some stability results for this functional equation in β-Banach spaces.

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References

  1. Aczél, J., Dhombres, J.: Functional Equations Inseveral Variables. Cambridge University Press, New York (1989)

    Book  Google Scholar 

  2. Almira, J.M., Lopez-Moreno, A.J.: On solutions of the Fréchet functional equation. J. Math. Anal. Appl. 332, 119–133 (2007)

    Article  MathSciNet  Google Scholar 

  3. Baker, J.A.: A general functional equation and its stability. Proc. Amer. Math. Soc. 133, 1657–1664 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brzd\cek, J., Popa, D., Xu, B.: The Hyers–Ulam stability of nonlinear recurrences. J. Math. Anal. Appl. 335, 443–449 (2007)

    Google Scholar 

  5. C\uadariu, L., Radu, V.: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, Art. ID 749392, (2008)

    Google Scholar 

  6. Chung, J.K., Sahoo, P.K.: On the general solution of a quartic functional equation. Bull. Korean Math. Soc. 40, 565–576 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific Publishing Company, New Jersey (2002)

    Book  MATH  Google Scholar 

  8. Djoković, D.\uZ.: A representation theorem for \((X_1-1)(X_2-1)\cdots(X_n-1)\) and its applications. Ann. Pol. Math. 22, 189–198 (1969)

    Google Scholar 

  9. Forti, G.L.: HyersUlam stability of functional equations in several variables. Aequ. Math. 50, 143–190 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  10. Fréchet, M.: Un definition fonctionnelle des polynômes. Nouv. Ann. de Math. 9, 145–162 (1909)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  12. Gordji, M.E., Khodaei, H.: A fixed point technique for investigating the stability of \((\alpha,\beta,\gamma)\)-derivations on Lie \(C^*\)-algebras. Nonlin. Anal. 76, 52–57 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hayes, W., Jackson, K.R.: A survey of shadowing methods for numerical solutions of ordinary differential equations. Appl. Numer. Math. 53, 299–321 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  15. Jun, K.W., Kim, H.M.: The generalized Hyers–Ulam–Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274, 867–878 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  18. Khodaei, H., Gordji, M.E., Kim, S.S., Cho, Y.J.: Approximation of radical functional equations related to quadratic and quartic mappings. J. Math. Anal. Appl. 395, 284–297 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  19. Köthe, G.: Topological Vector Spaces I. Springer-Verlag, Berlin (1969)

    MATH  Google Scholar 

  20. Kuczma, M.: Functional Equations in a Single Variable. PWN-Polish Scientific Publishers, Warszawa (1968)

    MATH  Google Scholar 

  21. Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality. Birkhäuser, Basel (2009)

    Book  MATH  Google Scholar 

  22. Lee, S.H., Im, S.M., Hawng, I.S.: Quartic functional equation. J. Math. Anal. Appl. 307, 387–394 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mazur, S., Orlicz, W.: Grundlegende Eigenschaften der Polynomischen Operationen, Erst Mitteilung. Studia Math. 5, 50–68 (1934)

    MATH  Google Scholar 

  24. Mazur, S., Orlicz, W.: Grundlegende Eigenschaften der Polynomischen Operationen, Zweite Mitteilung, ibidem. Studia Math. 5, 179–189 (1934)

    MATH  Google Scholar 

  25. Moszner, Z.: On the stability of functional equations. Aequ. Math. 77, 33–88 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Palmer, K.: Shadowing in Dynamical Systems. Theory and Applications. In: Mathematics and its Applications, vol. 501. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  27. Park, W.-G., Bae, J.-H.: On a bi-quadratic functional equation and its stability. Nonlin. Anal. 62, 643–654 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Park, C., O’Regan, D., Saadati, R.: Stability of some set-valued functional equations. Appl. Math. Lett. 24, 1910–1914 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Pólya, Gy., Szegő, G.: Aufgaben und Lehrsätze aus der Analysis. Springer, Berlin (1925)

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Sahoo, P.K.: On a functional equation characterizing polynomials of degree three. Bull. Inst. Math. Acad. Sin. 32, 35–44 (2004)

    MATH  MathSciNet  Google Scholar 

  32. Stević, S.: Bounded solutions of a class of difference equations in Banach spaces producing controlled chaos. Chaos Solitons Fractals. 35, 238–245 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Ulam, S.M.: A Collection of Mathematical Problems. Interscience Publ., New York, (1960); reprinted as: Problems in Modern Mathematics. Wiley, New York (1964)

    Google Scholar 

  34. Zhou, D.-X.: On a conjecture of Z. Ditzian. J. Approx. Theory 69, 167–172 (1992)

    Google Scholar 

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Authors and Affiliations

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

    M. E. Gordji

  2. Department of Mathematics, Malayer University, Malayer, P.O. Box 65719-95863, Iran

    H. Khodaei

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

    Themistocles M. Rassias

Authors
  1. M. E. Gordji
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  2. H. Khodaei
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  3. Themistocles M. Rassias
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Corresponding author

Correspondence to M. E. Gordji .

Editor information

Editors and Affiliations

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

    Themistocles M. Rassias

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Gordji, M., Khodaei, H., Rassias, T. (2014). A Functional Equation Having Monomials and Its Stability. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 96. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1286-5_9

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