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On the Stability of Additive, Quadratic, Cubic and Quartic Set-valued Functional Equations

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Abstract

For m = 1, 2, 3, 4, we study the following set-valued functional equation

$$\begin{array}{ll}f(ax + y) \oplus f(ax - y) = a^{m-2}[f(x + y) \oplus f(x - y)] \oplus 2(a^{2} - 1) [a^{m-2}f(x) \\ \quad \oplus \frac{(m - 2)(1 - (m - 2)^{2})}{6}f(y)]\end{array}$$

where a is a fixed positive integer with a > 1. We also prove the stability of this set-valued functional equation by using the Banach fixed point theorem.

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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. Arrow K.J., Debereu G.: Existence of an equilibrium for a competitive economy. Econometrica 22, 265–290 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aumann R.J.: Integrals of set-valued functions. J. Math. Anal. Appl. 12, 1–12 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aubin J.P., Frankowska H.: Set-Valued Analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  5. Brzdȩk J.: Hyperstability of the Cauchy equation on restricted domains. Acta Math. Hungar. 141, 58–67 (2013)

    Article  MathSciNet  Google Scholar 

  6. Brzdȩk J., Ciepliński K.: A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. Nonlinear Anal. TMA 74, 6861–6867 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brzdȩk J., Chudziak J., Páles Z.: A fixed point approach to stability of functional equations. Nonlinear Anal. TMA 74, 6728–6732 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brzdȩk J., Popa D., Xu B.: Selections of set-valued maps satisfying a linear inclusion in a single variable. Nonlinear Anal. TMA 74, 324–330 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cădariu L., Găvruţa L., Găvruţa P.: Weighted space method for the stability of some nonlinear equations. Appl. Anal. Discret. Math. 6, 126–139 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Castaing, C., Valadier, M.: Convex Analysis and Measurable Multifunctions. In: Lect. Notes in Math., vol. 580, Springer, Berlin (1977)

  11. Cho Y.J., Park C., Saadati R.: Functional inequalities in non-Archimedean Banach spaces. Appl. Math. Lett. 23, 1238–1242 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cho Y.J., Rassias T.M., Saadati R.: Stability of Functional Equations in Random Normed Spaces. Springer, New York (2013)

    Book  MATH  Google Scholar 

  13. Debreu, G.: Integration of correspondences. In: Le Cam, L.M., Neyman, J. (eds.) Proceedings of Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. II, part. I, pp. 351–372 (1966)

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Gajda Z., Ger R.: Subadditive multifunctions and Hyers–Ulam stability. Numer. Math. 80, 281–291 (1987)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Gilányi A., Kaiser Z., Páles Z.: Estimates to the stability of functional equations. Aequat. Math. 73, 125–143 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gordji M.E., Khodaei H.: A fixed point technique for investigating the stability of (α, β, γ)-derivations on Lie C *-algebras. Nonlinear Anal. TMA 76, 52–57 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gordji M.E., 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 

  20. Gordji M.E., Alizadeh Z., Khodaei H., Park C.: On approximate homomorphisms: a fixed point approach. Math. Sci. 6, 59 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hess C.: Set-valued integration and set-valued probability theory: an overview. In: Pap, E. (ed.) Handbook of Measure Theory, vol. I–II, pp. 617–673. North–Holland, Amsterdam (2002)

  22. Hindenbrand W.: Core and Equilibria of a Large Economy. Princeton Univ. Press, Princeton (1974)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  25. Jang, S.Y., Park, C., Cho, Y.: Hyers–Ulam stability of a generalized additive set-valued functional equation, J. Ineq. Appl. 2013:101, p 6 (2013)

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

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  28. Kenary, H.A., Rezaei, H., Gheisari, Y., Park, C.: On the stability of set-valued functional equations with the fixed point alternative. Fixed Point Theory Appl. 2012:81, p 17 (2012)

  29. Klein E., Thompson A.: Theory of Correspondence. Wiley, New York (1984)

    Google Scholar 

  30. Lu G., Park C.: Hyers–Ulam stability of additive set-valued functional equations. Appl. Math. Lett. 24, 1312–1316 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. McKenzie L.W.: On the existence of general equilibrium for a competitive market. Econometrica 27, 54–71 (1959)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Nikodem K.: On quadratic set-valued functions. Publ. Math. Debr. 30, 297–301 (1984)

    MathSciNet  MATH  Google Scholar 

  34. Nikodem K.: On Jensen’s functional equation for set-valued functions. Rad. Mat. 3, 23–33 (1987)

    MathSciNet  MATH  Google Scholar 

  35. Nikodem K., Popa D.: On selections of general linear inclusions. Publ. Math. Debr. 75, 239–249 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Páles Z.: Hyers–Ulam stability of the Cauchy functional equation on square-symmetric groupoids. Publ. Math. Debr. 58, 651–666 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Piszczek M.: The properties of functional inclusions and Hyers–Ulam stability. Aequat. Math. 85, 111–118 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pólya G., Szegő G.: Aufgaben und Lehrsätze aus der Analysis, vol I. Springer, Berlin (1925)

    Book  MATH  Google Scholar 

  40. Popa D.: A property of a functional inclusion connected with Hyers–Ulam stability. J. Math. Inequal. 4, 591–598 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rådström H.: An embedding theorem for spaces of convex sets. Proc. Am. Math. Soc. 3, 165–169 (1952)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. Rassias J.M.: On approximation of approximately linear mappings by linear mappings. J. Funct. Anal. 46, 126–130 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  45. Smajdor W.: Superadditive set-valued functions. Glas. Mat. 21, 343–348 (1986)

    MathSciNet  MATH  Google Scholar 

  46. Ulam, S.M.: A Collection of Mathematical Problems, Interscience Publishers, New York, 1960, Reprinted as: Problems in Modern Mathematics, Wiley, New York (1964)

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

    Hamid Khodaei

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  1. Hamid Khodaei
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Khodaei, H. On the Stability of Additive, Quadratic, Cubic and Quartic Set-valued Functional Equations. Results. Math. 68, 1–10 (2015). https://doi.org/10.1007/s00025-014-0416-0

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  • DOI: https://doi.org/10.1007/s00025-014-0416-0

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