Advertisement

Results in Mathematics

, 74:59 | Cite as

Polynomials of Arithmetically Homogeneous Functions: Stability and Hyperstability

  • Dan M. DăianuEmail author
  • Cristina Mîndruţă
Article

Abstract

We give large classes of control functions that provide generalized stability, respectively hyperstability for difference equations that characterize polynomials of arithmetically homogeneous functions. We also give a new technique to study the generalized stability and hyperstability of Fréchet’s equation, technique that allows us to expand and refine some of the known results in literature.

Keywords

Arithmetically homogeneous function h-polynomial Fréchet polynomial monomial stability hyperstability 

Mathematics Subject Classification

Primary 39B52 Secondary 39A70 39B82 47H10 

Notes

References

  1. 1.
    Albert, M., Baker, J.: Functions with bounded \(n\)th differences. Ann. Polon. Math. 43, 93–103 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brzdȩk, J., Popa, D., Raşa, I., Xu, B.: Ulam Stability of Operators. Academic Press, Elsevier (2018)zbMATHGoogle Scholar
  3. 3.
    Brzdȩk, J., Ciepliński, K.: Hyperstability and superstability. Abstr. Appl. Anal. 2013, ID 401756 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Dăianu, D.M.: Recursive procedure in the stability of Fréchet polynomials. Adv. Differ. Equ. 2014, 16 (2014)CrossRefGoogle Scholar
  5. 5.
    Dăianu, D.M.: A stability criterion for Fréchet’s first polynomial equation. Aequat. Math. 88(3), 233–241 (2014)CrossRefGoogle Scholar
  6. 6.
    Dăianu, D.M.: Fixed point approach to the stability of generalized polynomials. Fixed Point Theory 20(1), 135–156 (2019)Google Scholar
  7. 7.
    Dăianu, D.M., Mîndruţă, C.: Arithmetically homogeneous functions: characterizations, stability and hyperstability. Aequat. Math. 92(6), 1061–1077 (2018).  https://doi.org/10.1007/s00010-018-0579-y MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dăianu, D.M.: Taylor type formula with Fréchet polynomials. Aequat. Math. 92(4), 695–707 (2018).  https://doi.org/10.1007/s00010-018-0574-3 CrossRefzbMATHGoogle Scholar
  9. 9.
    Djoković, D.Ž.: A representation theorem for \(\left( X_{1}-1\right) \left( X_{2}-1\right) \cdots \left( X_{n}-1\right) \) and its applications. Ann. Polon. Math. 22, 189–198 (1969)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fréchet, M.: Une definition fonctionnelle des polynômes. Nouv. Ann. Math. 9, 145–182 (1909)zbMATHGoogle Scholar
  11. 11.
    Lijn, G.: Les polynômes abstraits. Bull. Sci. Math. 64, 55–80, 102–112, 183–198 (1940)Google Scholar
  12. 12.
    Marchoud, A.: Sur les dérivées et sur les différences des fonctions de variables réelles. J. Math. Pures Appl. T6(9), 337–426 (1927)Google Scholar
  13. 13.
    Prager, W., Schwaiger, J.: Stability of a functional equation for generalized polynomials. Aequat. Math. 90(1), 67–75 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsPolitehnica University of TimişoaraTimisoaraRomania
  2. 2.Department of InformaticsWest University of TimişoaraTimisoaraRomania

Personalised recommendations