Stability of Multi-Quadratic Functions in Lipschitz Spaces

  • Ismail NikoufarEmail author
Research Paper


The algebra of Lipschitz functions on a complete metric space plays a role in non-commutative metric theory similar to that played by the algebra of continuous functions on a compact space in non-commutative topology. The notion of stability of functional equations was posed by Ulam, and then, Hyers gave the first significant partial solution in 1941. The subject has been established and developed by an increasing number of mathematicians in various spaces, particularly during the last two decades. In Lipschitz spaces, the notion of stability-type problems was introduced by Tabor and Czerwik. It should be mentioned that this subject area has been considered less attention over the recent years. In this paper, we prove stability of multi-quadratic functional equations in Lipschitz spaces by introducing the notion of a multi-symmetric left invariant mean. Indeed, we prove under certain Lipschitz conditions a family of Lipschitz functions can be approximated by multi-quadratic functions.


Multi-quadratic functional equation Lipschitz space stability Approximation 

Mathematics Subject Classification

39B82 39B52 


  1. Cieplinski K (2010) Generalized stability of multi-additive mappings. Appl Math Lett 23:1291–1294MathSciNetCrossRefzbMATHGoogle Scholar
  2. Cieplinski K (2010) Stability of the multi-Jensen equation. J Math Anal Appl 363:249–254MathSciNetCrossRefzbMATHGoogle Scholar
  3. Cieplinski K (2011) On the generalized Hyers-Ulam stability of multi-quadratic mappings. Comput Math Appl 62(9):3418–3426MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cieplinski K (2011) Stability of multi-additive mappings in non-Archimedean normed spaces. J Math Anal Appl 373:376–383MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cieplinski K (2012) Stability of multi–additive mappings in \(\beta\)-Banach spaces.Nonlinear Anal 75:4205–4212Google Scholar
  6. Czerwik S, Dlutek K (2004) Stability of the quadratic functional equation in Lipschitz spaces. J Math Anal Appl 293:79–88MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ebadian A, Ghobadipour N, Nikoufar I, Gordji M (2014) Approximation of the cubic functional equations in Lipschitz spaces. Anal Theory Appl 30:354–362MathSciNetCrossRefzbMATHGoogle Scholar
  8. Ji P, Qi W, Zhan X (2014) Generalized stability of multi-quadratic mappings. J Math Res Appl 34:209–215MathSciNetzbMATHGoogle Scholar
  9. Liguang W, Bo L, Ran B (2010) Stability of a mixed type functional equation on multi-Banach spaces: a fixed point approach. Fixed Point Theory Appl 2010, Article ID 283827Google Scholar
  10. Liguang W, Kunpeng X, Qiuwen L (2014) On the stability a mixed functional equation deriving from additive, quadratic and cubic mappings. Acta Math Sin 30:1033–1049MathSciNetCrossRefGoogle Scholar
  11. Jung S-M, Lee Z-H (2008) A fixed point approach to the stability of quadratic functional equation with involution. Fixed Point Theory Appl 2008, Article ID 732086Google Scholar
  12. Jung S-M, Sahoo PK (2001) Hyers–Ulam stability of the quadratic equation of Pexider type. J Korean Math Soc 38(3):645–656Google Scholar
  13. Lee JR, Jang SY, Park C, Shin DY (2010) Fuzzy stability of quadratic functional equations. Adv Differ Equ 2010, Article ID 412160Google Scholar
  14. Nikoufar I (2015) Lipschitz approximation of the \(n\)-quadratic functional equations. Mathematica (Cluj) 57:67–74MathSciNetzbMATHGoogle Scholar
  15. Nikoufar I (2015) Quartic functional equations in Lipschitz spaces. Rend Circ Mat Palermo 64(2):171–176MathSciNetCrossRefzbMATHGoogle Scholar
  16. Nikoufar I (2016) Erratum to: quartic functional equations in Lipschitz spaces. Rend Circ Mat Palermo 65(2):345–350MathSciNetCrossRefzbMATHGoogle Scholar
  17. Nikoufar I (2016) Lipschitz criteria for bi-quadratic functional equations. Commun Korean Math Soc 31(4):819–825MathSciNetCrossRefzbMATHGoogle Scholar
  18. Park C (2002) On the stability of the quadratic mapping in Banach modules. J Math Anal Appl 276(1):135–144Google Scholar
  19. Park W-G, Bae J-H (2010) Approximate behavior of bi-quadratic mappings in quasinormed spaces. J Inequal Appl 2010, Article ID 472721Google Scholar
  20. Rassias JM (2002) On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings. Aequationes Math 64:62–69MathSciNetCrossRefzbMATHGoogle Scholar
  21. Skof F (1983) Local properties and approximations of operators. Rend Sem Mat Fis Milano 53:113–129Google Scholar
  22. Tabor J (1997) Lipschitz stability of the Cauchy and Jensen equations. Results Math 32:133–144MathSciNetCrossRefzbMATHGoogle Scholar
  23. Tabor J (1999) Superstability of the Cauchy, Jensen and isometry equations. Results Math. 35:355–379MathSciNetCrossRefzbMATHGoogle Scholar
  24. Zhao X, Yang X, Pang C-T (2013) Solution and stability of the multi-quadratic functional equation. Abs Appl Anal 2013, Article ID 415053Google Scholar

Copyright information

© Shiraz University 2018

Authors and Affiliations

  1. 1.Department of MathematicsPayame Noor UniversityTehranIran

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