Skip to main content
SpringerLink
Log in

On a Weak Version of Hyers–Ulam Stability Theorem in Restricted Domains

  • Chapter
  • First Online:
Handbook of Functional Equations

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

  • 1522 Accesses

Abstract

In this chapter we consider a weak version of the Hyers–Ulam stability problem for the Pexider equation, Cauchy equation satisfied in restricted domains in a group when the target space of the functions is a 2-divisible commutative group. As the main result we find an approximate sequence for the unknown function satisfying the Pexider functional inequality, the limit of which is the approximate function in the Hyers–Ulam stability theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 19.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 29.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 29.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aczél, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University Press, New York-Sydney (1989)

    MATH  Google Scholar 

  2. Aczél, J., Chung J.K.: Integrable solutions of functional equations of a general type. Studia Sci. Math. Hung. 17, 51–67 (1982)

    MATH  Google Scholar 

  3. Alsina, C., Garcia-Roig, J.L.: On a conditional Cauchy equation on rhombuses. In: Rassias, J.M. (ed.), Functional Analysis, Approximation Theory and Numerical Analysis. World Scientific, Singapore (1994)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Bahyrycz, A., Brzdȩk, J.: On solutions of the d’Alembert equation on a restricted domain. Aequat. Math. 85, 169–183 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  6. Batko, B.: Stability of an alternative functional equation. J. Math. Anal. Appl. 339, 303–311 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Batko, B.: On approximation of approximate solutions of Dhombres` equation. J. Math. Anal. Appl. 340, 424–432 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bourgin, D.G.: Multiplicative transformations. Proc. Nat. Acad. Sci. U S A 36, 564–570 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bourgin, D.G.: Class of transformations and bordering transformations. Bull. Amer. Math. Soc. 57, 223–237 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brzdȩk, J.: On the quotient stability of a family of functional equations, Nonlin. Anal. TMA 71, 4396–4404 (2009)

    Article  MathSciNet  Google Scholar 

  11. Brzdȩk, J.: On a method of proving the Hyers-Ulam stability of functional equations on restricted domains. Aust. J. Math. Anal. Appl. 6, 1–10 (2009)

    MathSciNet  Google Scholar 

  12. Brzdȩk, J., Sikorska, J.: A conditional exponential functional equation and its stability. Nonlin. Anal. TMA 72, 2929–2934 (2010)

    MathSciNet  Google Scholar 

  13. Chung, J.: Stability of functional equations on restricted domains in a group and their asymptotic behaviors. Comput. Math. Appl. 60, 2653–2665 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chung, J.: Stability of conditional Cauchy functional equations. Aequat. Math. 83, 313–320 (2012)

    Article  MATH  Google Scholar 

  15. Chung, J.: Stability of Jensen-type functional equation on restricted domains in a group and their asymptotic behaviors. J. Appl. Math. 2012, 12, Article ID 691981, (2012)

    Google Scholar 

  16. Czerwik, S.: Stability of Functional Equations of Hyers-Ulam-Rassias Type. Hadronic Press, Palm Harbor (2003).

    Google Scholar 

  17. Fochi, M.: An alternative functional equation on restricted domain. Aequat. Math. 70, 2010–212 (2005)

    Article  MathSciNet  Google Scholar 

  18. Ger, R., Sikorska, J.: On the Cauchy equation on spheres. Ann. Math. Sil. 11, 89–99 (1997)

    MathSciNet  Google Scholar 

  19. Gordji, M.E., Rassias, T. M.: Ternary homomorphisms between unital ternary \(C^*\)-algebras. Proc. Roman. Acad. Series A 12(3), 189–196 (2011)

    MathSciNet  Google Scholar 

  20. Hyers, D.H.: On the stability of the linear functional equations. Proc. Nat. Acad. Sci. U S A 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

  21. Hyers, D.H., Rassias T.M.: Approximate homomorphisms. Aequat. Math. 44, 125–153 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hyers, D.H., Isac, G., Rassias T.M.: Stability of Functional Equations in Several Variables. Birkhauser, Boston (1998)

    Book  MATH  Google Scholar 

  23. Jung, S.M.: On the Hyers–Ulam stability of functional equations that have the quadratic property. J. Math. Anal. Appl. 222, 126–137 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Jung, S.M.: Hyers–Ulam stability of Jensen’s equation and its application. Proc. Amer. Math. Soc. 126, 3137–3143 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  MATH  Google Scholar 

  26. Jung, S.-M., Rassias, T. M.: Ulam’s problem for approximate homomorphisms in connection with Bernoulli’s differential equation. Appl. Math. Comput. 187(1), 223–227 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Jung, S.-M., Rassias T.M.: Approximation of analytic functions by Chebyshev functions. Abst. Appl. Anal. 2011, 10 p, Article ID 432961, (2011)

    Google Scholar 

  28. Kuczma, M.: Functional equations on restricted domains. Aequat. Math. 18, 1–34 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  29. Park, C., W.-G. Park, Lee, J.R., Rassias, T.M.: Stability of the Jensen type functional equation in Banach algebras: A fixed point approach. Korean J. Math. 19(2), 149–161 (2011)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Rassias, T.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl. 251, 264–284 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  32. Rassias, J.M., Rassias, M.J.: On the Ulam stability of Jensen and Jensen type mappings on restricted domains. J. Math. Anal. Appl. 281, 516–524 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  33. Rassias, M.J., Rassias, J.M. On the Ulam stability for Euler-Lagrange type quadratic functional equations. Austral. J. Math. Anal. Appl. 2, 1–10 (2005)

    Google Scholar 

  34. Sikorska, J.: On two conditional Pexider functinal equations and their stabilities. Nonlin. Anal. TMA 70, 2673–2684 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Skof, F.: Sull’approssimazione delle applicazioni localmente \(\delta-\)additive. Atii Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 117, 377–389 (1983)

    MATH  MathSciNet  Google Scholar 

  36. Ulam, S.M.: A Collection of Mathematical Problems. Interscience, New York (1960)

    MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (no. 2012008507).

Author information

Authors and Affiliations

  1. Department of Mathematics, Kunsan National University, 573-701, Kunsan, Republic of Korea

    Jaeyoung Chung

  2. Department of Mathematics Education, Dankook University, 448-701, Yongin, Republic of Korea

    Jeongwook Chang

Authors
  1. Jaeyoung Chung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeongwook Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jaeyoung Chung .

Editor information

Editors and Affiliations

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

    Themistocles M. Rassias

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Chung, J., Chang, J. (2014). On a Weak Version of Hyers–Ulam Stability Theorem in Restricted Domains. 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_6

Download citation

Publish with us

Policies and ethics

Search

Navigation