Skip to main content
SpringerLink
Log in

Stability of a Cauchy-Jensen Additive Mapping in Various Normed Spaces

  • Chapter
  • First Online:
Applications of Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 134))

  • 1320 Accesses

Abstract

In this paper, using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the following Cauchy-Jensen additive functional equation in various normed spaces.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
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. T. Aoki, On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

    Article  MathSciNet  Google Scholar 

  2. L.M. Arriola, W.A. Beyer, Stability of the Cauchy functional equation over p-adic fields. Real Anal. Exch. 31, 125–132 (2005/2006)

    Google Scholar 

  3. T. Bag, S.K. Samanta, Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 11, 687–705 (2003)

    Google Scholar 

  4. T. Bag, S.K. Samanta, Fuzzy bounded linear operators. Fuzzy Sets Syst. 151, 513–547 (2005)

    Article  MathSciNet  Google Scholar 

  5. L. Cădariu, V. Radu, Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math. 4(1), Article ID 4 (2003)

    Google Scholar 

  6. L. Cădariu, V. Radu, On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346, 43–52 (2004)

    Google Scholar 

  7. L. Cădariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, Article ID 749392 (2008)

    Google Scholar 

  8. S.C. Cheng, J.N. Mordeson, Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86, 429–436 (1994)

    Google Scholar 

  9. P.W. Cholewa, Remarks on the stability of functional equations. Aequationes Math. 27, 76–86 (1984)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. S. Czerwik, Functional Equations and Inequalities in Several Variables (World Scientific, River Edge, 2002)

    Google Scholar 

  12. D. Deses, On the representation of non-Archimedean objects. Topol. Appl. 153, 774–785 (2005)

    Article  MathSciNet  Google Scholar 

  13. J. Diaz, B. Margolis, A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

    Article  MathSciNet  Google Scholar 

  14. M. Eshaghi Gordji, M. Bavand Savadkouhi, Stability of mixed type cubic and quartic functional equations in random normed spaces. J. Inequal. Appl. 2009, Article ID 527462, 9 pp. (2009)

    Google Scholar 

  15. M. Eshaghi Gordji, H. Khodaei, Stability of Functional Equations (Lap Lambert Academic Publishing, Saarbrücken, 2010)

    Google Scholar 

  16. M. Eshaghi Gordji, S. Abbaszadeh, C. Park, On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces. J. Inequal. Appl. 2009, Article ID 153084, 26 pp. (2009)

    Google Scholar 

  17. M. Eshaghi Gordji, M. Bavand Savadkouhi, C. Park, Quadratic-quartic functional equations in RN-spaces. J. Inequal. Appl. 2009, Article ID 868423, 14 pp. (2009)

    Google Scholar 

  18. M. Eshaghi Gordji, S. Zolfaghari, J.M. Rassias, M.B. Savadkouhi, Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces. Abstr. Appl. Anal. 2009, Article ID 417473, 14 pp. (2009)

    Google Scholar 

  19. W. Fechner, Stability of a functional inequality associated with the Jordan-von Neumann functional equation. Aequationes Math. 71, 149–161 (2006)

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  21. K. Hensel, Ubereine news Begrundung der Theorie der algebraischen Zahlen. Jahresber. Deutsch. Math. Verein 6, 83–88 (1897)

    Google Scholar 

  22. D.H. Hyers, On the stability of the linear functional equation. Proc. Natl. Acad. Sci. U. S. A. 27, 222–224 (1941)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  24. K. Jun, H. Kim, J.M. Rassias, Extended Hyers-Ulam stability for Cauchy-Jensen mappings. J. Differ. Equ. Appl. 13, 1139–1153 (2007)

    Article  MathSciNet  Google Scholar 

  25. I. Karmosil, J. Michalek, Fuzzy metric and statistical metric spaces. Kybernetica 11, 326–334 (1975)

    MathSciNet  Google Scholar 

  26. A.K. Katsaras, Fuzzy topological vector spaces. Fuzzy Sets Syst. 12, 143–154 (1984)

    Article  MathSciNet  Google Scholar 

  27. A.K. Katsaras, A. Beoyiannis, Tensor products of non-Archimedean weighted spaces of continuous functions. Georgian Math. J. 6, 33–44 (1999)

    Article  MathSciNet  Google Scholar 

  28. H.A. Kenary, On the stability of a cubic functional equation in random normed spaces. J. Math. Ext. 4, 1–11 (2009)

    MathSciNet  MATH  Google Scholar 

  29. H.A. Kenary, Stability of a Pexiderial functional equation in random normed spaces. Rend. Circ. Mat. Palermo 60, 59–68 (2011)

    Article  MathSciNet  Google Scholar 

  30. A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Mathematics and Its Applications, vol. 427 (Kluwer Academic Publishers, Dordrecht, 1997)

    Book  Google Scholar 

  31. Z. Kominek, On a local stability of the Jensen functional equation. Demonstratio Math. 22, 499–507 (1989)

    MathSciNet  MATH  Google Scholar 

  32. S.V. Krishna, K.K.M. Sarma, Separation of fuzzy normed linear spaces. Fuzzy Sets Syst. 63, 207–217 (1994)

    Article  MathSciNet  Google Scholar 

  33. D. Mihet, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)

    Article  MathSciNet  Google Scholar 

  34. M. Mohammadi, Y.J. Cho, C. Park, P. Vetro, R. Saadati, Random stability of an additive-quadratic-quartic functional equation. J. Inequal. Appl. 2010, Article ID 754210, 18 pp. (2010)

    Google Scholar 

  35. A. Najati, C. Park, The Pexiderized Apollonius-Jensen type additive mapping and isomorphisms between C ∗-algebras. J. Differ. Equ. Appl. 14, 459–479 (2008)

    Article  MathSciNet  Google Scholar 

  36. P.J. Nyikos, On some non-Archimedean spaces of Alexandrof and Urysohn. Topol. Appl. 91, 1–23 (1999)

    Article  MathSciNet  Google Scholar 

  37. C. Park, Generalized Hyers-Ulam-Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over C ∗-algebras. J. Comput. Appl. Math. 180, 279–291 (2005)

    Article  MathSciNet  Google Scholar 

  38. C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007, Article ID 50175 (2007)

    Article  MathSciNet  Google Scholar 

  39. C. Park, Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach. Fixed Point Theory Appl. 2008, Article ID 493751 (2008)

    Article  Google Scholar 

  40. C. Park, Fuzzy stability of a functional equation associated with inner product spaces. Fuzzy Sets Syst. 160, 1632–1642 (2009)

    Article  MathSciNet  Google Scholar 

  41. J.C. Parnami, H.L. Vasudeva, On Jensen’s functional equation. Aequationes Math. 43, 211–218 (1992)

    Article  MathSciNet  Google Scholar 

  42. V. Radu, The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91–96 (2003)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  44. Th.M. Rassias, Problem 16, in Report of the 27th International Symposium on Functional Equations. Aequations Mathematicae, vol. 39 (1990), pp. 292–293

    Google Scholar 

  45. Th.M. Rassias, On the stability of the quadratic functional equation and its applications. Stud. Univ. Babes-Bolyai. XLIII, 89–124 (1998)

    Google Scholar 

  46. Th.M. Rassias, The problem of S.M. Ulam for approximately multiplicative mappings. J. Math. Anal. Appl. 246, 352–378 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  48. Th.M. Rassias, Functional Equations, Inequalities and Applications (Kluwer Academic Publishers Co., Dordrecht, 2003)

    Book  Google Scholar 

  49. Th.M. Rassias, P. Semrl, On the behaviour of mappings which do not satisfy Hyers-Ulam stability. Proc. Am. Math. Soc. 114, 989–993 (1992)

    Article  Google Scholar 

  50. Th.M. Rassias, P. Semrl, On the Hyers-Ulam stability of linear mappings. J. Math. Anal. Appl. 173, 325–338 (1993)

    Article  MathSciNet  Google Scholar 

  51. J. Rätz, On inequalities associated with the Jordan-von Neumann functional equation. Aequationes Math. 66, 191–200 (2003)

    Article  MathSciNet  Google Scholar 

  52. R. Saadati, C. Park, Non-Archimedean \(\mathcal {L}\)-fuzzy normed spaces and stability of functional equations. Comput. Math. Appl. 60, 2488–2496 (2010)

    Google Scholar 

  53. R. Saadati, M. Vaezpour, Y.J. Cho, A note to paper “On the stability of cubic mappings and quartic mappings in random normed spaces”. J. Inequal. Appl. 2009, Article ID 214530 (2009). https://doi.org/10.1155/2009/214530

    Article  Google Scholar 

  54. R. Saadati, M.M. Zohdi, S.M. Vaezpour, Nonlinear L-random stability of an ACQ-functional equation. J. Inequal. Appl. 2011, Article ID 194394, 23 pages (2011). https://doi.org/10.1155/2011/194394

  55. B. Schewizer, A. Sklar, Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics (North-Holland, New York, 1983)

    Google Scholar 

  56. F. Skof, Local properties and approximation of operators. Rend. Sem. Mat. Fis. Milano 53, 113–129 (1983)

    Article  MathSciNet  Google Scholar 

  57. S.M. Ulam, Problems in Modern Mathematics, Science Editions (Wiley, New York, 1964)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Yasouj University, Yasuj, Iran

    Hassan Azadi Kenary

  2. Research Institute for Natural Sciences, Hanyang University, Seoul, South Korea

    Choonkil Park

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

    Themistocles M. Rassias

  4. Department of Mathematics, Daejin University, Pocheon, South Korea

    Jung Rye Lee

Authors
  1. Hassan Azadi Kenary
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Choonkil Park
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Themistocles M. Rassias
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jung Rye Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Choonkil Park .

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

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kenary, H.A., Park, C., Rassias, T.M., Lee, J.R. (2018). Stability of a Cauchy-Jensen Additive Mapping in Various Normed Spaces. In: Rassias, T. (eds) Applications of Nonlinear Analysis . Springer Optimization and Its Applications, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-89815-5_15

Download citation

Publish with us

Policies and ethics

Search

Navigation