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Approximation by Cubic Mappings

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Ulam Type Stability

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Abstract

Starting with a stability problem posed by Ulam for group homomorphisms, we characterize the functions with values in a Banach space, which can be approximated by cubic mappings with a given error.

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References

  1. Borelli, C., Forti, G.L.: On a general Hyers-Ulam stability result. Int. J. Math. Math. Sci. 18, 229–236 (1995)

    Article  MathSciNet  Google Scholar 

  2. Brzdek, J.: Note on stability of the Cauchy equation-an answer to a problem of Th.M. Rassias. Carphtian J. Math. 30(1), 47–54 (2004)

    Google Scholar 

  3. Brzdek, J., Popa, D., Raşa, I., Xu, B.: Ulam Stability of Opeartors. Academic Press, New York (2018)

    MATH  Google Scholar 

  4. Brzdek, J., Karapinar, E., Petruşel, A.: A fixed point theorem and the Ulam stability in generalized dq-metric spaces. J. Math. Anal. Appl. 467, 501–520 (2018)

    Article  MathSciNet  Google Scholar 

  5. Cădariu, L., Radu, V.: Fixed points in generalized metric spaces and the stability of a cubic functional equation. In: Fixed Point Theory and Applications, vol. 7, pp. 53–68. Nova Science Publishers, New York (2007)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  8. Eskandani, G.Z., Rassias, J.M., Găvruţa, P.: Generalized Hyers-Ulam stability for a general cubic functional equation in quasi-β-normed spaces. Asian Eur. J. Math. 4(3), 413–425 (2011)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Găvruţa, P.: An answer to a question of Th.M. Rassias and J. Tabor on mixed stability of mappings. Bul. Stiint. Univ. Politeh. Timis. Ser. Mat. Fiz. 42(56), 1–6 (1997)

    Google Scholar 

  11. Găvruţa, P.: On the Hyers-Ulam-Rassias stability of mappings. In: Milovanovic, G.V. (ed.) Recent Progress in Inequalities, pp. 465–469. Kluwer, Dordrecht (1998)

    Chapter  Google Scholar 

  12. Găvruţa, P.: An answer to a question of John M. Rassias concerning the stability of Cauchy equation. In: Advances in Equations and Inequalities. Hadronic Mathematics Series, pp. 67–71. Hadronic Press, Palm Harbor (1999)

    Google Scholar 

  13. Găvruţa, P.: On a problem of G. Isac and Th.M. Rassias concerning the stability of mappings. J. Math. Anal. Appl. 261, 543–553 (2001)

    Article  MathSciNet  Google Scholar 

  14. Găvruţa, P.: On the Hyers-Ulam-Rassias stability of the quadratic mappings. Nonlinear Funct. Anal. Appl. 9(3), 415–428 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Găvruţa, P., Cădariu, L.: General stability of the cubic functional equation. Bul. Ştiinţ. Univ. Politehnica Timişoara Ser. Mat.-Fiz. 47(61)(1), 59–70 (2002)

    Google Scholar 

  16. Găvruţa, P., Cădariu, L.: The generalized stability of a quadratic functional equation. Nonlinear Funct. Anal. Appl. 9(4), 513–526 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Găvruţa, P., Găvruţa, L.: Ψ-additive mappings and Hyers-Ulam stability. In: Pardalos, P.M., Rassias, Th.M., Khan, A.A. (eds.) Nonlinear Analysis and Variational Problems, pp. 81–86. Springer, New York (2010)

    Chapter  Google Scholar 

  18. Găvruţa, L., Găvruţa, P.: On a problem of John M. Rassias concerning the stability in Ulam sense of Euler-Lagrange equation. In: Rassias, J.M. (ed.) Functionl Equation, Difference Inequalities and Ulam Stability Notions, pp. 47–53. Nova Science Publishers, New York (2010)

    MATH  Google Scholar 

  19. Găvruţa, L., Găvruţa, P.: Ulam stability problem for frames, Chap.11. In: Rassias, Th.M., Brzdȩk, J. (eds.) Functional Equations in Mathematical Analysis. Springer Optimization and Its Applications, pp. 139–152. Springer, New York (2012)

    Chapter  Google Scholar 

  20. Găvruţa, L., Găvruţa, P.: Approximation of functions by additive and by quadratic mappings. In: Rassias, Th.M., Gupta, V. (eds.) Mathematical Analysis, Approximation Theory and Their Applications. Springer Optimization and Its Applications, vol. 111, pp. 281–292. Springer, Cham (2016)

    Chapter  Google Scholar 

  21. Găvruţa, P., Hossu, M., Popescu, D., Căprău, C.: On the stability of mappings and an answer to a problem of Th. M. Rassias. Ann. Math. Blaise Pascal 2(2), 55–60 (1995)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  24. Isac, G., Rassias, Th.M.: On the Hyers-Ulam stability of ψ-additive mappings. J. Approx. Theory 72, 131–137 (1993)

    Article  MathSciNet  Google Scholar 

  25. Jun, K.-W., Kim, H.-M.: The generalized Hyers-Ulam-Rassias stability of a cubic functional equation. J. Math. Anal. Appl. 274, 867–878 (2002)

    Article  MathSciNet  Google Scholar 

  26. Jun, K.-W., Lee, Y.-H.: On the stability of a cubic functional equation. J. Chungcheong Math. Soc. 21(3), 377–384 (2008)

    MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  28. Jung, S.-M., Rassias, M.Th.: A linear functional equation of third order associated to the Fibonacci numbers. Abstr. Appl. Anal. 2014, 7 pp. (2014). Article ID 137468

    Google Scholar 

  29. Jung, S.-M., Popa, D., Rassias, M.Th.: On the stability of the linear functional equation in a single variable on complete metric groups. J. Glob. Optim. 59, 165–171 (2014)

    Article  MathSciNet  Google Scholar 

  30. Lee, Y.-H., Jung, S.-M., Rassias, M.Th.: On an n-dimensional mixed type additive and quadratic functional equation. Appl. Math. Comput. 228, 13–16 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Mirmostafaee, A.K., Moslehian, M.S.: Fuzzy approximately cubic mappins. Inf. Sci. 178(19), 3791–3798 (2008)

    Article  Google Scholar 

  32. Mortici, C., Rassias, Th.M., Jung, S.-M.: On the stability of a functional equation associated with the Fibbonacci numbers. Abstr. Appl. Anal. 2014, 6 pp. (2014). Article ID 546046

    Google Scholar 

  33. Najati, A., Park, C.: On the stability of a cubic functional equation. Acta Math. Sin. 24(12), 1953–1964 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. Rassias, J.M.: Solution of the stability problem for cubic mappings. Glasnik Matematicki 36(56), 73–84 (2001)

    MathSciNet  Google Scholar 

  36. Saadati, R., Vaezpour, S.M., Park, C.: The stability of the cubic functional equations in various spaces. Math. Commun. 16, 131–145 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Skof, F.: Proprietá locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milani 53, 113–129 (1983)

    Article  Google Scholar 

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

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Politehnica University of Timişoara, Timişoara, Romania

    Paşc Găvruţa & Laura Manolescu

Authors
  1. Paşc Găvruţa
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  2. Laura Manolescu
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Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, AGH University of Science and Technology, Krakow, Poland

    Janusz Brzdęk

  2. Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

    Dorian Popa

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

    Themistocles M. Rassias

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Găvruţa, P., Manolescu, L. (2019). Approximation by Cubic Mappings. In: Brzdęk, J., Popa, D., Rassias, T. (eds) Ulam Type Stability . Springer, Cham. https://doi.org/10.1007/978-3-030-28972-0_8

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