DOI QR코드

DOI QR Code

On the Generalized Hyers-Ulam-Rassias Stability for a Functional Equation of Two Types in p-Banach Spaces

  • Received : 2006.07.24
  • Published : 2008.03.31

Abstract

We investigate the generalized Hyers-Ulam-Rassias stability in p-Banach spaces for the following functional equation which is two types, that is, either cubic or quadratic: 2f(x+3y) + 6f(x-y) + 12f(2y) = 2f(x - 3y) + 6f(x + y) + 3f(4y). The concept of Hyers-Ulam-Rassias stability originated essentially with the Th. M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297-300.

Keywords

stability;cubic mapping;quadratic mapping;quasi-normed spaces;p-Banach spaces

References

  1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, 1989.
  2. J. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc., 80(1980), 411-416. https://doi.org/10.1090/S0002-9939-1980-0580995-3
  3. Y. Benyamini and J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1, Colloq. Publ. 48, Amer. Math. Soc. Providence, 2000.
  4. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27(1984), 76-86. https://doi.org/10.1007/BF02192660
  5. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg, 62(1992), 59-64. https://doi.org/10.1007/BF02941618
  6. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific Publ. Co., New Jersey, London, Singapore, Hong Kong, 2002.
  7. S. Czerwik(ed), Stability of Functional Equations of Ulam-Hyers-Rassias Type, Hadronic Press, Inc., Palm Harbor, Florida, 2003.
  8. V. A. Faiziev, Th. M. Rassias and P. K. Sahoo, The space of (${\psi},{\gamma}$)-additive mappings on semigroups, Trans. Amer. Math. Soc., 364(11)(2002), 4455-4472.
  9. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci., 14(1991), 431-434. https://doi.org/10.1155/S016117129100056X
  10. P. Gavruta, A generalization of the Hyers-Ulam-Rassias Stability of approximately additive mappings, J. Math. Anal. Appl., 184(1994), 431-436. https://doi.org/10.1006/jmaa.1994.1211
  11. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27(1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  12. D. H. Hyers, G. Isac and Th. M. Rassias, "Stability of Functional Equations in Several Variables", Birkhauser, Basel, 1998.
  13. D. H. Hyers, G. Isac and Th. M. Rassias, On the asymptoticity aspect of Hyers-Ulam stability of mappings, Proc. Amer. Math. Soc., 126(1998), 425-430. https://doi.org/10.1090/S0002-9939-98-04060-X
  14. K. -W. Jun and H.-M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274(2)(2002), 867-878. https://doi.org/10.1016/S0022-247X(02)00415-8
  15. S. -M. Jung, On the Hyers-Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl., 222(1998), 126-137. https://doi.org/10.1006/jmaa.1998.5916
  16. S. -M. Jung, Hyers-Ulam-Rassias Stability of Functional equations in Mathematical Analysis, Hadronic Press, Inc., Palm Harbor, Florida, 2001.
  17. S. -M. Jung, On the Hyers-Ulam-Rassias stability of a quadratic functional equation, J. Math. Anal. Appl., 232(1999), 384-393. https://doi.org/10.1006/jmaa.1999.6282
  18. N. Kalton, N. T. Peck, and W. Roberts, An F-Space Sampler, London Mathematical Society Lecture Note Series 89, Cambridge University Press, (1984).
  19. Pl. Kannappan, Quadratic functional equation and inner product spaces, Results Math., 27(1995), 368-372. https://doi.org/10.1007/BF03322841
  20. C. Park, Generalized quadratic mappings in several variables, Nonlinear Anal. -TMA, 57(2004), 713-722. https://doi.org/10.1016/j.na.2004.03.013
  21. C. Park, Cauchy-Rassias stability of a generalized Trif 's mapping in Banach modules and its applications, Nonlinear Anal. -TMA, 62(2005), 595-613. https://doi.org/10.1016/j.na.2005.03.071
  22. J. M. Rassias, Solution of the Ulam stability problem for cubic mappings, Glas. Mat., 36(1)(2001), 63-72.
  23. J. M. Rassias, On the Hyers-Ulam stability problem for quadratic multi-dimensional mappings, Aequationes Math., 64(2002), 62-69. https://doi.org/10.1007/s00010-002-8031-7
  24. J. M. Rassias, On the Ulam stability of the mixed type mappings on restricted domains, J. Math. Anal. Appl., 276(2002), 747-762. https://doi.org/10.1016/S0022-247X(02)00439-0
  25. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  26. Th. M. Rassias, The problems of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl., 246(2000), 352-378. https://doi.org/10.1006/jmaa.2000.6788
  27. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251(2000), 264-284. https://doi.org/10.1006/jmaa.2000.7046
  28. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl., 62(2000), 23-130. https://doi.org/10.1023/A:1006499223572
  29. Th. M. Rassias (Ed.), "Functional Equations and inequalities", Kluwer Academic, Dordrecht, Boston, London, 2000.
  30. Th. M. Rassias (Ed.), "Functional Equations and Inequalities and Applications", Kluwer Academic, Dordrecht, Boston, London, 2003.
  31. Th. M. Rassias and J. Tabor, What is left of Hyers-Ulam stability?, Journal of Natural Geometry, 1(1992), 65-69.
  32. Th. M. Rassias and J. Tabor, "Stability of mappings of Hyers-Ulam type", Hadronic Press, Inc., Florida, 1994.
  33. Th. M. Rassias and P. Semrl, On the behavior of mappings which does not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc., 114(1992), 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
  34. S. Rolewicz, Metric Linear Spaces, Reidel and Dordrecht, and PWN-Polish Sci. Publ. 1984.
  35. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano, 53(1983), 113-129. https://doi.org/10.1007/BF02924890
  36. J. Tabor, Stability of the Cauchy functional equation in quasi-Banach spaces, Ann. Polon. Math., 83(2004), 243-255. https://doi.org/10.4064/ap83-3-6
  37. S. M. Ulam, Problems in Modern Mathematics, Chap. VI, Science ed., Wiley, New York, 1960.

Cited by

  1. A General System of Nonlinear Functional Equations in Non-Archimedean Spaces vol.53, pp.3, 2013, https://doi.org/10.5666/KMJ.2013.53.3.419