DOI QR코드

DOI QR Code

ON FUNCTIONAL INEQUALITIES CONNECTED WITH INTERTWINING MAPPINGS

  • Received : 2009.01.22
  • Accepted : 2009.06.04
  • Published : 2009.06.25

Abstract

For a mapping satisfying the inequality ${\parallel}{\lambda}f(x)+2{\lambda}f(y)+2f({\lambda}z){\parallel}{\leq}{\parallel}2f({\lambda}({\frac{2}{x}}+y+z)){\parallel}+{\phi}(x,y,z)$, we will study the stability problem of this mapping.

Keywords

References

  1. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64-66. https://doi.org/10.2969/jmsj/00210064
  2. W.G. Bade and P.C. Curtis Jr., Prime ideals and automatic continuity problems in Banach modules over $C^{\ast}$-algebras, J. Funct Anal. 29 (1978), 88-103. https://doi.org/10.1016/0022-1236(78)90048-4
  3. H.G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monogr(N.S). 24, Clarendon Press, Oxford University Press, Oxford 2000.
  4. H.G. Dales and A.R. Vilena, Continuity of derivations, intertwining maps, and cocycles from Banach algebras, J. London Math. Soc. 63 (2001), 215-225. https://doi.org/10.1112/S0024610700001770
  5. W. Fechner, Stability of a functional inequality associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149-161. https://doi.org/10.1007/s00010-005-2775-9
  6. Z. Gajda. On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), 431-434. https://doi.org/10.1155/S016117129100056X
  7. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  8. D.H. Hyers, G. Isac and Th.M. Rassias, Stability of functional equations in several variablers, Birkhauser, Basel, (1998).
  9. K.W. Jun and H.-M. Kim, Stability problem of Ulam for generalized forms of Cauchy functional equation, J. Math. Anal. Appl. 312 (2005), 535-547. https://doi.org/10.1016/j.jmaa.2005.03.052
  10. K.W. Jun and Y.H. Lee, A generalization of the Hyers-Ulam-Rassias stability of the Pexiderized quadratic equations, J. Math. Anal. Appl. 297 (2004), 70-86. https://doi.org/10.1016/j.jmaa.2004.04.009
  11. S.-M. Jung, Hyers-Ulam stability of a system of first order linear differential equations with constant coefficients, J. Math. Anal. Appl. 320 (2006), 549-561. https://doi.org/10.1016/j.jmaa.2005.07.032
  12. R. V. Kadison and G. Pederson. Means and convex combinations of unitary operators. Math. Scaud. 57 (1985), 249-266.
  13. K.B. Laursen, Automatic continuity of generalized intertwining operators, Dissertationes Math. (Rozprawy Mat.) 189 (1981).
  14. C. Park, Y. Cho and M. Han, Functional inequalities associated with Jordan-Von Neumann type additive functional equations, J. Inequal. Appl., 2007 (2007), Article ID 41820, 13 pages.
  15. J.M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), 126-130. https://doi.org/10.1016/0022-1236(82)90048-9
  16. 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
  17. Th.M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
  18. J. Roh and I.-S. Chang, Functional inequalities associated with additive mappings, Abstr. Appl. Anal. 2008 (2008), Article ID 136592, 1-10.
  19. S.M. Ulam, A collection of the mathematical problems, Interscience Publ. New York, (1960).