10.1186/2251-7456-6-59

On approximate homomorphisms: a fixed point approach

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  • Madjid Eshaghi Gordji*1,
  • Zahra Alizadeh1,
  • Hamid Khodaei1,
  • Choonkil Park2,
  1. Department of Mathematics, Semnan University, Semnan, IR
  2. Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, 133-791, KR

Published in Issue 2012-10-29

How to Cite

Gordji, M. E., Alizadeh, Z., Khodaei, H., & Park, C. (2012). On approximate homomorphisms: a fixed point approach. Mathematical Sciences, 6(1). https://doi.org/10.1186/2251-7456-6-59
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Abstract

Abstract Abstract Consider the functional equation ℑ 1 ( f ) = ℑ 2 ( f ) ( ℑ )in a certain general setting. A function g is an approximate solution of ( ℑ )if ℑ 1 ( g )and ℑ 2 ( g )are close in some sense. The Ulam stability problem asks whether or not there is a true solution of ( ℑ )near g . A functional equation is superstable if every approximate solution of the functional equation is an exact solution of it. In this paper, for each m = 1,2,3,4, we will find out the general solution of the functional equation f(ax+y)+f(ax-y)=am-2[f(x+y)+f(x-y)]+2(a2-1)[am-2f(x)+(m-2)(1-(m-2)2)6f(y)] for any fixed integer a with a ≠ 0, ± 1. Using a fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms in real Banach algebras for this functional equation. Moreover, we establish the superstability of this functional equation by suitable control functions. 2010 Mathematics Subject Classification 39B52, 47H10, 39B82

Keywords

  • Banach algebra,
  • Approximate homomorphism,
  • Additive,
  • Quadratic,
  • Cubic and quartic functional equation,
  • Fixed point approach
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References

  1. Ulam (1964) Wiley
  2. Hyers (1941) On the stability of the linear functional equation (pp. 222-224) https://doi.org/10.1073/pnas.27.4.222
  3. Aoki (1950) On the stability of the linear transformation in Banach spaces (pp. 64-66) https://doi.org/10.2969/jmsj/00210064
  4. Bourgin (1951) Classes of transformations and bordering transformations (pp. 223-237) https://doi.org/10.1090/S0002-9904-1951-09511-7
  5. Rassias (1978) On the stability of the linear mapping in Banach spaces (pp. 297-300) https://doi.org/10.1090/S0002-9939-1978-0507327-1
  6. Gajda (1991) On stability of additive mappings (pp. 431-434) https://doi.org/10.1155/S016117129100056X
  7. Rassias and Semrl (1992) On the behavior of mappings which do not satisfy Hyers-Ulam stability (pp. 989-993) https://doi.org/10.1090/S0002-9939-1992-1059634-1
  8. Isac and Rassias (1993) (pp. 131-137) https://doi.org/10.1006/jath.1993.1010
  9. Lee and Jun (2000) On the stability of approximately additive mappings (pp. 1361-1369) https://doi.org/10.1090/S0002-9939-99-05156-4
  10. Gávruta (1994) A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings (pp. 431-436) https://doi.org/10.1006/jmaa.1994.1211
  11. Aczel and Dhombres (1989) Cambridge University Press https://doi.org/10.1017/CBO9781139086578
  12. Kannappan (1995) Quadratic functional equation and inner product spaces (pp. 368-372) https://doi.org/10.1007/BF03322841
  13. Czerwik (1992) On the stability of the quadratic mapping in normed spaces (pp. 59-64) https://doi.org/10.1007/BF02941618
  14. Gordji Eshaghi and Khodaei (2009) Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces (pp. 5629-5643) https://doi.org/10.1016/j.na.2009.04.052
  15. Eshaghi Gordji and Khodaei (2009) On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations
  16. Forti (1980) An existence and stability theorem for a class of functional equations (pp. 23-30) https://doi.org/10.1080/17442508008833155
  17. Forti (2007) Elementary remarks on Ulam-Hyers stability of linear functional equations (pp. 109-118) https://doi.org/10.1016/j.jmaa.2006.04.079
  18. Jung (1998) Hyers-Ulam-Rassias stability of Jensen’s equation and its application (pp. 3137-3143) https://doi.org/10.1090/S0002-9939-98-04680-2
  19. Jung (2001) Hadronic Press Inc.
  20. Khodaei and Rassias (2010) Approximately generalized additive functions in several variables (pp. 22-41)
  21. Park (2004) On an approximate automorphism on a C∗-algebra (pp. 1739-1745) https://doi.org/10.1090/S0002-9939-03-07252-6
  22. Rassias (1982) On Approximation of approximately linear mappings by linear mappings (pp. 126-130) https://doi.org/10.1016/0022-1236(82)90048-9
  23. Rassias (1989) Solution of a problem of Ulam (pp. 268-273) https://doi.org/10.1016/0021-9045(89)90041-5
  24. Rassias (1984) New characterization of inner product spaces (pp. 95-99)
  25. Rassias (2000) On the stability of functional equations and a problem of Ulam (pp. 23-130) https://doi.org/10.1023/A:1006499223572
  26. Jun and Kim (2002) The generalized Hyers-Ulam-Rassias stability of a cubic functional equation (pp. 867-878) https://doi.org/10.1016/S0022-247X(02)00415-8
  27. Lee et al. (2005) Quartic functional equation (pp. 387-394) https://doi.org/10.1016/j.jmaa.2004.12.062
  28. Bourgin (1949) Approximately isometric and multiplicative transformations on continuous function rings (pp. 385-397) https://doi.org/10.1215/S0012-7094-49-01639-7
  29. Badora (2002) On approximate ring homomorphisms (pp. 589-597) https://doi.org/10.1016/S0022-247X(02)00293-7
  30. Baker et al. (1979) The stability of the equation f(x+y) = f(x)f(y) (pp. 242-246)
  31. Eshaghi Gordji et al. (2009) Approximately n-Jordan homomorphisms on Banach algebras
  32. Eshaghi Gordji and Najati (2010) Approximately J∗-homomorphisms: A fixed point approach (pp. 809-814) https://doi.org/10.1016/j.geomphys.2010.01.012
  33. Gordji and Savadkouhi (2009) Approximation of generalized homomorphisms in quasi–Banach algebras 17(2) (pp. 203-214)
  34. Park and Jang (2009) Cauchy–Rassias stability of sesquilinear n-quadratic mappings in Banach modules 39(6) (pp. 2015-2027) https://doi.org/10.1216/RMJ-2009-39-6-2015
  35. Park (2005) Homomorphisms between Lie JC∗-algebras and Cauchy-Rassias stability of Lie JC∗-algebra derivations (pp. 393-414)
  36. Park (2004) Lie ∗-homomorphisms between Lie C∗-algebras and Lie ∗-derivations on Lie C∗-algebras (pp. 419-434) https://doi.org/10.1016/j.jmaa.2003.10.051
  37. Gordgi and Ghobadipour (2010)
  38. Gordgi and Bavand Savadkouhi (2009) On approximate cubic homomorphisms
  39. Margolis and Diaz (1968) A fixed point theorem of the alternative for contractions on the generalized complete metric space (pp. 305-309)
  40. Hyers et al. (1998) https://doi.org/10.1007/978-1-4612-1790-9
  41. Radu (2003) The fixed point alternative and the stability of functional equations (pp. 91-96)
  42. Cădariu and Radu (2003) Fixed points and the stability of Jensen functional equation
  43. Cădariu and Radu (2004) On the stability of the Cauchy functional equation: a fixed point approach (pp. 43-52)
  44. Că dariu and Radu (2008) Fixed point methods for the generalized stability of functional equations in a single variable
  45. Bae and Park (2010) A functional equation having monomials as solutions (pp. 87-94) https://doi.org/10.1016/j.amc.2010.01.006
  46. Lee and Chung (2007) Stability for quadratic functional equation in the spaces of generalized functions (pp. 101-110) https://doi.org/10.1016/j.jmaa.2007.02.053
  47. Lee and Chung (2009) Stability of quartic functional equation in the spaces of generalized functions https://doi.org/10.1155/2009/838347
  48. Najati (2007) The generalized Hyers-Ulam-Rassias stability of a cubic functional equation (pp. 395-408)