10.1186/2251-7456-6-43

Stability and common stability for the systems of linear equations and itsapplications

HTML PDF
  • Mohsen Alimohammady*1,
  • Ali Sadeghi1,
  1. Department of Mathematics, University of Mazandaran, Babolsar, IR

Published in Issue 2012-10-04

How to Cite

Alimohammady, M., & Sadeghi, A. (2012). Stability and common stability for the systems of linear equations and itsapplications. Mathematical Sciences, 6(1). https://doi.org/10.1186/2251-7456-6-43
Crossref
Scopus
Google Scholar
Europe PMC

HTML views: 8

PDF views: 78

Abstract

Abstract In this paper some results about the Hyers-Ulam-Rassias stability for thelinear functional equations in general form and its Pexiderized can beproved for given functions on general domain to a complex Banach spacesunder some suitable conditions. In connection with the problem of G. L.Forti in the 13st ICFEI we consider the common stability for the systems offunctional equations and our aim is to establish some commonHyers-Ulam-Rassias stability for systems of homogeneous linear functionalequations. The results is applied to the study of some superstabilityresults for the exponential functional equation.

Keywords

  • Superstability,
  • Common stability,
  • Linear equation,
  • Fixed point
HTML PDF

References

  1. Ulam (1960) 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. 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
  5. Rassias (2001) On approximation of approximately quadratic mappings by quadraticmappings (pp. 67-78)
  6. Rassias (2001) Solution of a Cauchy-Jensen stability Ulam type prpblem 37(3) (pp. 161-177)
  7. Rassias (1989) Solution of a problem of Ulam (pp. 268-273) https://doi.org/10.1016/0021-9045(89)90041-5
  8. Gavruta (1999) An answer to a question of John M. Rassias concerning the stability of Cauchy equation (pp. 67-71)
  9. Jun et al. (2007) Extended Hyers-Ulam stability for a Cauchy-Jensen mappings 13(12) (pp. 1139-1153) https://doi.org/10.1080/10236190701464590
  10. Jun et al. (2007) Extended stability problem for alternative Cauchy-Jensen mappings 8(4)
  11. Rassias (1994) On the stability of a multi-dimensional Cauchy type functional equation (pp. 365-376) World Sic. Publ
  12. Rassias (2001) Solution of a quadratic stability Hyers-Ulam type problem 50(1) (pp. 9-17)
  13. Forti (1995) Hyers-Ulam: stability of functional equations in several variables (pp. 143-190) https://doi.org/10.1007/BF01831117
  14. Hyers and Rassias (1992) Approximate homomorphisms (pp. 125-153) https://doi.org/10.1007/BF01830975
  15. SiBaha et al. (2007) Ulam-Gavruta-Rassias stability of a linear functional equation 7(Fe07) (pp. 157-166)
  16. Baker (2005) A general functional equation and its stability (pp. 1657-1664) https://doi.org/10.1090/S0002-9939-05-07841-X
  17. Ger and Šemrl (1996) The stability of the exponential equation (pp. 779-787) https://doi.org/10.1090/S0002-9939-96-03031-6
  18. Borelli and Forti (1995) On a general Hyers-Ulam stability result (pp. 229-236) https://doi.org/10.1155/S0161171295000287
  19. Nakmahachalasint (2007) On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear andEuler-Lagrange-Rassias functional equations (pp. 1-10)
  20. Park and Najati (2010) Generalized additive functional inequalities in Banach algebras 1(2) (pp. 54-64)
  21. Gordji et al. (2009) Solution and stability of a Mixed type Cubic and Quartic functional eautionin Quasi-Banach spaces (pp. 1-14)
  22. Gordji and Khodaei (2009) On the generalized Hyers-Ulam-Rassias stablity of quadratic functionalequations (pp. 1-11) https://doi.org/10.1155/2009/923476
  23. Jarosz (1997) Almost multiplicative functionals (pp. 37-58)
  24. Johnson (1986) Approximately multiplicative functionals 34(2) (pp. 489-510) https://doi.org/10.1112/jlms/s2-34.3.489
  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. Radu (2003) The fixed point alternative and the stability of functional equations (pp. 91-96)
  27. Ravi and Arunkumar (2007) On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange typefunctional equation 7(Fe07) (pp. 143-156)
  28. Rassias (2000) The problem of S. M. Ulam for approximately multiplicative mappings (pp. 352-378) https://doi.org/10.1006/jmaa.2000.6788
  29. Isac and Rassias (1996) Stability of Ψ-additive mappings: Applications to nonlinear analysis 19(2) (pp. 219-228) https://doi.org/10.1155/S0161171296000324
  30. Hyers and Rassias (1983) The Stability of homomorphisms and related topics (pp. 140-153)
  31. Hyers and Ulam (1952) Approximately convex functions (pp. 821-828) https://doi.org/10.1090/S0002-9939-1952-0049962-5
  32. Czerwik (1992) On the stability of the homogeneous mapping (pp. 268-272)
  33. Nikodem (1988) Approximately quasiconvex functions (pp. 291-294)
  34. Tabor and Tabor (1994) Homogenity is superstable (pp. 123-130)
  35. Agarwal et al. (2003) Stability of functional equations in single variable (pp. 852-869) https://doi.org/10.1016/j.jmaa.2003.09.032
  36. Kuczma (1968) Functional equations in a single variable PWN, Warszawa PWN
  37. Baker (1979) The stability of certain functional equations 112(1991) (pp. 729-732)
  38. Lee and Jun (1998) The stability of the equation f(x + p) = kf(x) 4(35) (pp. 653-658)
  39. Jung (1997) On the general Hyers-Ulam stability of gamma functional equation (pp. 437-446)
  40. Jung (1997) On the modified Hyers-Ulam-Rassias stability of the functional equation forgamma function 39(62) (pp. 235-239)
  41. Jung (1998) On the stability of gamma functional equation (pp. 306-309) https://doi.org/10.1007/BF03322090
  42. Alzer (1999) Remark on the stability of the gamma functional equation (pp. 199-200) https://doi.org/10.1007/BF03322812
  43. Barnes (1899) The theory of the G-function (pp. 264-314)
  44. Jun et al. (2000) Stability of generalized Gamma and Beta functional equations (pp. 15-24) https://doi.org/10.1007/s000100050132
  45. Kim (2000) On the stability of generalized Gamma functional equation (pp. 513-520) https://doi.org/10.1155/S0161171200003598
  46. Kim and Lee (2000) The stability of the beta functional equation (pp. 89-96)
  47. Baker et al. (1979) The stability of the equation f(x + y) = f(x)f(y) (pp. 242-246)
  48. Alimohammady and Sadeghi (2011) On the superstability of the Pexider type of exponential equation in Banachalgebra
  49. Alimohammady and Sadeghi (2011) Some new results on the superstability of the Cuachy equation onsemigroups
  50. Bouikhalene et al. (2010) The superstability of d’ Alembert’s functional equation on theHeisenberg group (pp. 105-109) https://doi.org/10.1016/j.aml.2009.08.013
  51. Czerwik and Przybyla (2007) A general Baker superstability criterion for the, D’Alembert functional equation (pp. 303-306) Walter de Gruyter Gmbh @ Co.KG
  52. Baker (1980) The stability of the cosine equation (pp. 411-416) https://doi.org/10.1090/S0002-9939-1980-0580995-3
  53. Székelyhidi (1982) On a theorem of Baker, Lawrence and Zorzitto (pp. 95-96) https://doi.org/10.2307/2043816
  54. Rassias (1994) Problem 18, In: Report on the 31st ISFE (pp. 312-13)
  55. Diaz and Margolis (1968) A fixed point theorem of the alternative for contractions on a generalizedcomplete metric space (pp. 305-309) https://doi.org/10.1090/S0002-9904-1968-11933-0
  56. Gajda (1991) On stability of additive mappings (pp. 431-434) https://doi.org/10.1155/S016117129100056X
  57. Azadi Kenary and Cho (2011) Stability of mixed additive-quadratic Jensen type functional equation invarious spaces 61(9) (pp. 2704-2724) https://doi.org/10.1016/j.camwa.2011.03.024
  58. Azadi Kenary et al. (2011) A fixed point approach to Hyers-Ulam stability of a functional equation invarious normed spaces
  59. Azadi Kenary et al. (2011) Nonlinear approxiamtion of an ACQ functional equation in NAN-spaces
  60. Azadi Kenary and Park (2011) Direct and fixed point methods approach to the generalized Hyers - Ulamstability for a functional equation having monomials as solutions (pp. 301-307)
  61. Kenary (2011) Approximate Additive Functional Equations in Closed Convex Cone 5(2) (pp. 51-65)
  62. Forti (1985) Remark 11 in: Report of the 22nd Internat. Symposium on FunctionalEquations (pp. 90-91) https://doi.org/10.1007/BF02189806
  63. Nakmahachalasint (2007) Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities of additive functional equation in several variables