Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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APPROXIMATELY QUINTIC AND SEXTIC MAPPINGS ON THE PROBABILISTIC NORMED SPACES

  • Ghaemi, Mohammad Bagher (Department of Mathematics Iran University of Science and Technology) ;
  • Majani, Hamid (Department of Mathematics Iran University of Science and Technology) ;
  • Gordji, Majid Eshaghi (Department of Mathematics Semnan University, Center of Excellence in Nonlinear Analysis and Applications (CENAA) Semnan University)
  • Received : 2010.11.02
  • Published : 2012.03.31
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Abstract

We prove the stability for the systems of quadratic-cubic and additive-quadratic-cubic functional equations with constant coefficients on the probabilistic normed spaces (briefly PN spaces).

Keywords

References

  1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge, 1989.
  2. C. Alsina, B. Schweizer, and A. Sklar, On the definition of a probabilistic normed space, Aequationes Math. 46 (1993), no. 1-2, 91-98. https://doi.org/10.1007/BF01834000
  3. 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
  4. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86. https://doi.org/10.1007/BF02192660
  5. J. K. Chung and P. K. Sahoo, On the general solution of a quartic functional equation, Bull. Korean Math. Soc. 40 (2003), no. 4, 565-576. https://doi.org/10.4134/BKMS.2003.40.4.565
  6. 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
  7. S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, London, 2002.
  8. A. Ebadian, N. Ghobadipour, and M. E. Gordji, A fixed point method for perturbation of bimultipliers and Jordan bimultipliers in C*-ternary algebras, J. Math. Phys. 51 (2010), no. 1, 10 pages, doi:10.1063/1.3496391.
  9. A. Ebadian, A. Najati, and M. Eshaghi Gordji, On approximate additive-quartic and quadratic-cubic functional equations in two variables on abelian groups, Results Math. 58 (2010), no. 1-2, 39-53. https://doi.org/10.1007/s00025-010-0018-4
  10. M. Eshaghi Gordji, Stability of a functional equation deriving from quartic and additive functions, Bull. Korean Math. Soc. 47 (2010), no. 3, 491-502. https://doi.org/10.4134/BKMS.2010.47.3.491
  11. M. Eshaghi Gordji, Stability of an additive-quadratic functional equation of two variables in F-spaces, J. Nonlinear Sci. Appl. 2 (2009), no. 4, 251-259. https://doi.org/10.22436/jnsa.002.01.09
  12. M. Eshaghi Gordji, S. Abbaszadeh, and C. Park, On the stability of generalized mixed type quadratic and quartic functional equation in quasi-Banach spaces, J. Ineq. Appl. 2009 (2009), Article ID 153084, 26 pages.
  13. M. Eshaghi Gordji and M. Bavand-Savadkouhi, On approximate cubic homomorphisms, Adv. Difference Equ. 2009 (2009), Art. ID 618463, 11 pp., doi:10.1155/2009/618463.
  14. M. Eshaghi Gordji, S. Zolfaghari, J. M. Rassias, and M. B. Savadkouhi, Solution and stability of a mixed type cubic and quartic functional equation in quasi-Banach spaces, Abstr. Appl. Anal. 2009 (2009), Art. ID 417473, 1-14.
  15. M. Eshaghi Gordji, A. Ebadian, and S. Zolfaghari, Stability of a functional equation deriving from cubic and quartic functions, Abs. Appl. Anal. 2008 (2008), Article ID 801904, 17 pages.
  16. M. Eshaghi Gordji, M. B. Ghaemi, S. Kaboli Gharetapeh, S. Shams, and A. Ebadian, On the stability of J*-derivations, J. Geom. Phys. 60 (2010), no. 3, 454-459. https://doi.org/10.1016/j.geomphys.2009.11.004
  17. M. Eshaghi Gordji, M. B. Ghaemi, and H. Majani, Generalized Hyers-Ulam-Rassias Theorem in Menger Probabilistic Normed Spaces, Discrete Dyn. Nat. Soc. 2010 (2010), Art. ID 162371, 11 pp.
  18. M. Eshaghi Gordji, M. B. Ghaemi, H. Majani, and C. Park, Generalized Ulam-Hyers stability of Jensen functional equation in Sherstnev PN spaces, J. Inequal. Appl. 2010 (2010), Art. ID 868193, 14 pp.
  19. M. Eshaghi Gordji and N. Ghobadipour, Stability of ($\alpha,\beta,\gamma$)-derivations on Lie C*-algebras, Int. J. Geom. Methods Mod. Phys. 7 (2010), no. 7, 1093-1102. https://doi.org/10.1142/S0219887810004737
  20. M. Eshaghi Gordji, S. Kaboli Gharetapeh, T. Karimi, E. Rashidi, and M. Aghaei, Ternary Jordan *-derivations in C*-ternary algebras, J. Comput. Anal. Appl. 12 (2010), no. 2, 463-470.
  21. M. Eshaghi Gordji, S. Kaboli-Gharetapeh, C. Park, and S. Zolfaghri, Stability of an additive-cubic-quartic functional equation, Adv. Difference Equ. 2009 (2009), Article ID 395693, 20 pages.
  22. M. Eshaghi Gordji, S. Kaboli Gharetapeh, J. M. Rassias, and S. Zolfaghari, Solution and stability of a mixed type additive, quadratic and cubic functional equation, Adv. Difference Equ. 2009 (2009), Article ID 826130, 17 pages.
  23. M. Eshaghi Gordji, T. Karimi, and S. Kaboli Gharetapeh, Approximately n-Jordan homomorphisms on Banach algebras, J. Inequal. Appl. 2009 (2009), Article ID 870843, 8 pages.
  24. M. Eshaghi Gordji and H. Khodaei, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, Nonlinear Anal. 71 (2009), no. 11, 5629-5643. https://doi.org/10.1016/j.na.2009.04.052
  25. M. Eshaghi Gordji and H. Khodaei, On the generalized Hyers-Ulam-Rassias stability of quadratic functional equa- tions, Abstr. Appl. Anal. 2009 (2009), Article ID 923476, 11 pages.
  26. M. Eshaghi Gordji and H. Khodaei, The fixed point method for fuzzy approximation of a functional equation associated with inner product spaces, Discrete Dyn. Nat. Soc. 2010 (2010), Article ID 140767, 15 pages, doi:10.1155/2010/140767.
  27. M. Eshaghi Gordji, H. Khodaei, and R. Khodabakhsh, General quartic-cubic-quadratic functional equation in non-Archimedean normed spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 3, 69-84.
  28. M. Eshaghi Gordji and A. Najati, Approximately J*-homomorphisms: A fixed point approach, J. Geom. Phys. 60 (2010), no. 5, 809-814. https://doi.org/10.1016/j.geomphys.2010.01.012
  29. M. Eshaghi Gordji, J. M. Rassias, and N. Ghobadipour, Generalized Hyers-Ulam sta- bility of the generalized (n; k)-derivations, Abstr. Appl. Anal. 2009 (2009), Article ID 437931, 8 pages.
  30. M. Eshaghi Gordji and M. B. Savadkouhi, Approximation of generalized homomorphisms in quasi-Banach algebras, Analele Univ. Ovidius Constata, Math series 17 (2009), no. 2, 203-214.
  31. M. Eshaghi Gordji and M. B. Savadkouhi, Stability of cubic and quartic functional equations in non-Archimedean spaces, Acta Appl. Math. 110 (2010), no. 3, 1321-1329. https://doi.org/10.1007/s10440-009-9512-7
  32. M. Eshaghi Gordji and M. B. Savadkouhi, Stability of mixed type cubic and quartic functional equations in random normed spaces, J. Inequal. Appl. 2009 (2009), Article ID 527462, 9 pages.
  33. M. Eshaghi Gordji and M. B. Savadkouhi, Stability of a mixed type cubic-quartic functional equation in non-Archimedean spaces, Appl. Math. Lett. 23 (2010), no. 10, 1198-1202. https://doi.org/10.1016/j.aml.2010.05.011
  34. M. Eshaghi Gordji, M. B. Savadkouhim, and M. Bidkham, Stability of a mixed type additive and quadratic functional equation in non-Archimedean spaces, J. Comput. Anal. Appl. 12 (2010), no. 2, 454-462.
  35. M. Eshaghi Gordji, M. B. Savadkouhi, and C. Park, Quadratic-quartic functional equa- tions in RN-spaces, J. Inequal. Appl. 2009 (2009), Article ID 868423, 14 pages.
  36. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434. https://doi.org/10.1155/S016117129100056X
  37. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately ad- ditive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. https://doi.org/10.1006/jmaa.1994.1211
  38. P. Gavruta and L. Gavruta, A new method for the generalized Hyers-Ulam-Rassias stability, Int. J. Nonlinear Anal. Appl. 1 (2010), no. 2, 11-18.
  39. N. Ghobadipour, M. Eshaghi gordji, A. Ebadian, and Rassias, A perturbation of double derivations on Banach algebras, Commun. Math. Anal. 11 (2011), no. 1, 51-60.
  40. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  41. D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
  42. K. W. Jun and H. M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl. 274 (2002), 267-278.
  43. K. W. Jun, H. M. Kim, and I. S. Chang, On the Hyers-Ulam stability of an Euler- Lagrange type cubic functional equation, J. Comput. Anal. Appl. 7 (2005), no. 1, 21-33.
  44. S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press Inc., Palm Harbor, Florida, 2001.
  45. S.-M. Jung, Hyers-Ulam-Rassias stability of Jensen's equation and its application, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3137-3143. https://doi.org/10.1090/S0002-9939-98-04680-2
  46. S.-M. Jung, Stability of the quadratic equation of Pexider type, Abh. Math. Sem. Univ. Hamburg 70 (2000), 175-190. https://doi.org/10.1007/BF02940912
  47. P. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), no. 3-4, 368-372. https://doi.org/10.1007/BF03322841
  48. H. Khodaei and M. Kamyar, Fuzzy approximately additive mappings, Int. J. Nonlinear Anal. Appl. 1 (2010), no. 2, 44-53.
  49. H. Khodaei and Th. M. Rassias, Approximately generalized additive functions in several variables, Int. J. Nonlinear Anal. Appl. 1 (2010), 22-41.
  50. S. H. Lee, S. M. Im, and I. S. Hawng, Quartic functional equation, J. Math. Anal. Appl. 307 (2005), no. 2, 387-394. https://doi.org/10.1016/j.jmaa.2004.12.062
  51. A. K. Mirmostafaee, Stability of Quartic Mappings in Non-Archimedean Normed Spaces, Kyungpook Math. J. 49 (2009), 289-297. https://doi.org/10.5666/KMJ.2009.49.2.289
  52. A. Najati, On the stability of a quartic functional equation, J. Math. Anal. Appl. 340 (2008), no. 1, 569-574. https://doi.org/10.1016/j.jmaa.2007.08.048
  53. A. Najati, Hyers-Ulam-Rassias stability of a cubic functional equation, Bull. Korean Math. Soc. 44 (2007), no. 4, 825-840. https://doi.org/10.4134/BKMS.2007.44.4.825
  54. A. Najati and F. Moradlou, Stability of an Euler-Lagrange type cubic functional equa- tion, Turkish J. Math. 33 (2009), no. 1, 65-73.
  55. A. Najati and F. Moradlou, Stability of a mixed additive, quadratic and cubic functional equation in quasi- Banach spaces, Aust. J. Math. Anal. Appl. 5 (2008), no. 2, 10-21.
  56. A. Najati and G. Zamani Eskandani, Stability of a mixed additive and cubic functional equation in quasi-Banach spaces, J. Math. Anal. Appl. 342 (2008), no. 2, 1318-1331. https://doi.org/10.1016/j.jmaa.2007.12.039
  57. A. Najati and C. Park, On the stability of a cubic functional equation, Acta Math. Sin. (Engl. Ser.) 24 (2008) no. 12, 1953-1964. https://doi.org/10.1007/s10114-008-6560-2
  58. C. Park, On an approximate automorphism on a C*-algebra, Proc. Amer. Math. Soc. 132 (2004), no. 6, 1739-1745. https://doi.org/10.1090/S0002-9939-03-07252-6
  59. C. Park, On the stability of the orthogonally quartic functional equation, Bull. Iranian Math. Soc. 31 (2005), no. 1, 63-70.
  60. C. Park and M. E. Gordji, Comment on "Approximate ternary Jordan derivations on Banach ternary algebras" [Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)] [MR2513974], J. Math. Phys. 51 (2010), no. 4, 044102, 7 pp.
  61. C. Park and A. Najati, Generalized additive functional inequalities in Banach algebras, Int. J. Nonlinear Anal. Appl. 1 (2010), no. 2, 54-62.
  62. C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in C*- algebras: A Fixed Point Approach, Abstr. Appl. Anal. 2009 (2009), Art. ID 360432, 1-17, Doi:10.1155/2009/360432.
  63. C. Park and J. M. Rassias, Isomorphisms in unital C*-algebras, Int. J. Nonlinear Anal. Appl. 1 (2010), no. 2, 1-10.
  64. J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal. 46 (1982), no. 1, 126-130. https://doi.org/10.1016/0022-1236(82)90048-9
  65. J. M. Rassias, On Approximation of approximately linear mappings by linear mappings, Bull. Sci. Math. (2) 108 (1984), no. 4, 445-446.
  66. J. M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discuss. Math. 7 (1985), 193-196.
  67. J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory 57 (1989), no. 3, 268-273. https://doi.org/10.1016/0021-9045(89)90041-5
  68. J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese J. Math. 20 (1992), no. 2, 185-190.
  69. J. M. Rassias, Solution of a stability problem of Ulam, Discuss. Math. 12 (1992), 95-103.
  70. J. M. Rassias, On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces, J. Math. Phys. Sci. 28 (1994), no. 5, 231-235.
  71. J. M. Rassias, On the stability of a multi-dimensional Cauchy type functional equation, Ge- ometry, analysis and mechanics, 365-376, World Sci. Publ., River Edge, NJ, 1994.
  72. J. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
  73. J. M. Rassias, New characterization of inner product spaces, Bull. Sci. Math. 108 (1984), no. 1, 95-99.
  74. Th. M. Rassias, P. Semrl, On the behaviour 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
  75. K. Ravi, M. Arunkumar, and J. M. Rassias, Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Stat. 3 (2008), 36-46.
  76. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Elsevier North Holand, New York, 1983.
  77. A. N. Serstnev, On the motion of a random normed space, Dokl. Akad. Nauk SSSR 149 (1963), 280-283; English translation in Soviet Math. Dokl. 4 (1963), 388-390.
  78. S. Shakeri, R. Saadati, and C. Park, Stability of the quadratic functional equation in non-Archimedean L-fuzzy normed spaces, Int. J. Nonlinear Anal. Appl. 1 (2010), no. 2, 72-83.
  79. F. Skof, Local properties and approximation of operators, Geometry of Banach spaces and related topics (Italian) (Milan, 1983). Rend. Sem. Mat. Fis. Milano 53 (1983), 113- 129 (1986).
  80. S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Editions, Wiley, New York, 1964.

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