DOI QR코드

DOI QR Code

THE GENERAL SOLUTION AND APPROXIMATIONS OF A DECIC TYPE FUNCTIONAL EQUATION IN VARIOUS NORMED SPACES

  • Arunkumar, Mohan (Department of Mathematics Government Arts College) ;
  • Bodaghi, Abasalt (Department of Mathematics Garmsar Branch Islamic Azad University) ;
  • Rassias, John Michael (Pedagogical Department E.E. Section of Mathematics and Informatics National and Capodistrian University of Athens) ;
  • Sathya, Elumalai (Department of Mathematics Government Arts College)
  • Received : 2016.01.08
  • Accepted : 2016.05.09
  • Published : 2016.05.15

Abstract

In the current work, we define and find the general solution of the decic functional equation g(x + 5y) - 10g(x + 4y) + 45g(x + 3y) - 120g(x + 2y) + 210g(x + y) - 252g(x) + 210g(x - y) - 120g(x - 2y) + 45g(x - 3y) - 10g(x - 4y) + g(x - 5y) = 10!g(y) where 10! = 3628800. We also investigate and establish the generalized Ulam-Hyers stability of this functional equation in Banach spaces, generalized 2-normed spaces and random normed spaces by using direct and fixed point methods.

Keywords

References

  1. M. Acikgoz, A review on 2-normed structures, Int. J. Math. Anal. 1 (2007), no. 4, 187-191.
  2. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ, Press, 1989.
  3. C. Alsina, On the stability of a functional equation arising in probabilistic normed spaces, General Inequal., Oberwolfach 5, 263-271 (1986). Birkhuser, Basel (1987).
  4. 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
  5. M. Arunkumar, M. J. Rassias, and Y. Zhang, Ulam-Hyers stability of a 2-variable AC-mixed type functional equation: direct and fixed point methods, J. Mod. Math. Front. 1 (2012), no. 3, 10-26.
  6. M. Arunkumar and S. Karthikeyan, Solution and Intuitionistic Fuzzy stability of n-dimensional quadratic functional equation: Direct and Fixed Point Methods, Int. J. Adv. Math. Sci. 2 (2014), no. 1, 21-33.
  7. M. Arunkumar and T. Namachivayam, Stability of a n-dimensional additive functional equation in random normed space, Int. J. Math. Anal. 4 (2012), no. 2, 179-186.
  8. J. Baker, A general functional equation and its stability, Proc. Amer. Math. Soc. 133 (2005), no. 6, 1657-1664. https://doi.org/10.1090/S0002-9939-05-07841-X
  9. J. A. Baker, The stability of certain functional equations, Proc. Amer. Math. Soc. 112 (1991), 729-732. https://doi.org/10.1090/S0002-9939-1991-1052568-7
  10. A. Bodaghi, Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations, J. Intel. Fuzzy Syst. 30 (2016), 2309-2317. https://doi.org/10.3233/IFS-152001
  11. A. Bodaghi, Stability of a mixed type additive and quartic functional equation, Filomat. 28 (2014), no. 8, 1629-1640. https://doi.org/10.2298/FIL1408629B
  12. A. Bodaghi, Stability of a quartic functional equation, The Scientific World Journal. 2014, Art. ID 752146, 9 pages, doi:10.1155/2014/752146.
  13. A. Bodaghi, I. A. Alias, and M. Eshaghi Gordji, On the stability of quadratic double centralizers and quadratic multipliers: A fixed point approach, J. Inequal. Appl. 2012, Article ID 957541, 9 pages.
  14. A. Bodaghi, I. A. Alias, and M. H. Ghahramani, Ulam stability of a quartic functional equation, Abs. Appl. Anal. 2012, Art. ID 232630 (2012).
  15. A. Bodaghi, I. A. Alias, and M. H. Ghahramani, Approximately cubic functional equations and cubic multipliers, J. Inequal. Appl. 2011 (2011): 53. https://doi.org/10.1186/1029-242X-2011-53
  16. A. Bodaghi, S. M. Moosavi, and H. Rahimi, The generalized cubic functional equation and the stability of cubic Jordan *-derivations, Ann. Univ. Ferrara. 59 (2013), 235-250. https://doi.org/10.1007/s11565-013-0185-9
  17. A. Bodaghi, C. Park, and J. M. Rassias, Fundamental stabilities of the nonic functional equation in intuitionistic fuzzy normed spaces, Commun. Korean Math. Soc. To appear.
  18. D. G. Bourgin, Classes of transformations and bordering transformations, Bull. Amer. Math. Soc. 57 (1951), 223-237. https://doi.org/10.1090/S0002-9904-1951-09511-7
  19. L. Cadariu and V. Radu, Fixed points and the stability of quadratic functional equations,An. Univ. Timisoara, Ser. Mat. Inform. 41 (2003), 25-48.
  20. L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: A fixed point approach, Grazer Math. Ber. 346 (2004), 43-52.
  21. Y. J. Cho, M. E. Gordji, and S. Zolfaghari, Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces, J. Ineq. Appl. doi:10.1155/2010/403101
  22. P. W. Cholewa, Remarks on the stability of functional equations, Aequ. Math. 27 (1984), 76-86. https://doi.org/10.1007/BF02192660
  23. S. Czerwik, On the stability of the quadratic mappings in normed spaces, Abh. Math. Sem. Univ Hamburg. 62 (1992), 59-64. https://doi.org/10.1007/BF02941618
  24. G. L. Forti, Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations, J. Math. Anal. Appl. 295 (2004), 127-133. https://doi.org/10.1016/j.jmaa.2004.03.011
  25. Z. Gajda, On the stability of additive mappings, Inter. J. Math. Math. Sci. 14 (1991), 431-434. https://doi.org/10.1155/S016117129100056X
  26. P. G.avruta, 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
  27. M. E. Gordji, A. Bodaghi, and C. Park, A fixed point approach to the stability of double Jordan centralizers and Jordan multipliers on Banach algebras, U.P.B. Sci. Bull., Series A. 73, Iss. 2 (2011), 65-73.
  28. M. E. Gordji and M. B. Savadkouhi, Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces, J. Ineq. Appl. doi:10.1155/2009/527462.
  29. O. Hadzic and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 2001.
  30. O. Hadzic, E. Pap, and M. Budincevic, Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces, Kybernetika. 38 (2002), no. 3, 363-382.
  31. 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
  32. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of functional equations in several variables,Birkhauser, Basel, 1998.
  33. P. Kannappan, Quadratic functional equation and inner product spaces, Results Math. 27 (1995), 368-372. https://doi.org/10.1007/BF03322841
  34. O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets. Sys. 12 (1984), 215-229. https://doi.org/10.1016/0165-0114(84)90069-1
  35. L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions-a question of priority, Aequ. Math. 75 (2008), 289-296. https://doi.org/10.1007/s00010-007-2892-8
  36. B. Margolis and J. B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126 (1968), 305-309.
  37. D. Mihet and V. Radu, On the stability of the additive Cauchy functional equa- tion in random normed spaces, J. Math. Anal. Appl. 343 (2008), 567-572 . https://doi.org/10.1016/j.jmaa.2008.01.100
  38. D. Mihet, The probabilistic stability for a functional equation in a single variable, Acta Math. Hungar. 123, 249256 (2009), doi:10.1007/s10474-008-8101-y.
  39. D. Mihet, R. Saadati, and S. M. Vaezpour, The stability of the quartic functional equation in random normed spaces, Acta Appl. Math. 110 (2010), 797-803. https://doi.org/10.1007/s10440-009-9476-7
  40. D. Mihet, R. Saadati, and S. M. Vaezpour, The stability of an additive functional equation in Menger probabilistic normed spaces Math. Slovaca. 61 (2011), 817-826.
  41. J. H. Park, Intuitionistic fuzzy metric spaces, Chaos. Sol. Frac. 22 (2004), 10391046. https://doi.org/10.1016/j.chaos.2004.02.051
  42. J. Matina, Rassias, M. Arunkumar, and S. Ramamoorthi, Stability of the Leibniz additive-quadratic functional equation in quasi- $\beta$ normed spaces: direct and fixed point methods, J. Con. Appl. Math. 14 (2014), 22-46.
  43. J. M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA. 46 (1982), 126-130. https://doi.org/10.1016/0022-1236(82)90048-9
  44. J. M. Rassias, Solution of the Ulam problem for cubic mappings, An. Univ. Timisoara Ser. Mat. Inform. 38 (2000), 121-132.
  45. J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Math. 34(54) (1999), no. 2, 243-252.
  46. J. M. Rassias and M. Eslamian, Fixed points and stability of nonic functional equation in quasi-$\beta$-normed spaces, Cont. Anal. Appl. Math. 3 (2015), no. 2, 293-309.
  47. J. M. Rassias and H. M. Kim, Generalized Hyers-Ulam stability for general additive functional equations in quasi-$\beta$-normed spaces, J. Math. Anal. Appl. 356 (2009), no. 1, 302-309. https://doi.org/10.1016/j.jmaa.2009.03.005
  48. J. M. Rassias, E. Son, and H. M. Kim, On the Hyers-Ulam stability of 3D and 4D mixed type mappings, Far East J. Math. Sci. 48 (2011), no. 1, 83-102.
  49. J. M. Rassias, M. Arunkumar, E. Sathya, and N. M. Kumar, Solution And Stability Of A ACQ Functional Equation In Generalized 2-Normed Spaces, Intern. J. Fuzzy Math. Arch. 7 (2015), no. 2, 213-224.
  50. J. M. Rassias, M. Arunkumar, and T. Namachivayam, Stability Of The Leibniz Additive-Quadratic Functional Equation In Felbin's And Random Normed Spaces: A Direct Method, J. Acad. Res. J. Inter. (2015), 102-110.
  51. J. M. Rassias, M. Arunkumar, E. Sathya, and T. Namachivayam, Various Generalized Ulam-Hyers Stabilities of a Nonic Functional Equation, Tbilisi Math. J. (Submitted).
  52. 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
  53. Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), no. 1, 106-113. https://doi.org/10.1016/0022-247X(91)90270-A
  54. T. M. Rassias and P. Semrl, On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), no. 4, 989-993. https://doi.org/10.1090/S0002-9939-1992-1059634-1
  55. Th. M. Rassias, The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246 (2000), 352-378. https://doi.org/10.1006/jmaa.2000.6788
  56. Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan London, 2003.
  57. K. Ravi, M. Arunkumar, and J.M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, Int. J. Math. Sci. 3 (2008), no. 8, 36-47.
  58. K. Ravi, J. M. Rassias, M. Arunkumar, and R. Kodandan, Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, J. Inequal. Pure Appl. Math. 10 (2009), no. 4, Art. 114, 29 pages.
  59. R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos. Sol. Frac. 27 (2006), 3313-3344.
  60. R. Saadati, S. M. Vaezpour, and Y. Cho, A note on the "On the stability of cubic mappings and quadratic mappings in random normed spaces J. Inequal. Appl. 2009, Art. ID 214530 (2009).
  61. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics, North-Holland Publishing, New York, NY, USA, 1983.
  62. A. N. Sherstnev, On the notion of a random normed space, Doklady Akademii Nauk SSSR. 149 (1963), 280-283 (Russian).
  63. F. Skof, Proprieta locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano. 53 (1983), 113-129. https://doi.org/10.1007/BF02924890
  64. S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964.
  65. T. Z. Xu, J. M. Rassias, and W. X. Xu, Generalized Ulam-Hyers stability of a general mixed AQCQ-functional equation in multi-Banach spaces: a fixed point approach, Eur. J. Pure Appl. Math. 3 (2010), 1032-1047.
  66. T. Z. Xu, J. M. Rassias, M. J. Rassias, and W. X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-$\beta$-normed spaces, J. Inequal. Appl. 2010, Art. ID 423231, 23 page.
  67. T. Z. Xu, J. M. Rassias, and W. X. Xu, A fixed point approach to the stability of a general mixed AQCQ-functional equation in non-Archimedean normed spaces, Discrete Dyn. Nat. Soc. 2010, Art. ID 812545, 24 pages.
  68. T. Z. Xu and J. M. Rassias, Approximate Septic and Octic mappings in quasi-$\beta$-normed spaces, J. Comput. Anal. Appl. 15 (2013), no. 6, 1110-1119.
  69. S. Y. Yang, A. Bodaghi, and K. A. M. Atan, Approximate cubic *-derivations on Banach *-algebras, Abst. Appl. Anal. 2012, Article ID 684179, 12 pages, doi:10.1155/2012/684179.

Cited by

  1. GENERAL SOLUTION AND ULAM-HYERS STABILITY OF VIGINTI FUNCTIONAL EQUATIONS IN MULTI-BANACH SPACES vol.31, pp.2, 2018, https://doi.org/10.14403/jcms.2018.31.1.199
  2. DUOTRIGINTIC FUNCTIONAL EQUATION AND ITS STABILITY IN BANACH SPACES vol.28, pp.3, 2016, https://doi.org/10.11568/kjm.2020.28.3.525
  3. Hyers-Ulam stability quintic functional equation in F-spaces: direct method vol.13, pp.4, 2016, https://doi.org/10.32513/tbilisi/1608606046
  4. On Ulam Stability of Functional Equations in 2-Normed Spaces-A Survey vol.13, pp.11, 2016, https://doi.org/10.3390/sym13112200