Ulam-Hyers Stability of Additive and Reciprocal Functional Equations: Direct and Fixed Point Methods

Authors

  • M. Arunkumar  Department of Mathematics, Government Arts College, Tiruvannamalai, Tamilnadu, India
  • A. Vijayakumar  Department of Mathematics, R.M.K. Engineering College, Kavarapettai, Tamilnadu, India
  • S. Karthikeyan  Department of Mathematics, R.M.K. Engineering College, Kavarapettai, Tamilnadu, India

Keywords:

Additive functional equation, Reciprocal functional equation, generalized Ulam-Hyers stability, Intuitionistic fuzzy normed spaces, Non - Archimedean Fuzzy normed spaces, fixed point method.

Abstract

In this paper, the authors established the generalized Ulam - Hyers stability of additive functional equation


which is originating from arithmetic mean of n consecutive terms of  an arithmetic progression in Intuitionistic fuzzy normed spaces and  reciprocal functional equation

originating from n-consecutive terms of a harmonic progression in Non - Archimedean Fuzzy  normed spaces using direct and fixed point methods. Applications of the above functional equations are also given.

References

  1. J. Aczel and  J. Dhombres, Functional Equations in Several Variables, Cambridge Univ, Press, 1989.
  2. T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan,    2 (1950), 64-66.  
  3. M. Arunkumar, G. Vijayanandharaj, S. Karthikeyan, Solution and Stability of  a Functional Equation Originating From n- Consecutive Terms of an Arithmetic Progression, Universal Journal of Mathematics and Mathematical Sciences, 2, 2012, 161-171.
  4. M. Arunkumar, S. Karthikeyan, Solution And Stability Of A Reciprocal Functional Equation Originating From n-Consecutive Terms Of A Harmonic Progression: Direct And Fixed Point Methods, International Journal of Information Science and Intelligent system, Vol. 3(1), pp. 151-168, 2014.
  5. K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets Syst. 20, 87 ,1986.  
  6. S.S. Chang, Y. J. Cho, and S. M. Kang, Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publishers, Huntington, NY, USA, 2001. 
  7. S. Czerwik,  Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002.
  8. L. Cadariu, V. Radu, Fixed points and the stability of Jensen's functional equation. J. Inequal. Pure and Appl. Math. 4(1) (2003), Art. 4
  9. L. Cadariu, V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable, Fixed Point Theory and Applications, 2008, Article ID 749392, (2008), 15 pages.
  10. L. Cadariu, V. Radu, A generalized point method for the stability of Cauchy functional equation, In Functional Equations in Mathematical Analysis, Th. M. Rassias, J. Brzdek (Eds.), Series Springer Optimization and Its Applications 52, 2011.  
  11. G. Deschrijver, E.E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems, 23 (2003), 227-235.
  12. Z. Gajda, On stability of additive mappings, International Journal of Mathematics and Mathematical Sciences, vol. 14, no. 3, pp. 431–434, 1991.
  13. T. Gantner, R. Steinlage, R. Warren,  Compactness in fuzzy topological spaces, J. Math. Anal. Appl. 62 (1978) 547-562.  
  14. P.Gavruta,  A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings , J. Math. Anal. Appl., 184 (1994), 431-436.
  15. I. Golet, On Generalized Fuzzy Normed Spaces, International Mathematical Forum, 4, 2009, no. 25, 1237 - 1242.   
  16. D.H. Hyers,  On the stability of the linear functional equation, Proc.Nat. Acad.Sci.,U.S.A.,27 (1941) 222-224.
  17. D.H. Hyers, G. Isac, Th.M. Rassias, Stability of functional equations in several variables,Birkhauser, Basel, 1998.   
  18. U. Hoehle,  Fuzzy real numbers as Dedekind cuts with respect to a multiple-valued logic,  Fuzzy Sets Syst. 24 (1987) 263-278.   
  19. S. B. Hosseini, D. O'Regan, R. Saadati, Some results on intuitionistic fuzzy spaces, Iranian J. Fuzzy Syst, 4 (2007) 53- 64. 
  20. S.M. Jung,  Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
  21. S.M. Jung, A fixed point approach to the stability of the equation . The Australian J. Math. Anal. Appl. 6(1) (2009), 1-6.
  22. Pl. Kannappan,  Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, 2009.
  23. O. Kaleva, S. Seikkala,  On fuzzy metric spaces, Fuzzy Sets Syst. 12 (1984),    215-229.
  24. O. Kaleva , The completion of fuzzy metric spaces, J. Math. Anal. Appl. 109 (1985),       194-198. 
  25. O. Kaleva , A comment on the completion of fuzzy metric spaces, Fuzzy Sets Syst. 159(16) (2008), 2190-2192.
  26. H.A. Kenary, S.Y.Jang, C. Park,   Fixed point approach to the Hyers-Ulam stability of a functional equation in various normed spaces, Fixed Point Theory and Applications, doi:10.1186/1687-1812-2011-67.
  27. R. Lowen,  Fuzzy Set Theory, (Ch. 5 : Fuzzy real numbers), Kluwer, Dordrecht, 1996.
  28. B. Margoils, J.B. Diaz , A fixed point theorem of the alternative for contractions on a generalized complete metric space,  Bull.Amer. Math. Soc. 126 74 (1968), 305-309.
  29. A.K. Mirmostafaee, M.S. Moslehian,   Fuzzy version of Hyers-Ulam-Rassias theorem, Fuzzy sets Syst. 159(6) 2008, 720-729.
  30. J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22 (2004),      1039-1046.
  31. V. Radu, The Fixed point alternative and the stability of functional equations, Fixed Point Theory, 4, (2003), No.1, 91-96.
  32. J.M. Rassias, On approximately of approximately linear mappings by linear mappings, J. Funct. Anal. USA, 46, (1982) 126-130.
  33. J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bulletin des Sciences Math´ematiques, vol. 108, no. 4, pp. 445–446, 1984.
  34. J. M. Rassias, On a new approximation of approximately linear mappings by linear mappings, Discussiones Mathematicae, vol. 7, pp. 193–196, 1985.
  35. J. M. Rassias, Solution of a problem of Ulam, Journal of Approximation Theory, vol. 57, no. 3, pp. 268–273, 1989.
  36. J. M. Rassias, On the stability of the Euler-Lagrange functional equation, Chinese Journal of Mathematics, vol. 20, no. 2, pp. 185–190, 1992.
  37. Th.M. Rassias,  On the stability of the linear mapping in Banach spaces, Proc.Amer.Math. Soc., 72 (1978), 297-300.
  38. Th.M. Rassias,  Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan London, 2003.
  39. K. Ravi, M. Arunkumar, J.M. Rassias,  On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematical Sciences, Autumn 2008 Vol.3, No. 08, 36-47.
  40. K. Ravi, B.V. Senthil Kumar, Ulam-Gavruta-Rassias stability of Rassias  Reciprocal functional equation, Global J. of Appl. Math. and Math. Sci., 2008.
  41. K. Ravi, B.V. Senthil Kumar, Ulam stability of Generalized Reciprocal funtional equation in several variables, Int. J. App. Math. and Stat., 19 (D10) (2010),1-19.
  42. S.E. Rodabaugh,  Fuzzy addition in the L-fuzzy real line, Fuzzy Sets Syst. 8 (1982) 39-51.
  43. R. Saadati, J.H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons and Fractals, 27 (2006), 331-344.
  44. R. Saadati, J.H. Park, Intuitionstic fuzzy Euclidean normed spaces, Commun. Math. Anal., 1 (2006), 85-90.
  45. I. Sadeqi, M. Salehi,  Fuzzy compacts operators and topological degree theory, Fuzzy Sets Syst. 160(9) (2009), 1277-1285.
  46. S. Shakeri, Intuitionstic fuzzy stability of Jensen type mapping, J. Nonlinear Sci. Appli. Vol.2 No. 2 (2009), 105-112.
  47. S. M. Ulam,  Problems in Modern Mathematics, Science Editions, Wiley, NewYork, 1964.
  48. J. Xiao, X. Zhu,  On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst, 125 (2002), 153-161.
  49. J. Xiao, X. Zhu,  Topological degree theory and fixed point theorems in fuzzy normed space, Fuzzy Sets Syst, 147 (2004), 437-452.
  50.  Zhou .Ding -Xuan, On a conjecture of Z. Ditzian, J. Approx. Theory, 69 (1992),    167-172.

International Journal of Scientific Research in Science, Engineering and Technology

Downloads

Published

2015-08-25

Issue

Volume 1, Issue 4, July-August-2015

Section

Research Articles

License

Copyright (c) IJSRSET

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

[1]
M. Arunkumar, A. Vijayakumar, S. Karthikeyan, " Ulam-Hyers Stability of Additive and Reciprocal Functional Equations: Direct and Fixed Point Methods, International Journal of Scientific Research in Science, Engineering and Technology(IJSRSET), Print ISSN : 2395-1990, Online ISSN : 2394-4099, Volume 1, Issue 4, pp.59-77, July-August-2015.