FUZZY STABILITY OF A FUNCTIONAL EQUATION RELATED TO INNER PRODUCT SPACES

Abstract

The fuzzy stability problems for the Cauchy quadratic functional equation and the Jensen quadratic functional equation in fuzzy Banach spaces have been investigated by Moslehian et al. Th. M. Rassias introduced the following equality Xm i,j=1 kxi − xjk 2 = 2m Xm i=1 kxik 2 , Xm i=1 xi = 0, for a fixed integer m ≥ 3. By the above equality, we define the following functional equation
(0.1) Xm i,j=1 f(xi − xj ) = 2m Xmi=1 f(xi), Xm i=1 xi = 0. In this paper, we prove the generalized Hyers-Ulam stability of the functional equation (0.1) in fuzzy Banach spaces.

Keywords

• Fuzzy Banach space,Functional equation related to inner product space,Generalized Hyers-Ulam stability,2000 AMS Classification: 46 S 40

FUZZY STABILITY OF A FUNCTIONAL EQUATION RELATED TO INNER PRODUCT SPACES

Abstract

References

1. Aoki, T. On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2, 64–66, 1950.
2. Bag, T. and Samanta, S. K. Finite dimensional fuzzy normed linear spaces, J. Fuzzy Math. 11, 687–705, 2003.
3. Bag, T. and Samanta, S. K. Fuzzy bounded linear operators, Fuzzy Sets and Systems 151, 513–547, 2005.
4. Baktash, E., Cho, Y. J., Jalili, M., Saadati, R. and Vaezpour, S. M. On the stability of cubic mappings and quadratic mappings in random normed spaces, J. Inequal. Appl. 2008, Art. ID 902187, 2008.
5. Cheng, S. C. and Mordeson, J. M. Fuzzy linear operators and fuzzy normed linear spaces, Bull. Calcutta Math. Soc. 86, 429–436, 1994.
6. Cholewa, P. W. Remarks on the stability of functional equations, Aequationes Math. 27, 76–86, 1984.
7. Czerwik, P. Functional Equations and Inequalities in Several Variables (World Scientific Publishing Company, New Jersey, Hong Kong, Singapore and London, 2002).
8. Czerwik, S. On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62, 59–64, 1992.

9. Czerwik, S. The stability of the quadratic functional equation, in: (Stability of mappings of Hyers-Ulam type, Hadronic Press, Palm Harbor, FL, 1994), 81–91.
10. Felbin, C. Finite dimensional fuzzy normed linear spaces, Fuzzy Sets and Systems 48, 239– 248, 1992.
11. Gajda, Z. On stability of additive mappings, Internat. J. Math. Math. Sci. 14, 431–434, 1991.
12. G˘avruta, P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184, 431–436, 1994.
13. Hyers, D. H. On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27, 222–224, 1941.
14. Hyers, D. H., Isac, G. and Rassias, Th. M. Stability of Functional Equations in Several Variables(Birkh¨auser, Basel, 1998).
15. Jung, S. Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis (Hadronic Press lnc., Palm Harbor, Florida, 2001).
16. Katsaras, A. K. Fuzzy topological vector spaces II, Fuzzy Sets and Systems 12, 143–154, 1984.
17. Kramosil, I. and Michalek, J. Fuzzy metric and statistical metric spaces, Kybernetica 11, 326–334, 1975.
18. Krishna, S. V. and Sarma, K. K. M. Separation of fuzzy normed linear spaces, Fuzzy Sets and Systems 63, 207–217, 1994.
19. Mirmostafaee, A. K., Mirzavaziri, M. and Moslehian, M. S. Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems 159, 730–738, 2008.
20. Mirmostafaee, A. K. and Moslehian, M. S. Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems 159, 720–729, 2008.
21. Park, C. On the stability of the linear mapping in Banach modules, J. Math. Anal. Appl. 275, 711–720, 2002.
22. Park, C. Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equa- tions in Banach algebras, Fixed Point Theory and Applications 2007, Art. ID 50175, 2007. [23] Park, C. Generalized Hyers-Ulam-Rassias stability of quadratic functional equations: a fixed point approach, Fixed Point Theory and Applications 2008, Art. ID 493751, 2008.
23. Park, C. and Cui, J. Generalized stability of C*-ternary quadratic mappings, Abstract Appl. Anal. 2007, Art. ID 23282, 2007.
24. Park, C. and Najati, A. Homomorphisms and derivations in C*-algebras, Abstract Appl. Anal. 2007, Art. ID 80630, 2007.
25. Park, C., Cho, Y. and Han, M. Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, Art. ID 41820, 2007.
26. Park, C., Park, W. and Najati, A. Functional equations related to inner product spaces, Abstract Appl. Anal. 2009, Art. ID 907121, 2009.
27. Park, C. Fixed points, inner product spaces and functional equations (preprint).
28. Park, C. and Jang, S. Fuzzy stability of functional equations induced by inner product: a fixed point approach(preprint).
29. Rassias, Th. M. On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72, 297–300, 1978.
30. Rassias, Th. M. On characterizations of inner product spaces and generalizations of the H. Bohr inequality, in Topics in Mathematical Analysis (ed. Th.M. Rassias) (World Scientific Publ. Co., Singapore, 1989), 803–819.
31. Rassias, Th. M. On the stability of the quadratic functional equation and its applications, Studia Univ. Babes-Bolyai XLIII, 89–124, 1998.
32. Rassias, Th. M. The problem of S. M. Ulam for approximately multiplicative mappings, J. Math. Anal. Appl. 246, 352–378, 2000.
33. Rassias, Th. M. On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251, 264–284.
34. Rassias, Th. M. On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62, 23–130, 2000.
35. Rassias, Th. M. Problem 16; 2, Report of the 27thInternational Symp. on Functional Equa- tions, Aequationes Math. 39, 292–293; 309, 1990.
36. Rassias, Th. M. and ˇSemrl, P. On the behaviour of mappings which do not satisfy Hyers- Ulam stability, Proc. Amer. Math. Soc. 114, 989–993, 1992.
37. Rassias, Th. M. and ˇSemrl, P. On the Hyers-Ulam stability of linear mappings, J. Math. Anal. Appl. 173, 325–338, 1993.
38. Rassias, Th. M. and Shibata, K. Variational problem of some quadratic functionals in com- plex analysis, J. Math. Anal. Appl. 228, 234–253, 1998.
39. Skof, F. Propriet`a locali e approssimazione di operatori, Rend. Sem. Mat. Fis. Milano 53, 113–129, 1983.
40. Ulam, S. M. A Collection of the Mathematical Problems (Interscience Publ., New York, 1960).
41. Xiao, J. Z. and Zhu, X. H. Fuzzy normed spaces of operators and its completeness, Fuzzy Sets and Systems 133, 389–399, 2003.