# A NOTE ON THE HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC EQUATION

• Published : 2004.08.01
• 58 6

#### Abstract

In this paper we prove the Hyers-Ulam-Rassias stability by considering the cases that the approximate remainder ${\varphi}$ is defined by (x * y) ＋ (x * $y^{-1}$) - 2 (x) - 2 (y) =<${\varphi}$(x,y), (x*y*z)＋ (x)＋ (y)＋ (z)－ (x*y)－ (y*z)－ (z*x)＝${\varphi}$(x, y, z), where (G,*) is a group, X is a real or complex Hausdorff topological vector space, and is a function from G into X.

#### References

1. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), 76–86
2. S. Czerwik, On the stability of the quadratic mapping in the normed space, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64
3. Z. Gajda, On the stability of additive mappings, Internat. J. Math. & Math. Sci. 14 (1991), 431–434
4. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431–436
5. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222–224
6. D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequations Math. 44 (1992), 125–153
7. D. H. Hyers, G. Isac and Th. M. Rassias, On the asymptoticity aspect of Hyers- Ulam stability of mappings, Proc. Amer. Math. Soc. 126 (1998), 425–430
8. D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, $Birkh{\"{a}}User$, 1998
9. G. Isac and Th. M. Rassias, On the Hyers-Ulam stability of ${\psi}$-additive mappings, J. Approx. Theory 72 (1993), 131–137
10. G. Isac and Th. M. Rassias, Functional inequalities for approximately additive mappings, stability of mappings of Hyers-Ulam type, Hadronic Press, Inc., Florida (1994), 117–125
11. G. Isac and Th. M. Rassias, Stability of ${\psi}$-additive mappings: Applications to nonlinear analysis, Internat. J. Math. & Math. Sci. 19 (1996), no. 2, 219–228
12. W. Jian, Some Further Generalization of the Hyers-Ulam-Rassias Stability of functional equation, J. Math. Anal. Appl. 263 (2001), 406-423
13. K. -W. Jun and Y. H. Lee, On the Hyers-Ulam-Rassias stability of a Pexiderized quadratic inequality, Math. Inequal. Appl. 4 (2001), 93-118
14. K. -W. Jun, D. -S. Shin and B. -D. Kim, On the Hyers-Ulam-Rassias stability of the Pexider equation, J. Math. Anal. Appl. 239 (1999), 20–29
15. S.-M. Jung, On the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 204 (1996), 221–226
16. Y. H. Lee and K. W. Jun, A Generalization of the Hyers-Ulam-Rassias Stability of Jensen’s Equation, J. Math. Anal. Appl. 238 (1999), 305–315
17. Y. H. Lee and K. W. Jun, A Generalization of the Hyers-Ulam-Rassias Stability of Pexider Equation, J. Math. Anal. Appl. 246 (2000), 627–638
18. Y. H. Lee and K. W. Jun, A Note on the Hyers-Ulam-Rassias Stability of Pexider Equation, J. Korean Math. Soc. 37 (2000), 111–124
19. Y. H. Lee and K. W. Jun, On the Stability of Approximately Additive Mappings, Proc. Amer. Math. Soc. 128 (2000), 1361–1369
20. Th. M. Rassias, On a modified Hyers-Ulam sequence, J. Math. Anal. Appl. 158 (1991), 106–113
21. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300
22. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., to appear
23. Th. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl., to appear
24. Th. M. Rassias, On the stability of functional equations originated by a problem of Ulam, Stud. Univ. Babes-Bolyai Inform., to appear
25. Th. M. Rassias, On the stability of the quadratic functional equation, Mathematica, to appear
26. Th. M. Rassias, On the stability of the quadratic functional equation and its applications, Stud. Univ. Babes-Bolyai Inform., to appear
27. Th. M. Rassias, Report of the 27th Internat. Symposium on Functional Equations, Aequationes Math. 39 (1990), 292–293. Problem 16, $2^{\circ}$, Same report, p. 309
28. Th. M. Rassias and P. Semrl, On the behavior of mappings which does not satisfy Hyers-Ulam stability, Proc. Amer. Math. Soc. 114 (1992), 989–993
29. Th. M. Rassias and J. Tabor, What is left of Hyers-Ulam stability?, J. Nat. Geom. 1 (1992), 65–69
30. F. Skof, Local properties and approximations of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129
31. S. M. Ulam, Problems in Modern Mathematics, Chap. VI, Science eds., Wiley, New York, 1960

#### Cited by

1. Elementary remarks on Ulam–Hyers stability of linear functional equations vol.328, pp.1, 2007, https://doi.org/10.1016/j.jmaa.2006.04.079