# HYPERSTABILITY OF THE GENERAL LINEAR FUNCTIONAL EQUATION

• Published : 2015.11.30

#### Abstract

We give some results on hyperstability for the general linear equation. Namely, we show that a function satisfying the linear equation approximately (in some sense) must be actually the solution of it.

#### References

1. A. Bahyrycz and M. Piszczek, Hyperstability of the Jensen functional equation, Acta Math. Hungar. 142 (2014), no. 2, 353-365. https://doi.org/10.1007/s10474-013-0347-3
2. J. Brzdek, Hyperstability of the Cauchy equation on restricted domains, Acta Math. Hungar. 41, (2013), no. 1-2, 58-67.
3. J. Brzdek, J. Chudziak, and Zs. Pales, A fixed point approach to stability of functional equations, Nonlinear Anal. 74 (2011), no. 17, 6728-6732. https://doi.org/10.1016/j.na.2011.06.052
4. J. Brzdek and K. Cieplinski, Hyperstability and Superstability, Abstr. Appl. Anal. 2013 (2013), Art. ID 401756, 13 pp.
5. J. Brzdek and A. Pietrzyk, A note on stability of the general linear equation, Aequationes Math. 75 (2008), no. 3, 267-270. https://doi.org/10.1007/s00010-007-2894-6
6. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately ad- ditive mapping, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436. https://doi.org/10.1006/jmaa.1994.1211
7. P. Gavruta, An answer to a question of John M. Rassias concerning the stability of Cauchy equation, Advances in Equations and Inequalities, Hadronic Math. Ser. 1999.
8. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston-Basel-Berlin, 1998.
9. S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
10. M. Kuczma, An introduction to the Theory of Functional Equation and Inequalities, PWN, Warszawa-Krakow-Katowice, 1985.
11. P. Nakmahachalasint, Hyers-Ulam-Rassias stability and Ulam-Gavruta-Rassias stabili- ties of additive functional equation in several variables, Int. J. Math. Sci. 2007 (2007), Art. ID 13437, 6pp.
12. M. Piszczek, Remark on hyperstability of the general linear equation, Aequationes Math. 88 (2014), no. 1-2, 163-168. https://doi.org/10.1007/s00010-013-0214-x
13. D. Popa, Hyers-Ulam-Rassias stability of the general linear equation, Nonlinear Funct. Anal. Appl. 7 (2002), no. 4, 581-588.
14. D. Popa, On the stability of the general linear equation, Results Math. 53 (2009), no. 3-4, 383-389. https://doi.org/10.1007/s00025-008-0349-6
15. J. M. Rassias, On approximation of approximately linear mappings, J. Funct. Anal. 46 (1982), no. 1, 126-130. https://doi.org/10.1016/0022-1236(82)90048-9
16. 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
17. K. Ravi and M. Arunkumar, On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation, Int. J. Appl. Math. Stat. 7 (2007), 143-156.
18. K. Ravi and B. V. Senthil Kumar, Ulam-Gavruta-Rassias stability of Rassias Reciprocal functional equation, Glob. J. App. Math. Sci. 3 (2010), 57-79.
19. M. Ait Sibaha, B. Bouikhalene, and E. Elqorachi, Ulam-Gavruta-Rassias stability of a linear functional equation, Int. J. Appl. Math. Stat. 7 (2007), 157-166.

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