ON A COMPOSITE FUNCTIONAL EQUATION RELATED TO THE GOLAB-SCHINZEL EQUATION

• Gordji, Madjid Eshaghi (Department of Mathematics, Faculty of Sciences, South Tehran Branch, Islamic Azad University) ;
• Rassias, Themistocles M. (Department of Mathematics, National Technical University of Athens, Zofrafou Campus) ;
• Tial, Mohamed (Department of Mathematics, Faculty of Sciences, IBN Tofail University) ;
• Zeglami, Driss (Department of Mathematics, E.N.S.A.M, Moulay Ismail University)
• Published : 2016.03.31
• 157 8

Abstract

Let X be a vector space over a field K of real or complex numbers and $k{\in}{\mathbb{N}}$. We prove the superstability of the following generalized Golab-Schinzel type equation $f(x_1+{\limits\sum_{i=2}^p}x_if(x_1)^kf(x_2)^k{\cdots}f(x_{i-1})^k)={\limits\prod_{i=1}^pf(x_i),x_1,x_2,{\cdots},x_p{\in}X$, where $f:X{\rightarrow}K$ is an unknown function which is hemicontinuous at the origin.

Keywords

Hyers-Ulam stability;Golab-Schinzel equation;superstability

References

1. J. Aczel and S. Golab, Remarks on one-parameter subsemigroups of the affine group and their homo- and isomorphisms, Aequationes Math. 4 (1970), 1-10. https://doi.org/10.1007/BF01817739
2. J. A. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), no. 3, 411-416. https://doi.org/10.1090/S0002-9939-1980-0580995-3
3. J. A. Baker, J. Lawrence, and F. Zorzitto, The stability of the equation f(x + y) =f(x)f(y), Proc. Amer. Math. Soc. 74 (1979), no. 2, 242-246.
4. K. Baron, On the continuous solutions of the Golab-Schinzel equation, Aequationes Math. 38 (1989), no. 2-3, 155-162. https://doi.org/10.1007/BF01840001
5. N. Brillouet-Belluot, On some functional equations of Golab-Schinzel type, Aequationes Math. 42 (1991), no. 2-3, 239-270. https://doi.org/10.1007/BF01818494
6. N. Brillouet-Belluot, J. Brzdek, and K. Cieplinski, On some recent developments in Ulam's type stability, Abstr. Appl. Anal. 2012 (2012), Art. ID 716936, 41 pp.
7. N. Brillouet-Belluot and J. Dhombres, Equations fonctionnelles et recherche de sous-groupes, Aequationes Math. 31 (1986), no. 2-3, 253-293. https://doi.org/10.1007/BF02188194
8. J. Brzdek, Subgroups of the group Zn and a generalization of the Golab-Schinzel functional equation, Aequationes Math. 43 (1992), no. 1, 59-71. https://doi.org/10.1007/BF01840475
9. J. Brzdek, Some remarks on solutions of the functional equation $f(x+f(x)^ny)=tf(x)f(y)$, Publ. Math. Debrecen 43 (1993), no. 1-2, 147-160.
10. J. Brzdek, Golab-Schinzel equation and its generalizations, Aequationes Math. 70 (2005), no. 1-2, 14-24. https://doi.org/10.1007/s00010-005-2781-y
11. J. Brzdek, Stability of the generalized Golab-Schinzel equation, Acta Math. Hungar. 113 (2006), 115-126.
12. J. Brzdek, On the quotient stability of a family of functional equations, Nonlinear Anal. 71 (2009), no. 10, 4396-4404. https://doi.org/10.1016/j.na.2009.02.123
13. J. Brzdek, On stability of a family of functional equations, Acta Math. Hungar. 128 (2010), no. 1-2, 139-149. https://doi.org/10.1007/s10474-010-9169-8
14. J. Brzdek and K. Cieplinski, Hyperstability and superstability, Abstr. Appl. Anal. 2013 (2013), Article ID 401756, 13 pages.
15. A. Chahbi, On the superstability of the generalized Golab-Schinzel equation, Internat. J. Math. Anal. 6 (2012), no. 54, 2677-2682.
16. A. Charifi, B. Bouikhalene, S. Kabbaj, and J. M. Rassias, On the stability of Pexiderized Golab-Schinzel equation, Comput. Math. Appl. 59 (2010), no. 9, 3193-3202. https://doi.org/10.1016/j.camwa.2010.03.005
17. J. Chudziak, Approximate solutions of the Golab-Schinzel equation, J. Approx. Theory 136 (2005), no. 1, 21-25. https://doi.org/10.1016/j.jat.2005.04.011
18. J. Chudziak, Stability of the generalized Golab-Schinzel equation, Acta Math. Hungar. 113 (2006), no. 1-2, 133-144. https://doi.org/10.1007/s10474-006-0095-8
19. J. Chudziak, Approximate solutions of the generalized Golab-Schinzel equation, J. Inequal. Appl. 2006 (2006), Article ID 89402, 8 pp.
20. J. Chudziak, Stability problem for the Golab-Schinzel type functional equations, J. Math. Anal. Appl. 339 (2008), no. 1, 454-460. https://doi.org/10.1016/j.jmaa.2007.07.006
21. J. Chudziak and J. Tabor, On the stability of the Golab-Schinzel functional equation, J. Math. Anal. Appl. 302 (2005), no. 1, 196-200. https://doi.org/10.1016/j.jmaa.2004.07.053
22. R. Ger and P. Semrl, The stability of the exponential equation, Proc. Amer. Math. Soc. 124 (1996), no. 3, 779-787. https://doi.org/10.1090/S0002-9939-96-03031-6
23. S. Golab and A. Schinzel, Sur l'equation fonctionnelle f(x + yf(x)) = f(x)f(y), Publ. Math. Debrecen 6 (1959), 113-125.
24. D. H. Hyers, G. I. Isac, and Th. M. Rassias, Stability of Functional Equations in Sev-eral Variables, Progress in Nonlinear Differential Equations and their Applications 34, Birkhauser, Boston, Inc., Boston, MA, 1998.
25. E. Jablonska, On solutions of some generalizations of the Golab-Schinzel equation, In: Functional Equations in Mathematical Analysis, pp. 509-521. Springer Optimization and its Applications vol. 52. Springer 2012.
26. E. Jablonska, On continuous solutions of an equation of the Golab-Schinzel type, Bull. Aust. Math. Soc. 87 (2013), no. 1, 10-17. https://doi.org/10.1017/S0004972712000299
27. E. Jablonska, On locally bounded above solutions of an equation of the Golab-Schinzel type, Aequationes Math. 87 (2014), no. 1-2, 125-133. https://doi.org/10.1007/s00010-013-0195-9
28. S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Anal-ysis, Springer, New York, 2011.
29. Th. M. Rassias, On the stability of linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. https://doi.org/10.1090/S0002-9939-1978-0507327-1
30. A. Roukbi, D. Zeglami, and S. Kabbaj, Hyers-Ulam stability of Wilson's functional equation, Math. Sci. Adv. Appl. 22 (2013), 19-26.
31. D. Zeglami, The superstability of a variant of Wilson's functional equations on an ar-bitrary group, Afr. Mat. 26 (2015), 609-617. https://doi.org/10.1007/s13370-014-0229-z
32. D. Zeglami and S. Kabbaj, On the supesrtability of trigonometric type functional equa-tions, British J. Math. & Comput. Sci. 4 (2014), no. 8, 1146-1155. https://doi.org/10.9734/BJMCS/2014/7792
33. D. Zeglami, S. Kabbaj, A. Charifi, and A. Roukbi, ${\mu}$-Trigonometric functional equations and Hyers-Ulam stability problem in hypergroups, Functional Equations in Mathematical Analysis, pp. 337-358, Springer, New York, 2012.
34. D. Zeglami, A. Roukbi, and S. Kabbaj, Hyers-Ulam stability of generalized Wilson's and d'Alembert's functional equations, Afr. Mat. 26 (2015), 215-223. https://doi.org/10.1007/s13370-013-0199-6

Cited by

1. Stability problem for the composite type functional equations 2017, https://doi.org/10.1016/j.exmath.2017.08.002