Abstract

In this paper, we investigate a composite type functional equation f(x f(y)-y f(x))=f(x)-f(y)+x-y (1) on an Abelian group. This is a new composite functional equation introduced by authors and we are interested in finding the various properties of the function desired on an Abelian group which is uniquely divisible by 2.

Keywords

References

1. J. Aczel and J. Dhombers, Functional equation in several variables, In: Encyclopedia of Mathematics and its Applications, vol.31. Cambridge University Press, Cambridge, (1989).
2. Aczel, J., Golab, S. Remarks on one-parameter subsemigroups of the affine group and their homo- and isomorphisms, Aequationes Math. 4 (1970), 1-10.
3.Aichinger, E., Farag, M. On when the multiplicative center of a near-ring is a subnear-ring, Aequationes Math. 48 (2004), 46-59.
4.Z. Boros and Z. Daroczy, A composite functional equation with additive solutions, Publ. Math. Debrecen 69 (2006), no.1-2, 245-253.
5.Brillouet, N., Dhombres, J. Equations fonctionnelles et recherché de sousgroupes, Aequationes Math. 31 (1986), 253-293.
6.Choa, W. W. Problem 10854. Amer. Math. Monthly 108(2) (2001), 171. 7.A. Gilsnyi and Zs. Pales, A regularity theorem for composite functional equations, Arch. Math. (Basel) 77 (2001), 317 – 322.
8.Golab, S., Schinzel, A. Sur l’equation fonctionnelle f(x+f(x)y)=f(x)f(y), Publ. Math. Debrecen 6 (1959), 113-125.
9.M.E. Gordji, S.K. Gharetapeh, Ch. Park and S. Zolfaghri, Stability of an additive-cubic-quartic functional equation, Adv. in Difference Equations, 2009, Art. 395693 (2009), 20 pages.
10.M.E. Gordji, M.B. Savadkouhi, Stability of a mixed type additive, quadratic and cubic functional equation in random normed spaces, Filomat 25(3) (2011), 43-54.
11.M.E. Gordji, S. Zolfaghri, J.M. Rassias and M.B. Savadkouhi, Solution and stability of a mixed type cubic and quartic functional equation in Quas-Banach spaces, Abstract and Appl. Anal., 2009, Art. 417473, (2009), 14 pages.
12.Henderson, D Continuous additive functions; solutions to problem10854, Amer. Math. Monthly 111(8) (2004), 725-726.
13.Ilse, D., Lehmann, I., Schulz, W. Gruppoide und Funktionalgleichungen. VEB Deutscher Verlag der Wissenschaften, Berlin (1984).
14.S.H. Lee, S.M. IM and I.S. Hwang, Quartic functional equations, J. Math. Anal. Appl., 307 (2005), 387-394.
15.Luneburg, H., Plaumann, P. Die Funktionalgleichung von Golab und Schinzel in Galoisfeldern, Arch. Math. (Basel) 28 (1977), 55-59.
16.A. Najati, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, Turk. J. Math. 31 (2007), 395-408.
17.Zs. Pales, A regularity theorem for composite functional equations, Acta Sci. Math. (Szeged) 69 (2003), no. 3-4, 591-604.
18.Zs. Pales, Regularity problems and results concerning composite functional equations, Tatra Mt. Math. Publication 34 (2006), part II, 289-306.
19.Ch. Park, S.W. Jo and D.Y. Kho, On the stability of an AQCQ-Functional Equation, J. Chungcheong Math. Soc., 22(4) (2009), 757-770.
20.Ch. Park, Fuzzy stability of an Additive-Quadratic-Quartic functional equation, J. Inequ. Appl., 2010 (2010), Art. ID 253040, 22 pp.
21.K. Ravi, J.M. Rassias and P. Narasimman, Stability of a cubic functional equation in fuzzy normed space, J. Appl. Anal. Comp., 1(3) (2011), 411-425.
22.K. Ravi, B.V. Senthil kumar and S. Kandasamy, Solutiona and stability of a mixed type quadratic and additive functional equation in two variables, Int. J. Math. Comp., 13(D11) (2011), 51-68.
23.A. Tarski, Problem no. 83, Parametr 1 (1930), no. 6, 231; Solution: Mlody Matematyk 1 (1931), no. 1, 90 (in polish).
24.Wlodzimierz Fechner, On a composite functional equation on Abelian groups, Aequationes Math.78 (2009), 185-193.