Multiplicative symmetry and related functional equations
- 26 Downloads
Let (G, *) be a commutative monoid. Following J. G. Dhombres, we shall say that a functionf: G → G is multiplicative symmetric on (G, *) if it satisfies the functional equationf(x * f(y)) = f(y * f(x)) for allx, y inG. (1)
Equivalently, iff: G → G satisfies a functional equation of the following type:f(x * f(y)) = F(x, y) (x, y ∈ G), whereF: G × G → G is a symmetric function (possibly depending onf), thenf is multiplicative symmetric on (G, *).
In Section I, we recall the results obtained for various monoidsG by J. G. Dhombres and others concerning the functional equation (1) and some functional equations of the formf(x * f(y)) = F(x, y) (x, y ∈ G), (E) whereF: G × G → G may depend onf. We complete these results, in particular in the case whereG is the field of complex numbers, and we generalize also some results by considering more general functionsF.
In Section II, we consider some functional equations of the formf(x * f(y)) + f(y * f(x)) = 2F(x, y) (x, y ∈ K), where (K, +, ·) is a commutative field of characteristic zero, * is either + or · andF: K × K → K is some symmetric function which has already been considered in Section I for the functional equation (E). We investigate here the following problem: which conditions guarantee that all solutionsf: K → K of such equations are multiplicative symmetric either on (K, +) or on (K, ·)? Under such conditions, these equations are equivalent to some functional equations of the form (E) for which the solutions have been given in Section I. This is a partial answer to a question asked by J. G. Dhombres in 1973.
AMS (1991) subject classification39B22 39B32 39B52
Unable to display preview. Download preview PDF.
- Aczél, J.,Lectures on functional equations and their applications. [Math. Sci. Engrg. Vol. 19] Academic Press, New York—London, 1966.Google Scholar
- Aczél, J. andDhombres, J. G.,Functional equations in several variables. [Encyclopaedia of Mathematics, Vol. 31] Cambridge University Press, Cambridge, 1989.Google Scholar
- Bourbaki, N.,Eléments de mathématiques. Topologie générale. Chapitre VII, Diffusion C.C.L.S., Masson, 1974.Google Scholar
- Brillouēt-Belluot, N.,More about some functional equations of multiplicative symmetry. To appear.Google Scholar
- Daróczy, Z.,Über die Funktionalgleichung: ϕ(ϕ(x) · y) = ϕ(x) · ϕ(y). Act. Univ. Debrecen Ser. Fiz. Chem.8 (1962), 125–132.Google Scholar
- Dhombres, J. G.,Sur les opérateurs multiplicativement liés. Mém. Soc. Math. France, no.27 (1971).Google Scholar
- Dhombres, J. G.,Functional equations on semi-groups arising from the theory of means. Nanta Math.5 (1972), no. 3, 48–66.Google Scholar
- Dhombres, J. G.,Report of Meeting, Problem 122. Aequationes Math.11 (1974), 308.Google Scholar
- Matras, Y.,Sur l'équation fonctionnelle: f(x f(y)) = f(x) · f(y). Acad. Roy. Belg. Bull. Cl. Sci. (5)55 (1969), 731–751.Google Scholar