Functional equations for homogeneous polynomials arising from multilinear mappings and their stability. (English) Zbl 0820.39011

Schwaiger, Jens

Pr. Nauk. Uniw. Śląsk. Katowicach 1444, Ann. Math. Silesianae 8, 157-171 (1994).

Let \(H\) be an Abelian semigroup and \(G\) be an Abelian group uniquely divisible by \(n!\). A well-known result of Djokovic says that a function \(f: H\to G\) has the form \(f= g\circ \delta_ n\), where \(g: H^ n\to G\) is symmetric and \(n\)-additive and \(\delta_ n(x)= (x, x,\dots, x)\), if and only if \(\overset {n} {\underset {u}\Delta} f(x)= n! f(u)\) for all \(u\in H\).
Let now \(A\) be a commutative ring with identity, uniquely divisible by \(n!\), \(n\geq 2\) fixed; let \(M\), \(N\) be \(A\)-modules. One of the results of the present paper gives a characterization through a functional equation connected with the differences, of the functions \(f: M\to N\) of the form \(f= g\circ \delta_ n\), where \(g\) is \(n\)-linear and symmetric.
The stability of this functional equation and of other similar equations is then studied in Banach spaces over non-Archimedean valued fields.

Reviewer: G.L.Forti (Milano)

MSC:

39B52

Functional equations for functions with more general domains and/or ranges

Keywords:

difference equations; Abelian semigroup; Abelian group; commutative ring; \(A\)-modules; stability; Banach spaces; non-Archimedean valued fields