A generalized ring homomorphisms equation

A characterization of quadratic-multiplicative mappings

“…which is more general than (6), with arbitrary a, b, c, d, k, l ∈ R. This functional equation provide a joint generalization of (6) (k = l) and of a general equation discussed the paper [6] (d = 0). However, we will show that one cannot expected an elegant behavior of (21), which is characteristic for Eq.…”

“…In particular Dhombres provided sufficient conditions under which each solution of (4) is a ring homomorphism. Later this results has been developed and generalized in a few directions by Ger [3][4][5], Ger and Reich [6], Laohakosol, Pimsert and Udomkavanich [9], among others. The term "alienation phenomenon" was introduced in order to characterize the effect that each solution of a single equation satisfies the corresponding system of equations.…”

“…(5) it follows that for each nontrivial solution f of (5) On the other hand, it is possible to provide additional assumptions which imposed upon f will force that either d = 0 or k = l. For example, it is enough to assume that f : P → R is a solution of (21) and there exists a y 0 ∈ P such that f (y 0 ) = f (−y 0 ) = 0. Indeed, it suffices to substitute in ( Since f (y 0 ) = 0 and we keep assuming that R contains no zero divisors then k = l and thus we reduced (21) to (6).…”

Alienation of the logarithmic and exponential functional equations

“…The phenomenon of alienation was first discussed by Dhombres [2]. Later on, several papers and lectures have appeared on this subject (see [3][4][5][6][7][8][9][10][11][12]). …”

On the alienation of the exponential Cauchy equation and the Hosszú equation

“…This equation has been also considered by Ger and Reich in [12] but they were dealing with a more general form af (xy) + bf (x)f (y) + cf (x + y) + df (x) + kf (y) = 0.…”