ON THE SOLUTIONS OF THE EQUATION ƒ(xƒ(y)k +yƒ(x)l)=f(x)f(y)

Continuous solutions of a generalization of the Gołąb-Schinzel equation

“…Furthermore, whenever φ is of the form (15) or (16), then the general solution of (14) in the class of non-constant continuous functions f : R → R is given, respectively, by:…”

“…One can check that if φ has the form (15) or (16), then every function of the form (17), (18) or (19) satisfies (14) in the corresponding case. Assume that f : R → R is a non-constant continuous function satisfying (14). On account of Corollary 1 and Lemma 3(i),(iv), there exists an (8) holds.…”

“…which has been considered among others in [4]- [8], [12], [14], where in particular the continuous solutions f : R → R of that equation have been determined. In [3] it is proved that the cardinality of the set of discontinuous solutions f : R → R of this equation is 2 ℵ , where ℵ = card R. Some applications of this type of functional equations can be found for example in [2], [4], [6], [8] and [12].…”

Continuous solutions of a generalization of the Gołab–Schinzel equation II

“…Some information concerning that equation, recent results, applications and numerous references can be found in [1], [5]- [7] and [9]- [13]. In the case, when φ(x) = x k and ψ(x) = x l for x ∈ R, where k and l are fixed positive integers, equation (1) has been considered among others in [2]- [4] and [15].…”

On solutions of the Gołąb--Schinzel functional equation

“…studied by many authors in variuous case (see e.g. [5]- [7], [16], and [19]). Finally, we must mention that equation ( 1) is tightly connected with some classes of subgroups of the Lie groups L 1 s , the one-dimensional affine group, and some other groups (see [8], cf.…”