Semigroup-valued solutions of some composite equations

Abstract

Abstract. Let X be a linear space over the field K of real or complex numbers and (S, •) be a semigroup. We determine all solutions of the functional equationin the class of pairs of functions (f,g) such that f : X → S and g : X → K satisfies some regularity assumptions. Several consequences of this result are presented.

Functional equations of the Goła̧b-Schinzel type on a cone

“…We claim that in this case (d) is valid. In view of (29)-(30), in order to show this, it is enough to prove that g(x) = 0 for every (6) and so, in view of (31), we obtain that g(x) ≥ 0. Thus, taking a y ∈ C (3) , we get x + g(x)sy ∈ C for s ∈ [0, ∞).…”

“…Thus from (45) we derive that x + g(x)sy / ∈ C (1) ∪ C (3) for s ∈ [0, ∞) and so, as C (4) = C (5) = ∅, we get x + g(x)sy ∈ C (6) for s ∈ [0, ∞). Therefore, taking into account (29) and (45), we obtain Furthermore, L(y) > 0 because y ∈ C (3) .…”

“…Now, in order to complete the proof, it is enough to show that at most one of the sets C (i) for i ∈ {4, 5, 6} is nonempty. For the proof by contradiction suppose that C ( j 1 ) = ∅ and C ( j 2 ) = ∅ for some j 1 , j 2 ∈ {4, 5, 6}, (6) .…”

“…Further going pexiderization of the Gołab-Schinzel equation have been investigated in [5]- [6] and [10]- [11]. In a recent paper [8] the results of [1] and [19] have been generalized.…”

“…Then t x + g(t x)t y = t x + g x (t)t y = t x + t y ∈ C for t ∈ [0, ∞) and so, taking into account (6) and (21), we get…”