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1
Remarks concerning the pexiderized Gołąb-Schinzel functional equation
EN
This paper is devoted to proof of theorem concerning solutions of the pexiderized Gołąb-Schinzel functional equation. We provide explicite formulas expressing solutions of the equation. Our considerations refer to the paper [6].
2
Bounded solutions of a generalized Gołąb-Schinzel equation
EN
Let X be a linear space over the field K of real or complex numbers. We characterize solutions f : X - > K and M : K - > K of the equation f(x+M)(f)y)=f(x)f(y) in the case where the set {x is an element of X : f (x) = 0} has an algebraically interior point. As a consequence we give solutions of the equation such that f is bounded on this set.
3
One-to-one solutions of generalized Gołąb-Schinzel equation
EN
Let K be the field of real or complex numbers and let X be a nontrivial linear space over K. Assume that [...]. We give a necessary and sufficient condition for functions f and M to satisfy the equation The functional equation f(x+M(f(x))y)=f(x)f(y) is a generalization of the well-known Gołąb-Schinzel functional equation f(x+f(x)y)=f(x)f(y).
EN
Let k, n be positive integers and let f : Rn -> R be a solution of the functional equation f(x + f(x)ky)=f(x)f(y). We prove that, if there is a real positive a such that the set [x is an element of Rn : |f(x)| is an element of (0,a)} contains a subset of positive Lebesgue measure, then f is continuous. As a consequence of this we obtain that every Lebesgue measurable solution f : Rn -> R of the equation is continuous or equal zero almost everywhere (i.e. there is a set A C R of the Lebesgue measure zero with f(Rn \ A) = {0}).
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