Functional equations of the Gołąb-Schinzel type on a cone
Applying the results of \textit{A. Mureńko} [Publ. Math. 63, No. 4, 693--702 (2003; Zbl 1050.39028)], the authors deal with a similar problem, but in a much more general setting. Let \(X\) be a real linear space. A nonempty subset \(\mathcal{C}\) of \(X\) such that \(\alpha x + \beta y \in \mathcal{C}\) for each \(x, y \in \mathcal{C}\) and \(\alpha, \beta \geq 0\) is called a convex cone in \(X\). They determine the solutions of the functional equation \(f (x + g(x)y) = f (x) f (y)\) whenever \(x, y, x + g(x)y \in \mathcal{C}\) in the class of pairs \(( f, g)\) of functions continuous on rays mapping \(\mathcal{C}\) into \(\mathbb{R}\). Recall that a function \(f: \mathcal{C} \to \mathbb{R}\) is said to be continuous on rays, provided that for every \(x \in \mathcal{C}\) the function \(f_x : [0,\infty) \to \mathbb{R}\) given by \(f_x (t) = f (tx)\) is continuous for each \(t \in [0,\infty)\). The main result of the paper says that the pair \(( f, g)\) of functions continuous on rays satisfies in the above equation if and only if there exist continuous functions \(\tilde{f}, \tilde{g}: \mathbb{R} \to \mathbb{R}\) and a nontrivial linear functional \(L : L\in \mathcal{C} \to \mathbb{R}\) such that the pair \((\tilde{f}, \tilde{g})\) satisfies \(f (x + g(x)y) = f (x) f (y)\), whenever \(x, y, x + g(x)y \geq 0\) and \(f =\tilde{f}\circ L_{|\mathcal{C}}\) and either \(g = \tilde{g}\circ L_{|\mathcal{C}}\), or \(L(\mathcal{C}) = (- \infty, 0]\) and, for every \(x \in \mathcal{C}\), \(g(x) = (\tilde{g} \circ L)(x)\), whenever \(1+ L(x) > 0\) and \(g(x)\geq 0\), otherwise. Furthermore, they show that the similar assertion holds for the solution of the equation \(F(x + G(x)y) = F(x)F(y)\) for each \(x, y \in \mathcal{C}\) in the class of pairs \((F, G)\) of functions continuous on rays mapping \(X\) into \(\mathbb{R}\). Let us recall that a function \(F: X \to \mathbb{R}\) is said to be continuous on rays, provided that for every \(x \in X\) the function \(F_x : \mathbb{R} \to \mathbb{R}\) given by \(F_x(t)=F(tx)\) for \(t \in \mathbb{R}\) is continuous.
Gołąb-Schinzel-type functional equations
scientific article; zbMATH DE number 6515840
scientific article