Functional equations of the Gołąb-Schinzel type on a cone
type: Item
instance of: scholarly article
title: Functional equations of the Gołąb-Schinzel type on a cone
zbMATH Open document ID: 1331.39012
author: Jacek Chudziak
author: Zdeněk Kočan
published in: Monatshefte für Mathematik
publication date: 2015-12-03T00:00:00Z
review text: 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} …
reviewed by: Maryam Amyari
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 6515840
zbMATH Keywords: Gołąb-Schinzel-type functional equations
zbMATH Keywords: convex cone
zbMATH Keywords: continuity on rays
Wikidata QID: Q59406067
MaRDI profile type: Publication
full work available at URL: s00605-015-0784-x
OpenAlex ID: W2169377579
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DOI: 10.1007/S00605-015-0784-X
description: scientific article; zbMATH DE number 6515840