On non-negative measurable solutions of a difference functional equation
Consider the functional equation (1) \(\varphi(x)= \sum^k_{i=1} A_i\varphi (x+a_i)\), where \(A_1, \dots, A_k\) are positive reals and \(a_1, \dots, a_k\in \mathbb{R}^n \setminus \{0\}\). Let \({\mathcal S}\) be a translation invariant \(\sigma\)-field of subsets of \(\mathbb{R}^n\) containing Borel subsets and such that pre-images of sets from \({\mathcal S}\) by the addition and subtraction are product measurable. Given a proper \(\sigma\)-ideal \({\mathcal N}\subset {\mathcal S}\) satisfying some reasonable conditions (for instance a kind of the Steinhaus theorem as well as the Fubini theorem for \({\mathcal N}\) and a \(\sigma\)-ideal of subsets of \(\mathbb{R}^n \times \mathbb{R}^n\) are assumed to hold) the following is proved (cf. Theorem): Assume that the additive group generated by \(a_1, \dots, a_k\) is dense in \(\mathbb{R}^n\) and let \(\varphi: \mathbb{R}^n \to\mathbb{R}\) be a nonnegative \({\mathcal S}\)-measurable function satisfying (1) \({\mathcal N}\)-a.e. in \(\mathbb{R}^n\). Then either \(\varphi(x)=0\) \({\mathcal N}\)-a.e. in \(\mathbb{R}^n\), or there is a positive continuous logarithmically convex solution \(\varphi_0: \mathbb{R}^n\to \mathbb{R}\) of (1) such that \(\varphi(x)= \varphi_0( x)\) \({\mathcal N}\)-a.e. in \(\mathbb{R}^n\). In the proof the author has used some ideas of the reviewer [Prace Naukowe Uniwersytetu \`Sląskiegow Katowicach. 1206. Katowice: Uniwersytet \`Sląski (1991; Zbl 0741.39006)]. In particular, taking \({\mathcal S}= \{A\subset \mathbb{R}^n\): \(A\) has the Baire property\} and \({\mathcal N}= \{A\subset \mathbb{R}^n\): \(A\) is of the first category\} one can obtain a topological version of \textit{M. Laczkovich}'s theorem [J. Lond. Math. Soc., II. Ser. 34, 139-147 (1986; Zbl 0634.39001)] providing the form of Baire measurable solutions of (1) defined on the real line. Some consequences are also proved in the case where \({\mathcal N}\) is a so-called small system.
positive continuous logarithmically convex solution
scientific article; zbMATH DE number 1203810