Abstract
Using some results on convex and almost convex functions defined on a locally compact Abelian group, we prove a theorem showing a “measurability implies continuity” effect for non-negative solutions of the difference equation \({\varphi(x) = \sum_{i=1}^{k}p_{i}\varphi\left(x+a_{i} \right)}\), where \({p_{1}, \ldots, p_{k} \in (0, \infty)}\) and non-zero elements \({a_{1}, \ldots, a_{k}}\) of the group are given.
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Dedicated to Professor János Aczél on the occasion of his 90th birthday
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Jarczyk, W. Improving regularity of solutions of a difference equation. Aequat. Math. 89, 383–391 (2015). https://doi.org/10.1007/s00010-015-0351-5
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DOI: https://doi.org/10.1007/s00010-015-0351-5