We establish conditions under which Baire measurable solutions f of $$ \Gamma (x,y,|f(x) - f(y)|) = \Phi (x,y,f(x + \phi _1 (y)),...,f(x + \phi _N (y))) $$ defined on a metrizable topological group are continuous at zero.
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The scientific output of Marek Kuczma consists of 179 papers published in the years 1958-1993 and three books still used and quoted. Professor Marek Kuczma created and developed the theory of iterative functional equations but his name is also connected to important results on functional equations in several variables, in particular on Cauchy's equation and Jensen's inequality. In fact Marek Kuczma has founded a mathematical school: he supervised 13 Ph.D. dissertations, 10 his students have already their habilitation and 6 of them became full professors. The paper provides more information about the great teacher and some results of the outstanding mathematician.
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We show that the solution to the orthogonal additivity problem in real inner product spaces depends continuously on the given function and provide an application of this fact.
Let $\(E\) be a real inner product space of dimension at least 2. If \(f\) maps \(E\) onto \(E\) and both \(f\) and \(f \circ f \) are orthogonally additive, then \(f\) is additive.
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Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.
Let \(E\) be a real inner product space of dimension at least 2 and \(V\) a linear topological Hausdorff space. If \(\operatorname{card}E\leq \operatorname{card} V\), then the set of all orthogonally additive injections mapping \(E\) into \(V\) is dense in the space of all orthogonally additive functions from \(E\) into \(V\) with the Tychonoff topology. If \(\operatorname{card}V\leq \operatorname{card}E\), then the set of all orthogonally additive surjections mapping \(E\) into \(V\) is dense in the space of all orthogonally additive functions from \(E\) into \(V\) with the Tychonoff topology.
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New sufficient conditions for asymptotic stability of Markov operators are given. These criteria are applied to a class of Volterra type integral operators with advanced argument.
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We consider the family 𝓜 of measures with values in a reflexive Banach space. In 𝓜 we introduce the notion of a Markov operator and using an extension of the Fortet-Mourier norm we show some criteria of the asymptotic stability. Asymptotically stable Markov operators can be used to construct coloured fractals.