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dc.contributor.authorGer, Roman-
dc.identifier.citationAequationes Mathematicae, Vol. 93 (2019), s. 299-309pl_PL
dc.description.abstractRoger Cuculi`ere [Problem 11998, The American Mathematical Monthly 124 no. 7 (2017)] has posed the following problem: Find all continuous functions f : R −→ R that satisfy f(z) ≤ 1 for some nonzero real number z and f(x)2 + f(y)2 + f(x + y)2 − 2f(x)f(y)f(x + y) = 1 (C) for all real numbers x and y. We present the general Lebesgue measurable solution of (C) in the class of complex valued functions defined on the real line. Moreover, applying the invariant ideals method, we shall discuss a corresponding alternative d’Alembert equation f(x + y) = f(x − y) =⇒ f(x + y) + f(x − y) = 2f(x)f(y), (CA) stemming from Eq. (C) in the class of scalar valued functions defined on suitable groups. Equations (CA) seems to be of interest on its own.pl_PL
dc.rightsUznanie autorstwa 3.0 Polska*
dc.subjectAlternative (conditional) functional equationspl_PL
dc.subjectD’Alembert’s equationpl_PL
dc.subjectInvariant idealspl_PL
dc.subjectFubini’s Theorempl_PL
dc.titleOn an alternative d’Alembert’s equationpl_PL
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