On an alternative d’Alembert’s equation

Abstract

Roger 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.