An alternative-conditional functional equation and the orthogonally additive functionals

Abstract

Let X be a real inner product space of dimension greater than 2 and f be a real functional defined on X. Applying some ideas from the recent studies made on the alternative-conditional functional equation $$(x, y) = 0 \Rightarrow f(x + y)^2 = [f(x) + f(y)]^{2}$$ in this paper we study another alternative-conditional equation related to the previous, namely $$(x, y) = 0 \Rightarrow [f(x + y) - f(x)]^2 = f(y)^2$$ in order to describe its solutions. This research follows previous studies concerning alternative functional equations and the Cauchy functional equation on the restricted domain of the orthogonal vectors.