Abstract
Given real nonzero coefficients a, A, b, B and additive functions \(\phi,\psi : {\mathbb {R}}\to {\mathbb {R}}\), necessary and sufficient conditions for existence and uniqueness of additive solutions of the system ξ(ax)−Aξ(x)=ϕ(x), ξ(bx)−Bξ(x)=ψ(x) are presented. According to the various possibilities concerning the arithmetic nature of the four coefficients and the algebraic relationships between them, the additive solution(s) of the system are given explicitly for each case.
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Prager, W., Schwaiger, J. A system of two inhomogeneous linear functional equations. Acta Math Hung 140, 377–406 (2013). https://doi.org/10.1007/s10474-013-0315-y
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DOI: https://doi.org/10.1007/s10474-013-0315-y