On a class of linear functional equations without range condition

Gselmann, Eszter and Kiss, Gergely and Vincze, Csaba (2019) On a class of linear functional equations without range condition. AEQUATIONES MATHEMATICAE. ISSN 0001-9054

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Abstract

The main purpose of this work is to provide the general solutions of a class of linear functional equations. Let n≥2 be an arbitrarily fixed integer, let further X and Y be linear spaces over the field K and let αi,βi∈K, i=1,…,n be arbitrarily fixed constants. We will describe all those functions f,fi,j:X×Y→K, i,j=1,…,n that fulfill functional equation f(∑i=1nαixi,∑i=1nβiyi)=∑i,j=1nfi,j(xi,yj)(xi∈X,yi∈Y,i=1,…,n). Additionally, necessary and sufficient conditions will also be given that guarantee the solutions to be non-trivial.

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