On a sum form functional equation containing five unknown mappings

Abstract

The general solutions of the functional equation

$$\sum\limits_{i=1}^n \sum\limits_{j=1}^m F(p_{i}q_{j}) = \sum\limits_{i=1}^n G(p_{i})\sum\limits_{j=1}^m H(q_{j}) + \sum\limits_{i=1}^n K(p_{i}) \sum_{j=1}^m L(q_{j})$$

in which F,G,H,K and L are real-valued mappings with domain I = [0, 1]; L is a multiplicative mapping with L(0) =  0, L(1) =  1; ${(p_{1},\ldots,p_{n}) \in \Gamma_{n}}$, ${(q_{1},\ldots,q_{m}) \in \Gamma_{m}}$; ${n \ge 3}$; ${m \ge 3}$ being fixed integers; have been obtained.