General solution of a functional equation arisen
from characterization problems

Abstract. We give the general solution of the functional equation \[ h_{1}\left(\frac{x}{\lambda_{1}(\alpha+y)}\right)\frac{1}{\lambda_{1}(\alpha+y)}f_{Y}(y)= h_{2}\left(\frac{y}{\lambda_{2}(\beta+x)}\right)\frac{1}{\lambda_{2}(\beta+x)}f_{X}(x) \] for all \((x,y)\in\mathbb{R}^{2}_{+}\) with nonnegative functions \(h_{1}\), \(h_{2}\), \(f_{X}\), \(f_{Y}:\mathbb{R}_{+}\rightarrow\mathbb{R}\), such that there exist sets \(A_{1}\), \(A_{2}\), \(A_{3}\), \(A_{4}\subset\mathbb{R}_{+}\) with positive Lebesgue measure, on which these functions are positive.