On the modified entropy equation

Abstract

The object of this paper is to solve the so--called modified entropy equation \[ f\left(x, y, z\right)=f\left(x, y+z, \mathbf{0}\right)+ \mu\left(y+z\right)f\left(\mathbf{0}, \frac{y}{y+z}, \frac{z}{y+z}\right), \] on the positive cone of ${\mathbb R}^{k}$, where $\mu$ is a given multiplicative function on this cone. After that the regular solutions of this equation are determined. Furthermore we investigate its connection between the entropy equation and other equations, as well.