Homogeneous Functional Equation

Abstract

The functional equation $f(yx)=y^kf(x)$ (where k is a fixed real constant) is called the homogeneous functional equation of degree k. In the case when k D 1 in the above equation, the equation is simply called the homogeneous functional equation. In Section 5.1, the Hyers–Ulam–Rassias stability of the homogeneous functional equation of degree k between real Banach algebras will be proved in the case when k is a positive integer. It will especially be proved that every “approximately” homogeneous function of degree k is a real homogeneous function of degree k. Section 5.2 deals with the superstability property of the homogeneous equation on a restricted domain and an asymptotic behavior of the homogeneous functions. The stability problem of the equation between vector spaces will be discussed in Section 5.3. In the last section, we will deal with the Hyers–Ulam–Rassias stability of the homogeneous functional equation of Pexider type.