A conditional exponential functional equation and its stability

We study the conditional functional equation: γ(x+y)=γ(x−y)⇒f(x+y)=f(x)f(y), with a given function γγ; if γγ is a norm in a real linear space, then the equation becomes the well-known isosceles orthogonally exponential functional equation. We give the descriptions of its solutions and investigate the stability of it. We also consider the pexiderized version of the equation.