Some Comments on the Superstability of a General Functional Equation

Abstract

In this paper, we prove a superstability theorem for a general functional equation ∑j=1∞ajf(γj(t,s))=h(t)g(s), with the unknown functions g:T→X, h:S→K and f:S→X, such that the series ∑j=1∞f(γj(t,s)) is convergent for every (s,t)∈S×T, where S and T are nonempty sets, and X is a Banach space over a field K, which is either the set of real numbers R or the set of complex numbers C. Namely, we show that if h is unbounded, and the difference ∑j=1∞ajf(γj(t,s))−h(t)g(s) is bounded, then h and g satisfy the equation ∑j=1∞ajg(γj(t,s))=h(t)g(s). Next, we show that the superstability of pexiderizations and radical versions of several well-known functional equations (e.g., of Cauchy, d’Alembert, Wilson, Reynolds, and homogeneity) is a consequence of this simple outcome. In this way, we generalize several classical superstability results and, in particular, the first superstability outcome of J. Baker, J. Lawrence, and F. Zorzitto, which states that every unbounded mapping f, from a real vector space X into the set of real numbers R, satisfying the inequality |f(x+y)−f(x)f(y)|≤δ for every x,y∈X, with some real δ>0, is an exponential function, i.e., satisfies the equality f(x+y)=f(x)f(y) for all x,y∈X. In order to make this publication more accessible to a wider range of readers, we limit various related information, avoid very abstract generalizations, and provide some simple examples.