M-affine functions composing Sturm–Liouville families

Abstract

Given an n-variable mean M defined on a real interval I, an M-affine function is a solution to the functional equation [Equation not available: see fulltext.]When M is a quasilinear mean, the set of continuous M-affine functions is a Sturm–Liouville family on every compact interval [a, b] ⊆ I; i.e., for every α, β∈ [a, b] , there exists an M-affine function f such that f(a) = α and f(b) = β. The validity of the converse statement is explored in this paper and several consequences are derived from this study. New characterizations of quasilinear means and the solution to Eq. (1) under suitable conditions are among the more important ones.