On Approximate Solutions of Functional Equations in Vector Lattices

Abstract

We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra). The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equation $F(x+y)+F(x)+F(y)\ne \mathrm{0}\Rightarrow F(x+y)=F(x)+F(y)$ in Riesz spaces, the Cauchy equation with squares $F(x+y{)}^{\mathrm{2}}=(F(x)+F(y){)}^{\mathrm{2}}$ in $f$-algebras, and the quadratic functional equation $F(x+y)+F(x-y)=\mathrm{2}F(x)+\mathrm{2}F(y)$ in Riesz spaces.