On the stability of the multi-dimensional Euler-Lagrange functional equation. (English) Zbl 1141.39310

For the sake of simple exposition we discuss the result of the author in a very particular case. One considers for \(f: \mathbb{R}\to\mathbb{R}\) the functional equation \(f((x+y)/2)+f((x-y)/2)={1\over2}[f(x)+f(y)]\) (equivalent to \(f(x+y)+f(x-y)=2\,[f(x)+f(y)]\), a quadratic functional equation). One has the following stability theorem. If \(g:\mathbb{R}\to\mathbb{R}\) is a function for which there exists a constant \(c\geq0\) such that the functional inequality \(|g((x+y)/2)+g((x-y)/2)-{1\over2}[g(x)+g(y)]|\leq c\) holds for all \(x,y\in\mathbb{R}\), then \(\lim_{n\to\infty}2^{-2n}g(2^nx)\) exists and the function limit \(f(x)\) is the unique function satisfying the quadratic functional equation such that \(|g(x)-f(x)|\leq c\) and \(f(x)=2^{-2n}f(2^nx)\). In this paper the author performs a wide generalization of the above result by considering the quadratic functional equation with \(p\) independent variables \(p\in\mathbb{R}\) for functions \(f: X\to Y\) where \(X\) is a normed linear space and \(Y\) is “a real complete normed linear space”.


39B22 Functional equations for real functions
39B62 Functional inequalities, including subadditivity, convexity, etc.