## 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”.

### MSC:

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