On the stability of the Euler-Lagrange functional equation. (English) Zbl 0789.46036

Let \(X\) be a normed linear space, \(Y\) be a Banach space, and \(f\) and \(N\) be two mappings from \(X\) into \(Y\). We say \(N\) (resp. \(f\)) is a Euler- Lagrange mapping (resp. approximately Euler-Lagrange mapping) if and only if \[ N(x+ y)+ N(x- y)= 2[N(x)+ N(y)] \] (resp. \(\| f(x+ y)+ f(x- y)- 2[f(x)+ f(y)\|\leq C\| x\|^ a\| y\|^ b\) for any \(x,y\in X\) with some constants \(C\geq 0\), \(a\) and \(b\) such that \(0\leq a+ b<2\)).
The author proved that for any approximately Euler-Lagrange mapping \(f\) there is a unique nonlinear Euler-Lagrange mapping \(N\) such that \[ \| f(x)- N(x)\|\leq C(4- 2^{a+b})^{-1}\| x\|^{a+b}\quad\forall x\in X. \] {}.


46G05 Derivatives of functions in infinite-dimensional spaces