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 $\| f(x)- N(x)\|\leq C(4- 2^{a+b})^{-1}\| x\|^{a+b}\quad\forall x\in X.$ {}.

MSC:

 46G05 Derivatives of functions in infinite-dimensional spaces