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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. $\Vert f(x+ y)+ f(x- y)- 2[f(x)+ f(y)\Vert\le C\Vert x\Vert\sp a\Vert y\Vert\sp b$ for any $x,y\in X$ with some constants $C\ge 0$, $a$ and $b$ such that $0\le a+ b<2$). The author proved that for any approximately Euler-Lagrange mapping $f$ there is a unique nonlinear Euler-Lagrange mapping $N$ such that $$\Vert f(x)- N(x)\Vert\le C(4- 2\sp{a+b})\sp{-1}\Vert x\Vert\sp{a+b}\quad\forall x\in X.$${}.

46G05Derivatives, etc. (functional analysis)