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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. f(x+y)+f(x-y)-2[f(x)+f(y)Cx a y b for any x,yX with some constants C0, a and b such that 0a+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)C(4-2 a+b ) -1 x a+b xX·

.


MSC:
46G05Derivatives, etc. (functional analysis)