Let be a normed linear space, be a Banach space, and and be two mappings from into . We say (resp. ) is a Euler- Lagrange mapping (resp. approximately Euler-Lagrange mapping) if and only if
(resp. for any with some constants , and such that ).
The author proved that for any approximately Euler-Lagrange mapping there is a unique nonlinear Euler-Lagrange mapping such that