View source for Strong extremum

<!--
s0905501.png
$#A+1 = 23 n = 0
$#C+1 = 23 : ~/encyclopedia/old_files/data/S090/S.0900550 Strong extremum
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A minimal or maximal value  $  J ( \widetilde{y}  ) $
taken by a functional  $  J ( y) $
at a curve  $  \widetilde{y}  ( x) $, 
$  x _ {1} \leq  x \leq  x _ {2} $, 
for which one of the inequalities

$$ 
J ( \widetilde{y}  )  \leq   J ( y) \ \ 
\textrm{ or } \ \ 
J ( \widetilde{y}  )  \geq   J ( y)
$$

holds for all comparison curves  $  y ( x) $
in an  $  \epsilon $-neighbourhood of  $  y ( x) $. 
The curves  $  \widetilde{y}  ( x) $
and  $  y ( x) $
must satisfy given boundary conditions.

Since maximization of  $  J ( y) $
is equivalent to minimization of  $  - J ( y) $, 
instead of a strong maximum one often discusses only a strong minimum. The term  "strong"  emphasizes that only the condition of being  $  \epsilon $-near to  $  \widetilde{y}  ( x) $
is imposed on the comparison curves  $  y ( x) $:

$$ 
| y ( x) - \widetilde{y}  ( x) |  \leq   \epsilon
$$

on the whole interval  $  [ x _ {1} , x _ {2} ] $, 
whereas the derivatives of the curves  $  y ( x) $
and  $  \widetilde{y}  ( x) $
may differ as  "strongly"  as desired.

However, the very definition of a strong extremum is of a relative rather than absolute nature, since it gives an extremum not on the whole class of admissible comparison curves  $  y ( x) $
for which  $  J ( y) $
makes sense, but only relative to the subset of all admissible comparison curves belonging to the  $  \epsilon $-neighbourhood of  $  \widetilde{y}  ( x) $. 
However, for brevity, the term  "relative"  is often omitted and one speaks of a strong extremum, meaning a strong relative extremum (see also [[Strong relative minimum|Strong relative minimum]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,   L.A. Lyusternik,   "A course in variational calculus" , Moscow-Leningrad  (1950)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Cesari,   "Optimization - Theory and applications" , Springer  (1983)</TD></TR></table>