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Some xed point theorems. Suppose X is nonempty set and < is a complete linear ordering of X . Given x; y 2 X we write
 

Summary: Some xed point theorems.
Suppose X is nonempty set and < is a complete linear ordering of X . Given x; y 2 X we write
x  y
if x < y or x = y. Whenever a; b 2 X and a < b we let
(a; b) = fx 2 X : a < x < bg
and we assume that
(1) (a; b) 6= ;:
We let F be the set of f such that
(i) f : X ! X ;
(ii) if x; y 2 X , x  y then f(x)  f(y);
(iii) if x; y 2 X , x < y, w 2 X and f(x) < w < f(y) there is v 2 X such that
x < v < y and w = f(v):
We let G be the set of g such that
(iv) g : X ! X ;
(v) if x; y 2 X and x  y then g(y)  g(x);
(vi) if x; y 2 X , x < y, t 2 X and g(y) < w < g(x) there is v 2 X such that
x < v < y and w = f(v):
Theorem. Suppose S is a nonempty subset of X and f 2 F . If S has an upper bound then f [S] has an
upper bound and
f(sup S) = sup f [S]:

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics