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