Necessary Conditions for a Fixed Point of Maps in Non-Metric Spaces ()
1. Introduction
Let X denote a complete (or compact) metric space and also 
a continuous map of X onto Y, where Y is a bounded closed topological normal space with a countable base.
What must be the conditions, in the means of the meric space X, such that the continuous map 
from Y onto Y will have a fixed point?
We suppose that (see [1-3]):
the continuous map 
(not one to one) and the continuous map 
are given and the continuous inverse map of f, 
exists.
We remind that Banach contraction principle for multivalued maps is valid and also the next Theorem, proved by H. Covitz and S. B. Nadler Jr. (see [4]).
Theorem 1. Let 
be a complete metric space and 
a conraction map (
denotes the family of all nonempty closed bounded (compact) subsets of X). Then there exists 
such that
.
2. Main Result
We consider now the next theorem:
Theorem 2. Let 
denote a complete (or compact) metric space 
and also:
a continuous map of 
onto
, where 
is a bounded closed topological normal space with a countable base.
We suppose also that the maps:
is continuous and onto.
and
exists and it is continuous.
If 
is a point from 
and if we suppose also that
.
Then if the rest terms of the sequence 
are received from 
and the rest of the terms of the sequence 
are determined by 
and if also 
is a Cauchy sequence and therefore convergent to a fixed point 
in
, then the sequence 
will be also convergent to a fixed point 
in
.
Proof. Let 
is a point from 
and let us suppose also that 
and let the rest terms of the sequence 
are received from 
.
Let also the rest of the terms of the sequence 
are determined by 
.
If 
is a Cauchy sequence then for any 
there exists an integer
, such that for all integers i and k, 
and 
will be satisfied the inequality
and therefore the Cauchy sequence 
will be convergent with a fixed point 
in X, and because X is complete (or compact), i.e.
Since 
and 
and 
is a continuous map and 
is continuous map onto the closed and bounded space
, and also 
and
, therefore the sequence 
will be also convergent with a fixed point 
in
, such that 
and
, i.e.
Q.E.D.
3. Acknowledgements
We express our gratitude to Professor Alexander Arhangelskii from OU-Athens for creating the problem and to Professor Jonathan Poritz and Professor Frank Zizza from CSU-Pueblo for the precious help for solving this problem, and to Professor Darren Funk-Neubauer and Professor Bruce Lundberg for correcting some grammatical and spelling errors.