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.
![](https://www.scirp.org/html/3-5300241\ba19c5a2-83ce-4549-8bbd-e23f9fb90209.jpg)
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
![](https://www.scirp.org/html/3-5300241\bb2f937b-e988-4bd1-b04e-8c573d31670f.jpg)
and therefore the Cauchy sequence
will be convergent with a fixed point
in X, and because X is complete (or compact), i.e.
![](https://www.scirp.org/html/3-5300241\88322063-6f41-4bee-9286-2d2494b6651e.jpg)
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.
![](https://www.scirp.org/html/3-5300241\29240c5c-9c6d-4b04-bead-82bbdf9c0f7b.jpg)
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.