New Necessary Conditions for a Fixed-Point of Maps in Non-Metric Spaces ()
1. Introduction
Classification in non-metric spaces is considered before (ref. [1]). Fixed point sets of non-metric spaces were also under interest (ref. [2]).
With this work, we introduce new necessary conditions for a fixed point of maps on non-metric spaces. We use a contraction map on a metric topological space and a lately published definition of limit of a function between the metric topological space and the non-metric topological space. Then we show that we can create a functionh on the non-metric spaceY,
and present necessary conditions for a fixed point of this map on Y.
For that purpose, we denote by X a compact metric topological space and
a contraction map of X onto X.We suppose that Y is a bounded closed non-metric space and
is a map from X to Y satisfying Definition 3.
We remind next basic definitions and theorems:
Definition 1. Contraction Mapping
Let (X, d) be a complete metric space. Then the map
is called a contraction map on X if there exists
such that
for all
(ref. [3], ref. [4], ref. [5], ref [6], ref. [7], ref. [8], ref. [9]).
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. (ref. [9]).
Theorem 1. Let (X, d) be a complete metric space and
a contraction map. (B(X) denotes the family of all nonempty closed bounded (compact) subsets of X.) Then there exists
such that
.
Definition 2. Attracting Fixed Points
An attracting fixed point of a function f is a fixed point
of f such that for any value of x in the domain that is close enough to
, the iterated function sequence
converges to
(ref. [9]).
Theorem 2. Banach Fixed Point Theorem.
Let (X, d) be a non-empty complete metric space with a contraction mapping
. Then T admits a unique fixed-point
inX (i.e.
). Furthermore,
can be found as follows: start with an arbitrary element
and define a sequence
by
for
. Then
(ref. [3], ref. [4], ref. [5], ref. [6], ref. [7], ref. [8], ref. [9]).
Definition 3. Let
be a function between a metric topological spaceX and non-metric topological spaceY. We say that the limit of g at a point
is the point
if for all neighborhoodsN ofy in Y, there exists a neighborhoodM of x such that
(ref. [10]).
2. Main Result
We consider now the next theorem:
Theorem 3. Let X denote a non-empty compact metric topological space with a contraction set-valued map
.
Let Y is a bounded closed non-metric topological space.
We suppose also that the map:
exists and satisfies Definition 3.
Then we can construct a fixed-point of map in Y,
.
Proof. If
is a fixed-point for f (i.e.
),
is a neighborhood close enough of
. Let
close enough to
and we suppose that that the contracting mapf will satisfy Banach Fixed Point Theorem and the iterated function sequence
will satisfy Definition 2 and will converge to
. Therefore
is an attracting fixed point of f. Let us denote
,
,
, and so on, or
. Hence we created a sequence
such that
and
.
We suppose now that a function
exists and satisfies Definition 3 and the limit of
at the point
is the point
. According to Definition 3, a corresponding neighborhood
of
to a neighborhood
of
,
, can be chosen such that it will contain the sequence
. We can find also a neighborhood
of
containing only the sequence
, such that
and
, and also a neighborhood
of
containing only the sequence
, such that
, where
. This process of creating neighborhoods
of
can continue such that each
will contain only the corresponding sequence
,
,
, and so on. We created a sequence
of neighborhoods of
. According to their construction neighborhoods
are closer and closer to
wheni is larger and larger.
A correspondent sequence of neighborhoods
of
can be created also such that
.
We can choose
according to Definition 3, because by construction
and g(x) has the limit the
at the point
, and therefore
.
Therefore, we can choose a sequence of neighborhoods
of
such that
. Because the function g(x) has a limit
asx approaches
then
from the correspondent sequence of neighborhoods
becomes smaller and smaller and closer to
. By construction
,
, and therefore
.
It follows from Definition 3 that:
. It means that when
is the only
point
then
will be only the point
and then
.
Therefore, by using the sequence
, we can introduce the function
, where
.
If we denote
,
,
, and so on, or
, for which
. Therefore the iterated function sequence
will have a fixed point
, or
, if
contains the only point
.
Because every sequence
constructed by this way will have the same limit
then
will be the fixed point of the so constructed function
,
. □
Acknowledgements
We express our gratitude to Professor Alexander Arhangel’skii from OU-Athens for creating the problem.