New Necessary Conditions for a Fixed-Point of Maps in Non-Metric Spaces ()

Ivan Raykov^{}

Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, AR, USA.

**DOI: **10.4236/apm.2022.1210043
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Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, AR, USA.

Our purpose is to 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 function *h* on the non-metric space *Y*, *h* :*Y* →*Y* and present necessary conditions for a fixed point of this map on this map on *Y*. Therefore, this gives an opportunity to take a best conclusion in some sense, when non-metrizable matter is under consideration.

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Raykov, I. (2022) New Necessary Conditions for a Fixed-Point of Maps in Non-Metric Spaces. *Advances in Pure Mathematics*, **12**, 561-564. doi: 10.4236/apm.2022.1210043.

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 function*h* on the non-metric space*Y*,
$h:Y\to Y$ 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
$f:X\to X$ a contraction map of *X *onto *X*.We suppose that *Y *is a bounded closed non-metric space and
$g:X\to Y$ 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
$T:X\to X$ is called a contraction map on *X* if there exists
$q\in \left[0,1\right)$ such that

$d\left(T\left(x\right),T\left(y\right)\right)\le qd\left(x,y\right)$

for all $x,y\in X$ (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
$F:X\to B\left(X\right)$ a contraction map. (*B*(*X*) denotes the family of all nonempty closed bounded (compact) subsets of *X*.) Then there exists
$x\in X$ such that
$x\in F\left(x\right)$.

Definition 2. Attracting Fixed Points

An *attracting fixed point *of a function *f *is a fixed point
${x}_{0}$ of *f *such that for any value of *x *in the domain that is close enough to
${x}_{0}$, the iterated function sequence

$x,f\left(x\right),f\left(f\left(x\right)\right),f\left(f\left(f\left(x\right)\right)\right),\cdots $

converges to ${x}_{0}$ (ref. [9]).

Theorem 2. Banach Fixed Point Theorem.

Let (*X*, *d*) be a non-empty complete metric space with a contraction mapping
$T:X\to X$. Then T admits a unique fixed-point
${x}^{*}$ in*X* (*i.e*.
$T\left({x}^{*}\right)={x}^{*}$ ). Furthermore,
${x}^{*}$ can be found as follows: start with an arbitrary element
${x}_{0}\in X$ and define a sequence
${\left\{{x}_{n}\right\}}_{n\in N}$ by
${x}_{n}=T\left({x}_{n-1}\right)$ for
$n\ge 1$. Then
$\underset{n\to \infty}{\mathrm{lim}}{x}_{n}={x}^{*}$ (ref. [3], ref. [4], ref. [5], ref. [6], ref. [7], ref. [8], ref. [9]).

Definition 3. Let
$g:X\to Y$ be a function between a metric topological space*X* and non-metric topological space*Y*. We say that the limit of *g* at a point
$x\in X$ is the point
$y\in Y$ if for all neighborhoods*N* of*y* in *Y*, there exists a neighborhood*M* of *x *such that
$g\left(M\right)\subset N$ (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
$f:X\to X$.

Let *Y* is a bounded closed non-metric topological space.

We suppose also that the map:

$g:X\to Y$ exists and satisfies Definition 3.

Then we can construct a fixed-point of map in *Y*,*
$h:Y\to Y$ *.

Proof. If
${x}^{*}\in X$ is a fixed-point for *f* (*i.e*.
$f\left({x}^{*}\right)={x}^{*}$ ),*
$I\subset X$ *is a neighborhood close enough of
${x}^{*}$. Let
${x}_{0}\in I$ close enough to
${x}^{*}$ and we suppose that that the contracting map*f* will satisfy Banach Fixed Point Theorem and the iterated function sequence

${x}_{0},f\left({x}_{0}\right),f\left(f\left({x}_{0}\right)\right),f\left(f\left(f\left({x}_{0}\right)\right)\right),\cdots $

will satisfy Definition 2 and will converge to
${x}^{*}$. Therefore
${x}^{*}$ is an attracting fixed point of *f*. Let us denote
${x}_{1}\in f\left({x}_{0}\right)$,
${x}_{2}\in f\left({x}_{1}\right)=f\left(f\left({x}_{0}\right)\right)$,
${x}_{3}\in f\left({x}_{2}\right)=f\left(f\left(f\left({x}_{0}\right)\right)\right)$, and so on, or
${x}_{i+1}\in f\left({x}_{i}\right),i=0,1,2,3,\cdots $. Hence we created a sequence
$\left\{{x}_{i}\right\}$ such that
$\underset{i\to \infty}{\mathrm{lim}}{x}_{i}={x}^{*}$ and
$f\left({x}^{*}\right)={x}^{*}$.

We suppose now that a function
$g:X\to Y$ exists and satisfies Definition 3 and the limit of
$g\left(x\right)$ at the point
${x}^{*}\in X$ is the point
${y}^{*}\in Y$. According to Definition 3, a corresponding neighborhood
${M}_{0}$ of
${x}^{*}$ to a neighborhood
${N}_{0}\subset Y$ of
${y}^{*}\in Y$,
$g\left({M}_{0}\right)\subset {N}_{0}$, can be chosen such that it will contain the sequence
${\left\{{x}_{i}\right\}}_{i=0}^{\infty}$. We can find also a neighborhood
${M}_{1}\subset {M}_{0}$ of
${x}^{*}$ containing only the sequence
${\left\{{x}_{i}\right\}}_{i=1}^{\infty}$, such that
$g\left({M}_{0}\backslash {M}_{1}\right)\subset {N}_{0}$ and
${x}_{0}\in {M}_{0}\backslash {M}_{1}$, and also a neighborhood
${M}_{2}\subset {M}_{1}$ of
${x}^{*}$ containing only the sequence
${\left\{{x}_{i}\right\}}_{i=2}^{\infty}$, such that
$g\left({M}_{1}\backslash {M}_{2}\right)\subset {N}_{0}$, where
${x}_{1}\in {M}_{1}\backslash {M}_{2}$. This process of creating neighborhoods
${M}_{k}$ of
${x}^{*}$ can continue such that each
${M}_{k}$ will contain only the corresponding sequence
${\left\{{x}_{i}\right\}}_{i=k}^{\infty}$,
${x}_{i-1}\in {M}_{i-1}\backslash {M}_{i}$,
$g\left({M}_{i-1}\backslash {M}_{i}\right)\subset {N}_{0}$, and so on. We created a sequence
$\left\{{M}_{i}\right\}$ of neighborhoods of
${x}^{*}$. According to their construction neighborhoods
${M}_{i}$ are closer and closer to
${x}^{*}$ when*i* is larger and larger.

A correspondent sequence of neighborhoods $\left\{{N}_{i}\right\}$ of ${y}^{*}\in Y$ can be created also such that $g\left({M}_{i}\right)\subset {N}_{i}$.

We can choose
${N}_{i+1}\subset {N}_{i}$ according to Definition 3, because by construction
${M}_{i+1}\subset {M}_{i}$ and *g*(*x*) has the limit the
${y}^{*}\in Y$ at the point
${x}^{*}\in X$, and therefore
$g\left({M}_{i+1}\right)\subset g\left({M}_{i}\right)$.

Therefore, we can choose a sequence of neighborhoods
$\left\{{N}_{i}\right\}$ of
${y}^{*}\in Y$ such that
$g\left({M}_{i}\right)\subset {N}_{i}$. Because the function *g*(*x*) has a limit
${y}^{*}\in Y$ as*x* approaches
${x}^{*}\in X$ then
${N}_{i}$ from the correspondent sequence of neighborhoods
$\left\{{N}_{i}\right\}$ becomes smaller and smaller and closer to
${y}^{*}\in Y$. By construction
${y}_{i}\in g\left({x}_{i}\right)$,
${x}_{i}\in {M}_{i}\backslash {M}_{i+1}$, and therefore
${y}_{i}\in {N}_{i}\backslash {N}_{i+1}$.

It follows from Definition 3 that:

$\underset{{x}_{i}\to {x}^{*}}{\mathrm{lim}}g\left({x}_{i}\right)=g\left({x}^{*}\right)={y}^{*}=\underset{{x}_{i}\to {x}^{*}}{\mathrm{lim}}{y}_{i}={y}^{*}$. It means that when ${N}^{*}$ is the only

point ${y}^{*}$ then ${M}^{*}$ will be only the point ${x}^{*}$ and then $g\left({x}^{*}\right)={y}^{*}$.

Therefore, by using the sequence $\left\{{y}_{i}\right\}$, we can introduce the function $h:Y\to Y$, where ${y}_{0},h\left({y}_{0}\right),h\left(h\left({y}_{0}\right)\right),h\left(h\left(h\left({y}_{0}\right)\right)\right),\cdots $.

If we denote ${y}_{1}\in h\left({y}_{0}\right)$, ${y}_{2}\in h\left({y}_{1}\right)=h\left(h\left({y}_{0}\right)\right)$, ${y}_{3}\in h\left({y}_{2}\right)=h\left(h\left(h\left({y}_{0}\right)\right)\right)$, and so on, or ${y}_{i+1}\in h\left({y}_{i}\right),i=0,1,2,3,\cdots $, for which $h\left({y}_{i}\right)\to {y}^{*}$. Therefore the iterated function sequence $\left\{h\left({y}_{i}\right)\right\}$ will have a fixed point ${y}^{*}$, or $h\left({y}^{*}\right)={y}^{*}$, if ${N}^{*}$ contains the only point ${y}^{*}$.

Because every sequence $\left\{{y}_{i}\right\}$ constructed by this way will have the same limit ${y}^{*}$ then ${y}^{*}$ will be the fixed point of the so constructed function $h\left(y\right)$, $h\left({y}^{*}\right)={y}^{*}$. □

Acknowledgements

We express our gratitude to Professor Alexander Arhangel’skii from OU-Athens for creating the problem.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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