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

Abstract

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 :YY 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.

Keywords

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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 functionh on the non-metric spaceY, $h:Y\to Y$ 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 ${x}^{*}$ inX (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 spaceX and non-metric topological spaceY. We say that the limit of g at a point $x\in X$ is the point $y\in Y$ if for all neighborhoodsN ofy in Y, there exists a neighborhoodM 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 mapf 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 $g\left({M}_{0}\{M}_{1}\right)\subset {N}_{0}$ and ${x}_{0}\in {M}_{0}\{M}_{1}$, and also a neighborhood ${M}_{2}\subset {M}_{1}$ of ${x}^{*}$ containing only the sequence ${\left\{{x}_{i}\right\}}_{i=2}^{\infty }$, such that $g\left({M}_{1}\{M}_{2}\right)\subset {N}_{0}$, where ${x}_{1}\in {M}_{1}\{M}_{2}$. This process of creating neighborhoods ${M}_{k}$ of ${x}^{*}$ can continue such that each ${M}_{k}$ will contain only the corresponding sequence ${\left\{{x}_{i}\right\}}_{i=k}^{\infty }$, ${x}_{i-1}\in {M}_{i-1}\{M}_{i}$, $g\left({M}_{i-1}\{M}_{i}\right)\subset {N}_{0}$, 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 ${x}^{*}$ wheni 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 ${y}^{*}\in Y$ asx 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 ${y}_{i}\in g\left({x}_{i}\right)$, ${x}_{i}\in {M}_{i}\{M}_{i+1}$, and therefore ${y}_{i}\in {N}_{i}\{N}_{i+1}$.

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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