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.

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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. ). Fixed point sets of non-metric spaces were also under interest (ref. ).

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. , ref. , ref. , ref , ref. , ref. , ref. ).

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. ).

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. ).

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. , ref. , ref. , ref. , ref. , ref. , ref. ).

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. ).

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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