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                

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 :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.

Keywords

Topological Space, Compact Metric Space, Fixed Point, Contraction Map, Non-Metric Space

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 $\underset{n\to \infty }{\lim }{x}_n=x^{*}$ (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 $\underset{i\to \infty }{\lim }{x}_i=x^{*}$ 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 $x^{*}$ 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 $y_i\in g\left(x_i\right)$, $x_i\in M_i\backslash M_{i+1}$, and therefore $y_i\in N_i\backslash N_{i+1}$.

It follows from Definition 3 that:

$\underset{x_i\to x^{*}}{\lim }g\left(x_i\right)=g\left(x^{*}\right)=y^{*}=\underset{x_i\to x^{*}}{\lim }{y}_i=y^{*}$. 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.

References

[1]	Martin, J.R. and Weis, W. (1984) Fixed Point Sets of Metric and Non-Metric Spaces. Transactions of the American Mathematical Society, 284, 337-353. https://doi.org/10.1090/S0002-9947-1984-0742428-1
[2]	Weinshall, D., Jacobs, D. and Gdalyahu, Y. (1998) Classification in Non-Metric Spaces. NIPS, December.
[3]	Arhangel’skii, A.V. and Fedorchuck, V.V. (1990) The Basic Concepts and Constructions of General Topology. In: Arhangel’skii, A.V. and Pontrjagin, L.S., Eds., General Topology I, Encyclopedia of the Mathematical Sciences, Springer, Berlin, 1-90. https://doi.org/10.1007/978-3-642-61265-7_1
[4]	Arhangel’skii, A.V. and Ponomarev, V.I. (1984) Fundamentals of General Topology: Problems and Exercises. Mathematics and Its Applications, Vol. 13, Springer, Dordrecht, Boston.
[5]	Arhangel’skii, A.V. (1991) Topological Function Spaces. Kluwer Academic Publishers, Dordrecht, Boston.
[6]	Arhangel’skii, A.V. and Tkachenko, M. (2008) Topological Groups and Related Structures. Atlantic Press, Dordrecht. https://doi.org/10.2991/978-94-91216-35-0
[7]	Dugundji, J. (1966) Topology. Allyn and Bacon, Inc., Boston.
[8]	Davis, S.W. (2005) Topology. McGraw-Hill, Boston.
[9]	Gorniewicz, L. (2008) Topological Fixed Point Theory of Multivalued Mappings. Springer Verlag, Berlin.
[10]	Eckman, E. (2012) Connect the Shapes Crochet Motifs: Creative Techniques for Joining Motifs of All Shapes. Storey Publishing, North Adams, Massachusetts.