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 → 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 → X a contraction map of X onto X.We suppose that Y is a bounded closed non-metric space and g : X → 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 → X is called a contraction map on X if there exists q ∈ [ 0 , 1 ) such that

d ( T ( x ) , T ( y ) ) ≤ q d ( x , y )

for all x , y ∈ 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 → B ( X ) a contraction map. (B(X) denotes the family of all nonempty closed bounded (compact) subsets of X.) Then there exists x ∈ X such that x ∈ F ( x ) .

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 ( x ) , f ( f ( x ) ) , f ( f ( f ( x ) ) ) , ⋯

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 → X . Then T admits a unique fixed-point x * inX (i.e. T ( x * ) = x * ). Furthermore, x * can be found as follows: start with an arbitrary element x 0 ∈ X and define a sequence { x n } n ∈ N by x n = T ( x n − 1 ) for n ≥ 1 . Then lim n → ∞ x n = x * (ref. [3], ref. [4], ref. [5], ref. [6], ref. [7], ref. [8], ref. [9]).

Definition 3. Let g : X → 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 ∈ X is the point y ∈ Y if for all neighborhoodsN ofy in Y, there exists a neighborhoodM of x such that g ( M ) ⊂ N (ref. [10]).

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 → X .

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

We suppose also that the map:

g : X → Y exists and satisfies Definition 3.

Then we can construct a fixed-point of map in Y, h : Y → Y .

Proof. If x * ∈ X is a fixed-point for f (i.e. f ( x * ) = x * ), I ⊂ X is a neighborhood close enough of x * . Let x 0 ∈ 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 ( x 0 ) , f ( f ( x 0 ) ) , f ( f ( f ( x 0 ) ) ) , ⋯

will satisfy Definition 2 and will converge to x * . Therefore x * is an attracting fixed point of f. Let us denote x 1 ∈ f ( x 0 ) , x 2 ∈ f ( x 1 ) = f ( f ( x 0 ) ) , x 3 ∈ f ( x 2 ) = f ( f ( f ( x 0 ) ) ) , and so on, or x i + 1 ∈ f ( x i ) , i = 0 , 1 , 2 , 3 , ⋯ . Hence we created a sequence { x i } such that lim i → ∞ x i = x * and f ( x * ) = x * .

We suppose now that a function g : X → Y exists and satisfies Definition 3 and the limit of g ( x ) at the point x * ∈ X is the point y * ∈ Y . According to Definition 3, a corresponding neighborhood M 0 of x * to a neighborhood N 0 ⊂ Y of y * ∈ Y , g ( M 0 ) ⊂ N 0 , can be chosen such that it will contain the sequence { x i } i = 0 ∞ . We can find also a neighborhood M 1 ⊂ M 0 of x * containing only the sequence { x i } i = 1 ∞ , such that g ( M 0 \ M 1 ) ⊂ N 0 and x 0 ∈ M 0 \ M 1 , and also a neighborhood M 2 ⊂ M 1 of x * containing only the sequence { x i } i = 2 ∞ , such that g ( M 1 \ M 2 ) ⊂ N 0 , where x 1 ∈ 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 { x i } i = k ∞ , x i − 1 ∈ M i − 1 \ M i , g ( M i − 1 \ M i ) ⊂ N 0 , and so on. We created a sequence { M i } 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 { N i } of y * ∈ Y can be created also such that g ( M i ) ⊂ N i .

We can choose N i + 1 ⊂ N i according to Definition 3, because by construction M i + 1 ⊂ M i and g(x) has the limit the y * ∈ Y at the point x * ∈ X , and therefore g ( M i + 1 ) ⊂ g ( M i ) .

Therefore, we can choose a sequence of neighborhoods { N i } of y * ∈ Y such that g ( M i ) ⊂ N i . Because the function g(x) has a limit y * ∈ Y asx approaches x * ∈ X then N i from the correspondent sequence of neighborhoods { N i } becomes smaller and smaller and closer to y * ∈ Y . By construction y i ∈ g ( x i ) , x i ∈ M i \ M i + 1 , and therefore y i ∈ N i \ N i + 1 .

It follows from Definition 3 that:

lim x i → x * g ( x i ) = g ( x * ) = y * = lim x i → x * y i = y * . It means that when N * is the only

point y * then M * will be only the point x * and then g ( x * ) = y * .

Therefore, by using the sequence { y i } , we can introduce the function h : Y → Y , where y 0 , h ( y 0 ) , h ( h ( y 0 ) ) , h ( h ( h ( y 0 ) ) ) , ⋯ .

If we denote y 1 ∈ h ( y 0 ) , y 2 ∈ h ( y 1 ) = h ( h ( y 0 ) ) , y 3 ∈ h ( y 2 ) = h ( h ( h ( y 0 ) ) ) , and so on, or y i + 1 ∈ h ( y i ) , i = 0 , 1 , 2 , 3 , ⋯ , for which h ( y i ) → y * . Therefore the iterated function sequence { h ( y i ) } will have a fixed point y * , or h ( y * ) = y * , if N * contains the only point y * .

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

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

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

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