On Coincidence Points of Mappings Between Partially Ordered Sets

Abstract

A coincidence point theorem for mappings between partially ordered sets is obtained. This result is compared with some known coincidence point theorems and fixed point theorems.

1 Introduction

Given nonempty sets X, Y and mappings $\psi ,\varphi : X \rightarrow Y,$ a point $x \in X$ is called a coincidence point of $\psi$ and $\varphi$ if

$$ \psi (x)=\varphi (x). $$

In this paper, we derive sufficient conditions for the existence of coincidence points for the case when X and Y are partially ordered sets.

The coincidence point problem for mappings between partially ordered sets was investigated in the papers [1,2,3,4]. Here we obtain a more general coincidence point existence condition than one in [1, 3]. We also show that the Caristi fixed point theorem (see, for example, [5]) follows from the results of this paper.

2 Preliminaries

Recall that a relation $\preceq$ is called a partial order on X if it is reflexive (i.e., $x \preceq x$ for all $x\in X$), antisymmetric, (i.e., $x_1 \preceq x_2$ and $x_2 \preceq x_1$ imply $x_1=x_2$), and transitive, (i.e., $x_1 \preceq x_2$ and $x_2 \preceq x_3$ imply $x_1 \preceq x_3$). The set X with a partial order $\preceq$ is called a partially ordered set (or poset) and is denoted by $(X,\preceq ).$

Let $(X,\preceq )$ be a partially ordered set. A subset $S \subset X$ is called a chain if any two elements $x_1,x_2 \in S$ are comparable (i.e., either $x_1\preceq x_2$ or $x_2\preceq x_1$). A point $x \in X$ is called a lower bound of a set $A\subset X$ if $x \preceq a$ for every $a \in A.$ A lower bound $\bar{x} \in X$ of A is called the infimum of A,  and is denoted by $\inf A,$ if $x \preceq \bar{x}$ for every lower bound x of A. A point $\bar{a}\in A$ is called a minimal point in the set A if there is no point $a \in A$ such that $a \prec \bar{a}.$

A subset $A\subset X$ is called orderly complete in X, if for any chain $S\subset A$ there exists $\inf S\in X$ and $\inf S\in A.$ If X is orderly complete in X,  then we say that $(X, \preceq )$ orderly complete. Hence, X is orderly complete if and only if any chain $S\subset X$ has an infimum.

Let $(Y,\preceq )$ be a partially ordered set. A mapping $\varphi :X \rightarrow Y$ is called isotone if for any $x_1,x_2 \in X$ the relation $x_1 \preceq x_2$ implies $\varphi (x_1) \preceq \varphi (x_2).$

For arbitrary $x \in X$ denote

$$ O_X(x)= \{ u \in X : \, u \preceq x\} . $$

A mapping $\psi : X \rightarrow Y$ is called orderly covering a set $W \subset Y$ if

$$ O_{Y}\bigl ( \psi (x) \bigr ) \cap W \subset \psi \bigl ( O_{X}(x) \bigr ) $$

for every $x \in X.$ In this case, we will also say that $\psi$ covers W.

The definition of covering was introduced in [1, 3]. The rest of the above mentioned notions are standard and can be found in [6]. For other definitions of covering an properties of covering mappings in various spaces see, for example, [7,8,9].

In [3], the following coincidence point theorem was obtained. Let the mappings $\psi ,\varphi : X\rightarrow Y$ and sets $U\subset X,$ $W\subset Y$ be given.

Denote by $\mathcal {S}(\psi ,\varphi ,U,W)$ the set of all chains $S \subset X$ such that

$$\begin{aligned} \begin{array}{cc} {\displaystyle S\subset U, \quad \psi (S)\subset W, \quad \psi (x)\succeq \varphi (x) \quad \forall \, x\in S,} \\ {\displaystyle \psi (x_1)\preceq \varphi (x_2) \quad \forall \,x_1, \,x_2\in S: \,\,\, x_1 \prec x_2.} \end{array} \end{aligned}$$

(1)

Theorem 1

([3, Theorem 1]) Given a point $x_0 \in X$ satisfying the relation $\psi (x_0) \succeq \varphi (x_0),$ assume that

• (a) $\varphi$ is isotone;

• (b) $\psi$ orderly covers the set $W:= \varphi (O_X(x_0));$

• (c) for any chain $S\in \mathcal {S}= \mathcal {S}(\psi ,\varphi ,O_X(x_0),W)$ there exists a lower bound $u \in X$ of S such that $\psi (u)\succeq \varphi (u).$

Then, there exists $\xi \in X$ such that $\psi (\xi )=\varphi (\xi )$ and $\xi \preceq x_0.$ Moreover, the set $\{x\in O_X(x_0):\,\psi (x)=\varphi (x)\}$ has a minimal element.

3 Coincidence Point Theorem

Let $(X,\preceq ),$ $(Y,\preceq )$ be partially ordered sets, mappings $\psi ,\varphi : X\rightarrow Y$ be given. Denote

$$\begin{aligned} \mathcal {C}(\varphi ,\psi ):=\{x\in X: \, \varphi (x)\preceq \psi (x)\}. \end{aligned}$$

(2)

Theorem 2

Assume that

• (d) $\forall \, x\in X: \quad \varphi (x)\prec \psi (x) \quad \exists \, x'\in X: \quad x' \prec x, \quad \varphi (x') \preceq \psi (x').$

• (e) every chain $S\in \mathcal {C}(\varphi ,\psi )$ has a lower bound $u\in X$ such that $\varphi (u) \preceq \psi (u).$

Then, for every $x_0\in \mathcal {C}(\varphi ,\psi )$ there exists $\xi \in X$ such that $\psi (\xi )=\varphi (\xi )$ and $\xi \preceq x_0,$ and moreover the set $\{x \in O_X(x_0):\,\psi (x)=\varphi (x)\}$ has a minimal point.

Proof

Take an arbitrary $x_0\in \mathcal {C}(\varphi ,\psi ).$ The Hausdorff maximal principle implies that there exists a maximal (in the partially ordered set $(\mathcal {C}(\varphi ,\psi ),\preceq )$) chain S that contains $x_0.$ Assumption (e) implies that there exists a lower bound $\xi \in \mathcal {C}(\varphi ,\psi )$ of S. Since S is a maximal chain, we have $\xi =\inf S.$

Let us show that $\xi$ is the desired point. Consider the contrary: $\varphi (\xi )\prec \psi (\xi ).$ Then, there exists $\xi '\prec \xi$ such that $\varphi (\xi ')\preceq \psi (\xi ).$ Hence, $\xi ' \in \mathcal {C}(\varphi ,\psi ).$ Moreover, $\xi '\prec \xi \preceq x$ for all $x\in S.$ Thus, the chain S is a proper subset of the chain $S\cup \{\xi '\}.$ This contradicts to the maximality of the chain S. This contradiction implies that $\psi (\xi )=\varphi (\xi ).$ Inequality $\xi \preceq x_0$ follows from the relations $\xi =\inf S,$ $x_0\in S.$

Let us show that $\xi$ is a minimal point of the set $\{\xi \in O_X(x_0):\,\psi (x)=\varphi (x)\}.$ Consider the contrary: there exists $\xi '\in X$ such that $\xi '\prec \xi$ and $\varphi (\xi ) = \psi (\xi ).$ Then, $\xi ' \in \mathcal {C}(\varphi ,\psi )$ and $\xi '\prec \xi \preceq x$ for all $x\in S.$ Thus, the chain S is a proper subset of the chain $S\cup \{\xi '\}.$ This contradicts to the maximality of the chain S. This contradiction implies that $\xi$ is a minimal point of the set $\{\xi \in O_X(x_0):\,\psi (x)=\varphi (x)\}.$ $\square$

The concept of a fixed point is a partial case of coincidence point. Indeed, given a mapping $\varphi :X\rightarrow X,$ a fixed point $\xi \in X$ of $\varphi$ is a coincidence point of $\varphi$ and the identity map. Let us formulate a simple assertion on the fixed point existence that directly follows from Theorem 2.

Corollary 1

Let $(X,\preceq )$ be orderly complete. Given a mapping $\varphi : X\rightarrow X,$ assume that $\varphi (x)\preceq x$ for every $x\in X.$ Then, for every $x_0\in X$ there exists $\xi \in X$ such that $\xi =\varphi (\xi ),$ $\xi \preceq x_0.$ Moreover, the set $\{x \in O_X(x_0):\,x=\varphi (x)\}$ has a minimal point.

4 Discussion

Let us show that Theorem 1 follows from Theorem 2.

Let the assumptions of Theorem 1 hold. Define a partial order $\unlhd$ in X as follows: $x_1\lhd x_2$ $\Leftrightarrow$ $x_1 \prec x_2,$ $\psi (x_1) \preceq \varphi (x_2),$ $\psi (x_1)\in W.$ Show that the assumptions of Theorem 2 hold for mappings $\psi ,\varphi$ and the partial order $\unlhd$ in X and $\preceq$ in Y.

Take an arbitrary chain $S\subset \mathcal {C}(\varphi ,\psi )$ with respect to partial order $\unlhd .$ Then, $S\in \mathcal {S}(\psi ,\varphi ,O_X(x_0),W).$ Thus, assumption (c) implies that (e) holds. Let us verify (d). Take an arbitrary $x\in X$ such that $\varphi (x)\prec \psi (x).$ Assumption (b) implies that

$$ \varphi (x) \in O_Y(\psi (x)) \subset \psi (O_X(x)). $$

Hence, there exists $x'\in X$ such that $x'\prec x$ and $\psi (x')= \varphi (x).$ This equality and assumption (a) imply $\varphi (x')\preceq \varphi (x)=\psi (x').$ By definition of the relation $\unlhd$ we obtain $x' \unlhd x.$ Thus, (d) holds. So, we have shown that Theorem 1 follows from Theorem 2.

In [3], it was proved that some known fixed point theorems including the Knaster–Tarski theorem (see, for example, [10, Sect. 2.1]) and the Birkhoff–Tarski theorem (see, for example, [6, p. 266]) follow from Theorem 1. Hence, these assertions follow also from Theorem 2.

Let us now consider the fixed point problem and coincidence point problem for mappings between metric spaces. In [3], it was shown that the coincidence point theorem for mappings between metric spaces [7, Theorem 1] and some similar results follow from Theorem 1. Hence, these assertions as well as Banach contraction mapping principle and some of its generalizations follow from Theorem 2. Let us show that one more result on fixed points in metric spaces can be deduced from Theorem 2.

Recall the Caristi fixed point theorem. Let $(X,\rho )$ be a metric space, $\varphi :X\rightarrow X$ and $U:X\rightarrow \mathbb {R}_+$ be given.

Theorem 3

(see [5]) Assume that the space $(X,\rho )$ is complete, the function U is lower semicontinuous, the mapping $\varphi$ satisfies the relation

$$\begin{aligned} \rho (x,\varphi (x))\le U(x)-U(\varphi (x)) \quad \forall \, x\in X. \end{aligned}$$

(3)

Then, there exists $\xi \in X$ such that $\xi =\varphi (\xi ).$

Let us deduce this proposition from Theorem 2. Set

$$ P:=\{(x,r)\in X\times \mathbb {R}_+: \, r\ge U(x)\}. $$

Since U is lower semicontinuous, the set $P\subset X\times \mathbb {R}_+$ is closed. Define a binary relation $\preceq$ on $X\times \mathbb {R}_+$ assuming

$$ (x_1,r_1)\preceq (x_2,r_2) \, \Leftrightarrow \, \rho (x_1,x_2)\le r_2-r_1. $$

This relation is a partial order, the partially ordered set $(P,\preceq )$ is orderly complete (see [3, Lemma 3]) (this construction was introduced in papers [11, 12] and became a useful tool for reducing some problems in metric spaces and normed spaces to problems in partially ordered sets). Define a mapping $\omega : P \rightarrow P$ by formula

$$ \omega (x,r):= \bigl (\varphi (x),U(\varphi (x))\bigr ), \quad (x,r)\in P. $$

The mapping $\omega$ satisfies all the assumptions of Corollary 1. Indeed, $(P,\preceq )$ is orderly complete and $\omega (x,r)=\bigl (\varphi (x),U(\varphi (x))\bigr ) \preceq (x,U(x))\preceq (x,r)$ in virtue of (3) and the definition of the relation $\preceq .$ So, Corollary 1 implies that there exists $(\xi ,r)\in P$ such that $\omega (\xi ,r)=(\xi ,r).$ Hence, $\xi$ is a fixed point of $\omega .$

We have shown that the Caristi fixed point theorem follow from Theorem 2. The introduced coincidence point theorem can also be applied to various problems including control problems, ordinary differential equations and optimization problems. An examples of application of a coincidence point theorems and the concept of covering to control problems and ordinary differential equations can be found in [13,14,15,16]. For application of close order-theoretic results in optimization see [17, 18].