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.
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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
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
A mapping \(\psi : X \rightarrow Y \) is called orderly covering a set \(W \subset Y\) if
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
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
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
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
Then, there exists \(\xi \in X\) such that \(\xi =\varphi (\xi ).\)
Let us deduce this proposition from Theorem 2. Set
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
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
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].
References
Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: On coincidence points of mappings in partially ordered spaces. Dokl. Math. 88(3), 710–713 (2013)
Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Coincidence points of set-valued mappings in partially ordered spaces. Dokl. Math. 88(3), 727–729 (2013)
Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Coincidence points principle for mappings in partially ordered spaces. Topol. Appl. 179, 13–33 (2015)
Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Coincidence points principle for set-valued mappings in partially ordered spaces. Topol. Appl. 201, 178–194 (2016)
Caristi, J.: Fixed point theorems for mappings satisfying the inwardness condition. T. Am. Math. Soc. 215, 241–251 (1976)
Lyusternik, L.A., Sobolev, V.I.: Brief Course in Functional Analysis. Vishaya Shkola, Moscow (1982) (in Russian)
Arutyunov, A.V.: Covering mappings in metric spaces and fixed points. Dokl. Math. 76(2), 665–668 (2007)
Arutyunov, A.V., Izmailov, A.F.: Directional stability theorem and directional metric regularity. Math. Oper. Res. 31(3), 526–543 (2006)
Arutyunov, A.V., Avakov, E.R., Izmailov, A.F.: Directional regularity and metric regularity. SIAM J. Optimiz. 18(3), 810–833 (2007)
Granas, A., Dugundji, D.: Fixed Point Theory. Springer, New York (2003)
DeMarr, R.: Partially ordered spaces and metric spaces. Amer. Math. Mon. 72(6), 628–631 (1965)
Bishop, E., Phelps, R.R.: The support functionals of a convex set. Proc. Symp. Pure Math. 7, 27–35 (1963)
Arutyunov, A.V., Zhukovskiy, S.E.: Existence of local solutions in constrained dynamic systems. Appl. Anal. 90(6), 889–898 (2011)
Zhukovskiy, E.S.: On ordered-covering mappings and implicit differential inequalities. Diff. Equ. 52(12), 1539–1556 (2016)
Arutyunov, A.V., Zhukovskii, E.S., Zhukovskii, S.E.: On the well-posedness of differential equations unsolved for the derivative. Diff. Equ. 47(11), 1541–1555 (2011)
Arutyunov, A.A.: On derivations associated with different algebraic structures in group algebras. Eurasian Math. J. 9(3), 8–13 (2018)
Arutyunov, A.V.: Second-order conditions in extremal problems. The abnormal points. T. Am. Math. Soc. 350(11), 4341–4365 (1998)
Arutyunov, A.V., Vinter, R.B.: A simple “finite approximations” proof of the Pontryagin maximum principle under reduced differentiability hypotheses. Set-Valued Anal. 12(1–2), 5–24 (2004)
Acknowledgements
The research was supported by RFBR grant (Project No. 19-01-00080). Theorem 2 was obtained under the support of the Russian Science Foundation (Project No. 17-11-01168).
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Zhukovskiy, S.E. (2020). On Coincidence Points of Mappings Between Partially Ordered Sets. In: Pinelas, S., Kim, A., Vlasov, V. (eds) Mathematical Analysis With Applications. CONCORD-90 2018. Springer Proceedings in Mathematics & Statistics, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-030-42176-2_5
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