ISSN 1064�5624, Doklady Mathematics, 2013, Vol. 88, No. 3, pp. 727–729. © Pleiades Publishing, Ltd., 2013.Original Russian Text © A.V. Arutyunov, E.S. Zhukovskiy, S.E. Zhukovskiy, 2013, published in Doklady Akademii Nauk, 2013, Vol. 453, No. 6, pp. 595–598.

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Let X and Y be nonempty sets and Ψ, Φ: X Y beset�valued mappings. A point ξ ∈ X is said to be a coin�cidence point of Ψ and Φ if Ψ(ξ) ∩ Φ(ξ) ≠ . In thecase of metric spaces X and Y, the existence and prop�erties of coincidence points of set�valued mappingswere fairly completely investigated in [1–3]. Below, westudy the existence of coincidence points of set�valuedmappings in the case when X and Y are partiallyordered spaces. More specifically, the results of [4] areextended to set�valued mappings.

Used in this paper, the concepts from the theory ofpartially ordered spaces (comparable elements, chain,lower bound, greatest lower bound, minimal element,etc.) are not defined, since they are well known (see,e.g., [5, 6]). The order relation in partially orderedspaces is denoted by , while (X, ) is used to desig�nate such a space.

A set A ⊂ X is said to be orderly σ�complete in X if,for any nonincreasing sequence {xn}, there existsinf{xn} ∈ X and inf{xn} ∈ A.

Along with the partially ordered space (X, ), weconsider a partially ordered space (Y, ). Given an arbi�trary set U ⊂ X and mappings Φ: X Y and ϕ: X → Y,we set Φ(U) = (u) and ϕ(U) = ϕ(u)}.

A mapping ϕ: X → Y is called monotone if, for anyx1, x2 ∈ X, the relation x1 x2 implies ϕ(x1) ϕ(x2).By analogy with “single�valued” mappings, a set�val�ued mapping Φ: X Y is said to be monotone if, forany x1, x2 ∈ X and y2 ∈ Φ(x2), the relation x1 x2

implies the existence of y1 ∈ Φ(x1) such that y1 y2.

→→

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→→

Φu U∈

∪ {u U∈

∪

� �

→→

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The single�valued selector of a set�valued mappingΦ: X Y is a single�valued mapping ϕ: X → Y suchthat ϕ(x) ∈ Φ(x) ∀x ∈ X.

A mapping Φ: X Y is said to be orderly σ�closedwith respect to a set W if

for any nonincreasing sequences {xn} ⊂ X and {yn} ⊂ Wsuch that yn ∈ Φ(xn) ∀n and there exist inf{xn} ∈ X andinf{yn} ∈ W.

For an arbitrary set U ⊂ X and a point x ∈ X, define

Definition 1. We will say that a set�valued mapping Ψ:X Y orderly covers a set W ⊂ Y if, for any x ∈ X,

or, equivalently,

(1)

If W = Y, then the set�valued mapping Ψ that orderlycovers W is called orderly covering.

The property of ordered covering is preserved if theset W is decreased; i.e., if Ψ: X → Y orderly covers W

and ⊂ W, then Ψ orderly covers . Note also that,given any x ∈ X, if Ψ(x) is a singleton (i.e., Ψ is single�valued), then the above definition turns into Defini�tion 1 in [4].

Recall that the inverse of a set�valued surjectivemapping Ψ: X Y is defined as

The following result is a direct consequence of the def�inition of an ordered covering.

→→

→→

inf yn{ } Φ inf xn{ }( )∈

OX x( ) ξ X: ξ x∈{ },=

OX U( ) OX u( ).u U∈

∪=

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→→

OY Ψ x( )( ) W∩ Ψ OX x( )( )⊂

x0∀ X, y0∀ Ψ x0( ), y∀ W: y y0∈ ∈ ∈

x∃ X: x x0, y Ψ x( ).∈ ∈�

�

W̃ W̃

→→

Ψ1–: Y X,

Ψ1– y( ) x X: y Ψ x( )∈ ∈{ } y∀ Y.∈=

→→

Coincidence Points of Set�Valued Mappingsin Partially Ordered Spaces

A. V. Arutyunova, E. S. Zhukovskiyb, and S. E. Zhukovskiyc

Presented by Academician A.B. Kurzhanskii May 23, 2013

Received May 30, 2013

DOI: 10.1134/S106456241306029X

a Peoples’ Friendship University of Russia, ul. Miklukho�Maklaya 6, Moscow, 119198 Russiab Derzhavin Tambov State University,Tambov, ul. Internatsional’naya 13, 392000 Russiac Universidade do Porto, Porto, Portugale�mail: arutun@orc.ru, zhukovskys@mail.ru,s�e�zhuk@yandex.ru

MATHEMATICS

728

DOKLADY MATHEMATICS Vol. 88 No. 3 2013

ARUTYUNOV et al.

Proposition 1. Let Ψ: X Y be a surjective set�val�ued mapping. Then Ψ is orderly covering if and only ifthe inverse mapping Ψ–1: Y → X is monotone.

Now we pass to theorems on coincidence points oftwo set�valued mappings.

Let Ψ, Φ: X Y, ψ, ϕ: X → Y be mappings, whileU ⊂ X and V ⊂ Y be sets. Denote by �(ψ, ϕ, U, V) theset of all chains S ⊂ X such that

(2)

Note that, if S ∈ �(ψ, ϕ, U, W), then the restrictionsof ψ and ϕ to S are monotone.

Theorem 1. Let

(3)

and let the following conditions hold:(A) The mapping Φ is monotone.(B) The mapping Ψ orderly covers the set W =

Φ(OX(x0)).(C) For any selectors ψ, ϕ: X → Y of Ψ and Φ,

respectively, and for any chain S ∈ �(ψ, ϕ, OX(x0), W),the chains S, ψ(S), and ϕ(S) have, respectively lowerbounds u ∈ X, y ∈ Ψ(u), and z ∈ Φ(u) such that y z.

Then the set�valued mappings Ψ and Φ have a coin�cidence point in the set OX(x0); i.e.,

In the special case where X = Y and Ψ: X X is anidentity mapping, Theorem 1 turns into Smithson’sfixed�point principle [7].

It should also be noted that Theorem 1 cannot bededuced from Theorem 1 in [4] by applying the latterto some selectors ψ and ϕ of the set�valued mappingsΨ and Φ, respectively. The cause is that the monotoneset�valued mapping Φ may not have a monotoneselector. Below is a relevant example.

Example 1. Let X = Y = �+, Φ: X Y, Φ(x) = {x}∀x ∈ X\{0}, and Φ(0) = (0, ∞). The set�valued map�ping Φ is monotone. Let ϕ: X → Y be an arbitraryselector of Φ. Then ϕ(0) > 0 and ϕ(x) = x ∀x ∈ X\{0}.Obviously, ϕ is not monotone.

In contrast to similar results for single�valued map�pings in [4], Theorem 1 does not guarantee the exist�ence of a minimal element in the set of coincidencepoints of Ψ and Φ. Let us give an example of set�valuedmappings for which all the conditions of Theorem 1hold, but the set of their coincidence points does nothave a minimal element.

Example 2. Let X = {a, x0, x1, x2, …} be a countablelinearly ordered space such that a � … � x2 � x1 � x0,and let Y = {w, h, y0, y1, y2, …} be a countable space inwhich any two distinct elements are not comparable.The set�valued mappings Ψ, Φ: X Y are defined as

→→

→→

S U, ψ S( ) V, ψ x( ) ϕ x( ) x∀ S,∈⊂ ⊂

ψ x1( ) ϕ x2( ) x1∀ x2, S: x1 � x2.∈

�

�

x0∃ X, y0∃ Ψ x0( ),∈ ∈

z0∃ Φ x0( ): y0 z0∈ �

�

ξ∃ X: Ψ ξ( ) Φ ξ( )∩ and ξ x0.≠∈ ��

→→

→→

→→

n = 0, 1, … . Let us show that all the assumptions ofTheorem 1 hold for these mappings.

Condition (3) is satisfied, since y0 ∈ Ψ(x0) and y0 ∈Φ(x0). The mapping Φ is monotone, since, for any

x ∈ X and ∈ OX(x), we have Φ(x) ⊂ Φ( ). The map�ping Ψ orderly covers Y, since relation (1) holds: for

any x ∈ X, y ∈ Ψ(x), and ∈ Y, if y ≠ , then the ele�

ments and y are not comparable.

We verify assumption (C). Let ψ, ϕ: X → Y be selec�tors of Ψ and Φ, respectively, and S be an arbitrary chainfrom �(ψ, ϕ, OX(x0), ϕ(OX(x0))). Since ψ(x) ϕ(x)∀x ∈ X and distinct elements of Y are not comparable,we have ψ(x) = ϕ(x) ∀x ∈ S. Consequently, S ⊂ {x0, x1,x2, …} and ψ(xj) = ϕ(xj) ∀j: xj ∈ S. Therefore, it followsfrom (2) that S consists of a single point xn. Therefore,assumption (C) holds for u = xn and y = z = yn.

For the set�valued mappings Ψ and Φ under con�sideration, the set of coincidence points is the chainΞ = {x0, x1, x2, …}, which does not have a minimal ele�ment.

Some results guaranteeing the existence of minimalfixed points of set�valued mappings were proved, forexample, in [6, Section 2.2.1]. The following theoremgives the conditions under which set�valued mappingshave minimal coincidence points.

Theorem 2. Let all the assumptions of Theorem 1hold and, additionally,

(4)

Then the set Ξ = {ξ ∈ OX(x0): Ψ(ξ) ∩ Φ(ξ) ≠ } ofcoincidence points of the mappings Ψ and Φ has a mini�mal element.

Under certain additional assumptions (to those inTheorem 2), we can guarantee the existence of notonly a minimal coincidence point but also a least coin�cidence point.

Given arbitrary x1, x2 ∈ X, let ΩX(x1, x2) denote theset of all lower bounds for the two�element set {x1, x2}.The similar notation ΩY is introduced for Y. Note thatΩX(x1, x2) = OX(x1) ∩ OX(x2) ∀x1, x2 ∈ X.

Theorem 3. Let assumptions (3), (A), and (C) hold,

and, moreover, for any x1, x2 ∈ OX(x0), there exists apoint ω(x1, x2) ∈ ΩX(x1, x2) such that

(D1) for any y1 ∈ Ψ(x1), y2 ∈ Ψ(x2), and y ∈ W ∩ΩY(y1, y2), where W = Φ(OX(x0)), there exists x ω(x1, x2)such that y ∈ Ψ(x);

(D2) for any y1 ∈ Φ(x1) and y2 ∈ Φ(x2), there existsy ∈ Φ(ω(x1, x2)) such that y ∈ ΩY(y1, y2).

Ψ xn( ) yn{ }, Ψ a( ) h{ };= =

Φ xn( ) y0 y1 … yn, , ,{ }, Φ a( ) w y0 y1 …, , ,{ },= =

x x

y y

z

�

x1 x2∀ OX x0( ), y1∀ Φ x1( ), y2∀ Φ x2( )∈ ∈ ∈

y∃ Φ x1( ): y y1, y y2.∈

�

��

�

ΩX x1 x2,( ) x1∀ x2, OX x0( ),∈≠ �

�

DOKLADY MATHEMATICS Vol. 88 No. 3 2013

COINCIDENCE POINTS OF SET�VALUED MAPPINGS 729

Then the set of coincidence points {ξ ∈ OX(x0): Ψ(ξ) ∩Φ(ξ) ≠ } is not empty and there is a least element in it.

Note that conditions (4) and (D2) hold automati�cally if the monotone mapping Φ is single�valued.Therefore, Theorems 2 and 3 imply Theorems 1 and 3in [4], which are their respective single�valued ver�sions.

By modifying the assumptions of Theorem 1, acoincidence point of set�valued mappings can be con�structed by an iteration method (by analogy with thesuccessive approximations considered in [4] for “sin�gle�valued” mappings).

Let Ψ and Φ: X Y be set�valued mappings suchthat condition (3) holds, Φ is monotone, and Ψ is anordered covering of the set W = Φ(OX(x0)).

Step 1. Since the set�valued mapping Ψ orderlycovers W, there exists an element x1 ∈ X for which z0 ∈Ψ(x1), x1 x0. Since Φ is monotone, there is z1 ∈Φ(x1) such that z1 z0.

Step 2. There exists an element x2 ∈ X for whichz1 ∈ Ψ(x2), x2 x1. Since Φ is monotone, there is z2 ∈Φ(x2) such that z2 z1.

Continuing these constructions, by induction, weobtain sequences {xn} ⊂ X and {zn} ⊂ Y such that

(5)

The graph of Φ: X → Y is denote by gph(Φ), whilethe order in the Cartesian product X × Y is define bythe relation

Theorem 4. Suppose that the set�valued mappings Ψ,Φ: X Y satisfy condition (3) and at least one of the setsgph(Ψ) and gph(Φ) is orderly σ�complete in X × Y.Assume that

(E) the set�valued mapping Φ is monotone andorderly σ�closed with respect to the set V = Ψ(OX(x0));

(F) the set�valued mapping Ψ orderly covers theset W = Φ(OX(x0)) and is orderly σ�closed with respectto W.

Then the sequence in (5) has a greatest lower boundξ = inf{xn} that is a coincidence point of Ψ and Φ.

Theorems 1 and 4 do not follow from each other.While all the assumptions of Theorem 1 are satisfied,assumption (E) may be violated. While all the assump�tions of Theorem 4 hold, assumption (C) may be vio�lated.

From the results obtained in this work, we candeduce the theorems in [1, 2] on coincidence points of

covering and Lipschitz set�valued mappings in metricspaces and, specifically, classical fixed�point theoremsfor contraction mappings. This can be done by apply�ing the order relation proposed by DeMarr, Bishop,and Phelps for sets X × �+ and Y × �+, where X and Yare given metric spaces and �+ = [0, ∞) (see [8, Sec�tion 7.5]).

Suppose that x0 ∈ X is a point and Ψn, Φn: X Yare sequences of set�valued mappings. The followingresult is a consequence of Theorem 1.

Proposition 2. Assume that each of the mappings Ψnand Φn satisfies assumptions (A)–(C) and, additionally,there are sequences of points {xn} ⊂ X and {yn}, {zn} ⊂ Ysuch that

for any n. Then, for any n, the mappings ξn and Ψn havea coincidence point Φn such that ξn x0.

By applying the technique from [8, Section 7.5](see above), this proposition yields a correspondingstability theorem for coincidence points in metricspaces (see Theorem 3 in [3]).

ACKNOWLEDGMENTS

This work was supported by the Russian Founda�tion for Basic Research (project nos. 12�01�00427, 12�01�91150) and by a Marie Curie International Incom�ing Fellowship within the 7th European CommunityFramework Programme.

REFERENCES

1. A. V. Arutyunov, Dokl. Math. 76, 665–668 (2007).

2. A. Arutyunov, E. Avakov, B. Gel’man, A. Dmitruk, andV. Obukhovskii, J. Fixed Points Theory Appl. 5 (1),105–127 (2009).

3. A. V. Arutyunov, Math. Notes 86, 153–158 (2009).

4. A. V. Arutyunov, E. S. Zhukovskyi, and S. E. Zhu�kovskyi, Dokl. Math. 88, 475–478 (2013).

5. A. N. Kolmogorov and S. V. Fomin, Elements of theTheory of Functions and Functional Analysis (Nauka,Moscow, 1981; Dover, New York, 1999).

6. S. Carl and S. Heikkilä, Fixed Point Theory in OrderedSets and Applications: From Differential and IntegralEquations to Game Theory (Springer, Berlin, 2011).

7. R. E. Smithson, Proc. Am. Math. Soc. 27, 304–310(1971).

8. F. Clarke, Optimization and Nonsmooth Analysis (Wiley,New York, 1983; Nauka, Moscow, 1988).

Translated by I. Ruzanova

�

→→

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zn 1– Ψ xn( ) Φ xn 1–( ),∩∈

xn xn 1– , zn zn 1– n.∀� �

x1 y1,( ) x2 y2,( )

x1 y1,( )⇔ x2 y2,( ) or x1 � x2 and y1 y2( ).=

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xn x0, yn zn, yn Ψ xn( ), zn Φ xn( )∈ ∈� �

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