ISSN 1064�5624, Doklady Mathematics, 2014, Vol. 89, No. 3, pp. 346–348. © Pleiades Publishing, Ltd., 2014.Original Russian Text © A.V. Arutyunov, S.E. Zhukovskiy, 2014, published in Doklady Akademii Nauk, 2014, Vol. 456, No. 5, pp. 514–517.

346

The following natural question arises in the study offixed points of contraction mappings acting in metricspaces. Given two mappings Φ1 and Φ2 from a com�plete metric space (X, ρX) with metric ρX into itself thatsatisfy the Lipschitz condition with the same constantβ ∈ [0, 1), how far do their respective fixed points x1and x2 differ? They satisfy the estimate

(1)

This is a special case of Lim’s lemma [1] (see below),which provides a similar estimate for set�valued map�pings.

A natural generalization of a fixed point of a map�ping is a coincidence point of two mappings. Specifi�cally, in addition to the metric space (X, ρX), considera metric space (Y, ρY) with metric ρY, and let Ψ, Φ: X → Ybe given mappings. A coincidence point of ψ and φ is apoint x ∈ X such that Ψ(x) = Φ(x).

If Ψ and Φ are set�valued mappings, i.e., they takeeach point x ∈ X to nonempty closed subsets Ψ(x) ⊂ Yand Φ(x) ⊂ Y, respectively, which is denoted by Ψ:X Y and Φ: X Y, then their coincidence point xis defined by the relation

The set of coincidence points of Ψ and Φ is denoted byCoin(Ψ, Φ). Sufficient conditions for the existence ofcoincidence points were obtained in [2].

The solution of the coincidence point problem is apair of points (ξ, η) ∈ X × Y such that

Let gph(Ψ) = {(x, y) ∈ X × Y: y ∈ Ψ(x)} be the graphof the set�valued mapping Ψ. Then, the set Γ(Ψ, Φ) ofsolutions (ξ, η) to the coincidence point problemobviously coincides with the intersection of the corre�sponding graphs; i.e.,

ρX x1 x2,( )1

1 β–���������� ρX Φ1 x( ) Φ2 x( ),( ).

x X∈

sup≤

→→

→→

Ψ x( ) Φ x( )∩ .≠ �

η Ψ ξ( ) Φ ξ( ).∩∈

Moreover, ξ ∈ Coin(Ψ, Φ) if and only if there existsη ∈ Y such that (ξ, η) ∈ Γ(Ψ, Φ).

The goal of this paper is to estimate the distancebetween the solutions of the coincidence point prob�lem for two pairs of mappings, providing a significantstrengthening and generalization of Lim’s lemma. Forthis purpose, we introduce the necessary notation andrecall some concepts from set�valued analysis.

Let BX(x, r) = {ξ ∈ X: ρX(x, ξ) ≤ r} denote the closedball of radius r centered at the point x in X; similarnotation is used in Y. For an arbitrary subset M ⊂ Y,define BY(M, r) = (y, r). In the Cartesian prod�

uct X × Y of metric spaces, the metric is defined by theformula ρ = ρX + ρY. Thus, if gph(Ψ) and gph(Φ) areboth closed, then the solution set Γ(Ψ, Φ) of the coin�cidence point problem is closed as well.

Given nonempty sets M, N ⊂ Y, the distancebetween them is defined as

and the excess from M to N is defined by the relation

The Hausdorff distance h between M and N is defined as

Obviously, h+(M, N) ≤ h(M, N) for all M, N.The excess h+(M, N) and the Hausdorff distance

h(M, N) can both take the infinite value +∞. TheHausdorff distance h is symmetric, whereas h+ is not(i.e., it may happen that h+(M, N) ≠ h+(N, M)),although it satisfies the triangle inequality. On thespace of nonempty closed subsets of a metric space,the function h defines a generalized (i.e., taking thevalue +∞) Hausdorff metric h.

The triangle inequality can be violated for dist, butdist and h+ satisfy the following inequality, which willbe used below. Specifically, for any sets E, M, N ⊂ Y,

Γ Ψ Φ,( ) gph Ψ( ) gph Φ( ).∩=

BYy M∈

∪

dist M N,( ) inf ρY y1 y2,( ): y1 M, y2 N∈ ∈{ },=

h+ M N,( ) inf r 0: M> BY N r,( )⊂{ }=

= sup dist y N,( ): y M∈{ }.

h M N,( ) max h+ M N,( ) h+ N M,( ),{ }.=

Perturbation of Solutions of the Coincidence Point Problemfor Two Mappings

A. V. Arutyunov and S. E. ZhukovskiyPresented by Academician V.A. Il’in December 5, 2013

Received December 16, 2013

DOI: 10.1134/S1064562414030247

Peoples' Friendship University of Russia, ul. Miklukho�Maklaya 6, Moscow, 119198 Russiae�mail: arutun@orc.ru, s�e�zhuk@yandex.ru

MATHEMATICS

DOKLADY MATHEMATICS Vol. 89 No. 3 2014

PERTURBATION OF SOLUTIONS OF THE COINCIDENCE POINT PROBLEM 347

(2)Let us prove (2). For an arbitrary ε > 0, there exist

e ∈ E and n ∈ N such that ρY(n, e) ≤ dist(E, N) + ε. Bydefinition, there exists m ∈ M such that ρY(m, n) ≤h+(N, M) + ε. By the triangle inequality, ρY(m, e) ≤ρY(m, n) + ρY(n, e). Moreover, dist(M, E) ≤ ρY(m, e).From this, we obtain dist(E, M) ≤ dist(E, N) +h+(N, M) + 2ε. Since ε > 0 is arbitrary, this inequalityyields (2).

Definition 1. Given α > 0, a set�valued mapping Ψ:X Y is said to be α�covering if

(3)

A set�valued mapping Φ is called β�Lipschitz if itsatisfies the Lipschitz condition with respect to theHausdorff metrics h with a Lipschitz constant β ≥ 0.

Given α > β ≥ 0, let �α, β denote the set of ordered

pairs (Ψ, Φ) of set�valued mappings Ψ, Φ: X Y thatsatisfy the following basic conditions:

(i) Ψ is α�covering and its graph is closed.(ii) Φ is β�Lipschitz.(iii) At least one of the graphs gph(Ψ) and gph(Φ)

is complete.Let us present Theorem 3 from [3]. Given (x, y) ∈

X × Y, a subset Γ ⊂ X × Y, and a two�dimensional vec�tor A = (AX, AY) ∈ �2, the notation

means that, for any ε > 0, there exist (ξ, η) ∈ Γ suchthat

Theorem 1. Let (Ψ, Φ) ∈ �α, β. Then the set Γ(Ψ, Φ)is nonempty. Moreover, for arbitrary x ∈ X and y ∈ Y,

, (4)where

Let us derive the main result of this paper fromTheorem 1.

Given arbitrary sets Γ, ⊂ X × Y and a vector A =(AX, AY) ∈ �2, the notation

means that, for arbitrary (x, y) ∈ , we have Dist((x,y), Γ) ≤ A. This concept is a vector generalization of

the fact that the excess h+( , Γ) is at most a.

Let Ψ, , Φ, : X Y be arbitrary set�valuedmappings. For x ∈ X, define

dist E M,( ) dist E N,( ) h+ N M,( ).+≤

→→

Ψ BX x r,( )( ) BY Ψ x( ) αr,( ) r∀ 0, x∀ X.∈≥⊇

→→

Dist x y,( ) Γ,( ) A≤

ρX x ξ,( ) AX ε, ρY y η,( ) AY ε.+≤+≤

Dist x y,( ) Γ Ψ Φ,( ),( ) A x y,( )≤

A x y,( ) α β–( )1– r1 r2+ βr1 αr2+,( ),=

r1 dist y Ψ x( ),( ), r2 dist y Φ x( ),( ).= =

Γ̃

H+Γ̃ Γ,( ) A≤

Γ̃

Γ̃

Ψ̃ Φ̃ →→

AX x( )h+

Ψ̃ x( ) Ψ x( ),( ) h+Φ̃ x( ) Φ x( ),( )+

α β–����������������������������������������������������������������������,=

AY x( )βh+

Ψ̃ x( ) Ψ x( ),( ) αh+Φ̃ x( ) Φ x( ),( )+

α β–����������������������������������������������������������������������������,=

Given a set Ξ, let supξ ∈ ΞA(ξ) denote the set of two�dimensional vectors Λ such that Λ ≥ A(ξ) for any ξ ∈ Ξ,where the inequality is understood in the coordinate�wise sense.

Theorem 2. Let (Ψ, Φ) ∈ �α, β. Then

(5)

(6)

Proof. It is similar to the proof of Theorem 2 in [4].

Fix ( , ) ∈ Γ( , ) and consider an arbitrary ε > 0.By Theorem 1 as applied to the mappings Ψ and Φ at

x = and y = , there exist (ξ, η) ∈ Γ(Ψ, Φ) such that

(7)

(8)

Applying inequality (2) yields

since ∈ ( ) and, hence, dist( , ( )) = 0.Similarly, we have

Combining this with (7) and (8), we obtain

(9)

However, as was noted above, ξ ∈ Coin(Ψ, Φ).Therefore, since ε > 0 is arbitrary, the first inequality

implies that dist( , Coin(Ψ, Φ)) ≤ AX( ). Since ∈

Coin( , ) is arbitrary, we obtain (6). From (9),since ε > 0 is arbitrary, we derive

From this, since ( , ) ∈ Γ( , ) are arbitrary, weobtain (5). The theorem is proved.

The following corollaries are straightforward con�sequences of Theorem 3.

Corollary 1. Let (Ψ, Φ) ∈ �α, β. Then

Note that the function h+ is heavily used in fixedpoint theorems for set�valued contraction mappings(see [5, Section 5E]).

A x( ) AX x( ) AY x( ),( ).=

H+Γ Ψ̃ Φ̃,( ) Γ Ψ Φ,( ),( ) Λ≤

Λ∀ A ξ̃( ),ξ̃ Coin Ψ̃ Φ̃,( )∈

sup∈

h+Coin Ψ̃ Φ̃,( ) Coin Ψ Φ,( ),( ) AX ξ̃( ).

ξ̃ Coin Ψ̃ Φ̃,( )∈

sup≤

ξ̃ η̃ Ψ̃ Φ̃

ξ̃ η̃

ρX ξ̃ ξ,( )dist η̃ Ψ ξ̃( ),( ) dist η̃ Φ ξ̃( ),( )+

α β–������������������������������������������������������������ ε,+≤

ρY η̃ η,( )βdist η̃ Ψ ξ̃( ),( ) αdist η̃ Φ ξ̃( ),( )+

α β–������������������������������������������������������������������ ε.+≤

dist η̃ Ψ ξ̃( ),( ) dist η̃ Ψ ξ̃( ),( ) h+Ψ̃ ξ̃( ) Ψ ξ̃( ),( )+≤

= h+Ψ̃ ξ̃( ) Ψ ξ̃( ),( ),

η̃ Ψ̃ ξ̃ η̃ Ψ̃ ξ̃

dist η̃ Φ̃ ξ̃( ),( ) h+Φ̃ ξ̃( ) Φ ξ̃( ),( ).≤

ρX ξ̃ ξ,( ) AX ξ̃( ) ε, ρY η̃ η,( ) AY ξ̃( ) ε.+≤+≤

ξ̃ ξ̃ ξ̃

Ψ̃ Φ̃

Dist ξ̃ η̃,( ) Γ Ψ Φ,( ),( ) A ξ̃( ).≤

ξ̃ η̃ Ψ̃ Φ̃

h+Coin Ψ̃ Φ̃,( ) Coin Ψ Φ,( ),( )

≤h+

Ψ̃ x( ) Ψ x( ),( ) h+Φ̃ x( ) Φ x( ),( )+

α β–���������������������������������������������������������������������� .

x X∈

sup

348

DOKLADY MATHEMATICS Vol. 89 No. 3 2014

ARUTYUNOV, ZHUKOVSKIY

Corollary 2. Let (Ψ, Φ), ( , ) ∈ �α, β. Then

In other words, the excess h+ in the correspondingestimate can be replaced with the Hausdorff distance h.

Specifically, if X = Y and Ψ(x) ≡ (x) ≡ x, this assertionyields Lim’s lemma (see [1]).

Corollary 3. Given a pair (Ψ, Φ) and a sequence ofpairs {(Ψn, Φn)} ⊂ �α, β, if for some x ∈ Coin(Ψ, Φ),

as n → ∞, then, for any sequence {δn} of positive num�bers converging to zero, there exists a sequence {xn} suchthat

.

Moreover, for all n,

This assertion is a strengthening of Theorem 2 in[4], since h+ ≤ h.

Let us return to Lim’s lemma. For some pairs of

contraction mappings (Φ, ), the supremum on theright�hand side of estimate (1) can take the value +∞;as a result, this estimate becomes not informative.

Indeed, if Φ, : � → �, Φ(x) = 2–1x, (x) ≡ 0, β =2–1, then inequality (1) takes the form 0 ≤ +∞. Thesame is true for Theorem 3 and its corollaries. Below,we present a partial solution of this problem.

Let U ⊂ X and V ⊂ Y be given sets. For x ∈ X, define

Theorem 3. Let (Ψ, Φ) ∈ �α, β, and let sets U ⊂ Xand V ⊂ Y be such that

Then

(10)

(11)

Ψ̃ Φ̃

h Coin Ψ̃ Φ̃,( ) Coin Ψ Φ,( ),( )

≤h Ψ̃ x( ) Ψ x( ),( ) h Φ̃ x( ) Φ x( ),( )+

α β–�����������������������������������������������������������������.

x X∈

sup

Ψ̃

h+Ψ x( ) Ψn x( ),( ) 0, h+

Φ x( ) Φn x( ),( ) 0,→ →

Ψn xn( ) Φn xn( )∩ , xn x→≠ �

ρX xn x,( )h+

Ψ x( ) Ψn x( ),( ) h+Φ x( ) Φn x( ),( )+

α β–������������������������������������������������������������������������� δn.+≤

Φ̃

Φ̃ Φ̃

AXV x( )

= h+

Ψ̃ x( ) V∩ Ψ x( ),( ) h+Φ̃ x( ) V∩ Φ x( ),( )+

α β–����������������������������������������������������������������������������������������,

AYV x( )

= βh+Ψ̃ x( ) V∩ Ψ x( ),( ) αh+

Φ̃ x( ) V∩ Φ x( ),( )+α β–

�����������������������������������������������������������������������������������������������,

AV x( ) AXV x( ) AY

V x( ),( ).=

Ψ̃ x( ) Φ̃ x( ) V∩ ∩ ≠

x∀ Coin Ψ̃ x( ) Φ̃ x( ),( ) U.∩∈

�

H+Γ Ψ̃ Φ̃,( ) U V×( )∩ Γ Ψ Φ,( ),( ) Λ≤

Λ∀ AV x( ),x U∈

sup∈

h+Coin Ψ̃ Φ̃,( ) U∩ Coin Ψ Φ,( ),( ) AX

V x( ).x U∈

sup≤

The proof of this theorem repeats the argumentused in the proof of Theorem 3.

Setting X = Y, Ψ(x) ≡ (x) ≡ {x}, α = 1, and U =V, we obtain the following generalization of Lim’slemma.

Corollary 4. Let X be a complete space; Φ, : X Xbe given mappings; and Φ be a contraction with a con�stant β < 1. Then

Here, Fix(Φ) ⊂ X denotes the set of fixed points of Φ.It is easy to see that, if U is bounded, then the supre�

mum on the right�hand side of this estimate is finite.The above results can be interpreted from the point of

view of stability theory. Specifically, let � be a given classof ordered pairs of set�valued mappings Ψ, Φ: X Y.

As is customary (see [3, 6–8]), the solution of thecoincidence point problem is called Ulam–Hyers sta�ble in the class � if there exists a two�dimensional pos�itive vector A such that, for any two pairs (Ψ1, Φ1),(Ψ2, Φ2) ∈ �, if

then

Thus, Theorem 3 means that, for any α > β ≥ 0, thesolution of the coincidence point problem is Ulam–Hyers stable in the class �α, β. In a similar manner, wecan interpret the other results and Lim’s lemma.

ACKNOWLEDGMENTS

This work was supported by the Russian Founda�tion for Basic Research (project no. 14�01�31185) andby grants from the Russian Science Foundation andthe Ministry of Education and Science of the RussianFederation.

REFERENCES1. T.�C. Lim, J. Math. Anal. Appl. 110, 436–441 (1985).2. A. V. Arutyunov, Dokl. Math. 76, 665–668 (2007).3. A. V. Arutyunov, Dokl. Math. 89, (2014).4. A. V. Arutyunov, Math. Notes 86, 153–158 (2009).5. A. L. Dontchev and R. T. Rockafellar, Implicit Func�

tions and Solution Mappings (Springer, Berlin, 2009).6. S. M. Ulam, A Collection of Mathematical Problems (Inter�

science, New York, 1960; Nauka, Moscow, 1964).7. D. H. Hyers, Proc. Nat. Acad. Sci. USA 27 (4), 222–

224 (1941).8. I. A. Rus, Fixed Points Theory 10 (2), 305–320 (2009).

Translated by I. Ruzanova

Ψ̃

Φ̃ →→

h+Fix Φ̃( ) U∩ Fix Φ( ),( )

h+Φ̃ x( ) U∩ Φ x( ),( )

1 �����������������������������������������

x U∈

sup .≤

→→

h Ψ1 x( ) Ψ2 x( ),( ) t,≤

h Φ1 x( ) Φ2 x( ),( ) t x∀ X,∈≤

H+Γ Ψ1 Φ1,( ) Γ Ψ2 Φ2,( ),( ) tA,≤

H+Γ Ψ2 Φ2,( ) Γ Ψ1 Φ1,( ),( ) tA.≤