Coincidence points in the cases of metric spaces and metric maps

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Coincidence points in the cases of metric spaces and metric maps

Thi HongVan Nguyena, B.A Pasynkovb, ∗

of Arutyunov’s theorem on coincidence points.

Obtaining the main results of the paper is based on the use of the class of almost exactly(α, β)-searchfunctionals that is wider than Fomenko’s class of(α, β)-search

f X ⊂ Y  ⊂ Y ,the corestriction corY  f of f to Y  isthe mappingof X to Y suchthat(cor Y  f )(x) = f x

forany x ∈ X.If Y = f X then cor f is usedinsteadof corf X f

* Corresponding author.

E-mail addresses:nguyenhongvan@mail.ru (T.H.V Nguyen), bpasynkov@gmail.com (B.A Pasynkov).

http://dx.doi.org/10.1016/j.topol.2015.12.026

0166-8641/© 2015 Published by Elsevier B.V.

In Sections 1and2,we consider theproblems of theexistenceand searchingof coincidencepoints andthe commonpreimageof aclosed subset(in particular,thecommonroot) inthecaseof afinite systemofmappingsofonemetricspace toanotherone.

In[7],T.N.Fomenko used(α, β)-search functionals(i.e.mappingsto[0,+∞))onmetricspacestosolvetheproblemsmentionedabove.An(α, β)-search functional ϕ on ametricspace X allows toobtain,forany

x ∈ X,afundamentalsequenceofpoints x0= x, x1, , x n , in X such that ϕ(xn)−−−−→ 0 n →∞ Undersomeadditional conditions (for example, if X is complete and ϕ is continuous), there exists ξ = lim

In Section 3, a fiberwise variants of Arutyunov’s theorem on coincidence points ([2], Theorem 1) areobtained.Theproofsofourtheoremsare basedontheuseofalmostexactly(α, β)-search functionals

Weconsider onlyone-valuedmappings

1 Searchfunctionalsonmetricspaces

Fixaspace(X, ρ).

Let F (X) be thesetofall(notnecessarilycontinuous)mappingsof X to itselfand CF (X) the setofallcontinuous mappingsof X to itself For A, B∈ F (X),set

d F (A, B) = sup {ρ(Ax, Bx): x ∈ X}.

For A ∈ F (X) and C ∈ CF (X),set F (X, A) ={B ∈ F (X): d F (A, B) < + ∞} and CF (X, A) ={D ∈

CF (X) : d F (C, D) < + ∞}.It isevidentthat:

for A, B ∈ F (X),either F (X, A) ∩ F (X, B) =∅ or F (X, A) = F (X, B) and this equalityis equivalent

to theinequality dF (A, B) < +∞;

for C, D ∈ CF (X),either CF (X, C) ∩ CF (X, D) = ∅ or CF (X, C) = CF (X, D) and this equality isequivalent totheinequality dF (C, D) < +∞;

forevery A ∈ F (X) (respectively, A ∈ CF (X)),thefunction dF isametricon F (X, A) (respectively, on

Recall thedefinitionof(α, β)-search functionalsgiven byT.N Fomenko in[7]

Further,letR+={x ∈ R: x  0} andN+={0} ∪ N.

Definition1.1 Afunctional ϕ : X → R+ iscalled(α, β)-search, α, β ∈ R, 0 < β < α, if

(*)forany x ∈ X,there existsapoint x = x  (x) ∈ X suchthat

ρ(x, x )ϕ(x)

α and ϕ(x

) β

Thefollowingassertionisevident

Proposition1.1 A functional ϕ : X → R+ is (α, β)-search, 0 < β < α, if and only if

(∗ A ) there exists a mapping A : X → X (i.e., A ∈ F (X)) such that for any x ∈ X,

ρ(x, Ax)ϕ(x)

α and ϕ(Ax) β

Weextendtheclass of(α, β)-search functionals inthefollowing way

Definition1.2 Afunctional ϕ : X → R+ iscalled almost exactly (α, β)-search for α, β ∈ R,0 < β < α, if(**)forany x ∈ X andevery δ > 0 there isapoint x = x  (x, δ) ∈ X suchthat

ρ(x, x )ϕ(x)

α + δ and ϕ(x

) β

Thenextassertionisalsoevident

Proposition 1.2 For a functional ϕ : X → R+ and α, β ∈ R, 0 < β < α, the following conditions are equivalent:

1 ϕ is almost exactly (α, β)-search;

2 for any δ > 0, there exists a mapping A δ ∈ F (X) such that for any x ∈ X,

inItem3)ofProposition 1.2,forall δ > 0 (respectively, forall k), onecantakethesame A ∈ F (X) instead

ofall Aδ (respectively, Ak)

Foranyfunctional ϕ(x) on X, let ϕ(x) = min(ϕ(x),1), x ∈ X.

Definition 1.3 For analmost exactly (α, β)-search functional ϕ : X → R+, anumber ε > 0, aconvergentseries Σ = ∞

k −1

· γσ k ϕ(x k−1),

(b k) ϕ(x k) β

α ϕ(x k−1)+



β α

k−1

· (ϕ0+ γσ), (b  k) ϕ(x k)



β α

k

· (ϕ0+ γ(σ1+ + σ k))



β α

k

· (ϕ0+ γσ),

k ∈ N.

Remark1.2.Itfollowsfrom(a k) and(b k) thatif ϕ(xk−1)= 0,then xl = x k−1 and ϕ(xl)= 0 forany  k.

Inparticular, itistruefor k− 1= 0,i.e.,if ϕ(x) = ϕ(x0)= 0,then xk = x and ϕ(xk)= 0 for all k∈ N+.Theorem 1.1 If a functional ϕ : X → R+ is almost exactly (α, β)-search, then for every convergent series

Σ= ∞

k=1

σ k with nonnegative terms and the sum σ > 0, any ε > 0 and any x ∈ X,

(0) there exists at least one (ϕ, Σ, x, ε)-generated sequence;

(1) any (ϕ, Σ, x, ε)-generated sequence x0= x, x1, , x k , is fundamental;

(2) for this sequence, ϕ(x k)→ 0 as k→ ∞;

(3) if there is the limit r(x) of this sequence, then

ρ(x k , r(x))



β α

(4) if x ∈ ϕ −1 (0), then the limit r(x) from (3) exists and r(x) = x.

Proof (0) Points of the required sequence x0 = x, x1, , x k , must be chosen in thefollowing way:if

ϕ(x k−1) then xk istakenso that(1.1δ)istruefor x = x k−1,x  = x k−1 andfor δ that is equalto theminimum ofthesecondsummandsintherightpartsof (a k) and(b k);if ϕ(xk−1)= 0,then xk = x k−1.(1) Let p, q∈ N+,p < q Then(see(a  k))

i (ϕ0+ γσ) 1

p

1− β α

=



β α

=



β α

Definition 1.4 For analmost exactly (α, β)-search functional ϕ : X → R+, anumber ε > 0, aconvergentseries Σ = ∞

Corollary1.1 If ϕ is an almost exactly (α, β)-search functional on X, then

(0) for any ε > 0 and any convergent series Σ = ∞

k=1

σ k with nonnegative terms and the sum σ > 0,

there exists a (ϕ, Σ, ε)-generated sequence of A (k) ∈ F (X), k∈ N+;

(5) if a sequence s ={A (k) ∈ F (X) : k ∈ N+} is (ϕ, Σ, ε)-generated and the functional ϕ is bounded, then

all A (k) are contained in the metric part F (X, id X ) of F (X) and s is fundamental with respect to d F Suppose now that a (ϕ, Σ, ε)-generated sequence A (k) ∈ F (X), k∈ N+, pointwise converges to a mapping

(8) if ϕ is locally bounded (for example, continuous), and all A (k) are continuous, then r is continuous;

if, additionally, r(X) = ϕ −1 (0) (for example, if the functional ϕ is continuous), then r is a retraction of X

onto ϕ −1 (0);

(9) if ϕ is bounded, then the sequence A (k) , k ∈ N+, converges uniformly on X to r.

Proof (0) Foreach x ∈ X,take(seepoint(0)ofTheorem 1.1)a(ϕ, Σ, x, ε)-generated sequence xk,k ∈ N+,andlet A(k) (x) = x k,x ∈ X, k∈ N+.Thesequence{A (k) : k ∈ N+} isrequired

(5) Supposethat ϕ(x)  C > 0 for any x ∈ X.Then (as intheproof of point (1)of Theorem 1.1)forany p, q∈ N+,p < q, andany x ∈ X,

ρ(A (p) (x), A (q) (x))



β α

p

C

α − β + ε



hold.Wehaveobtainedthefirstassertionofpoint(7).Thecontinuityofall A(k)andtheuniformconvergence

of A(k),k ∈ N,to r on O imply thecontinuityof r on O

(8) Itfollows frompoint (7)thatr is locally continuousandso r is continuous If ϕ is continuous, then(see point 2of Theorem 1.1) ϕ(r(x)) = lim

k→∞ ϕ(A

(k) (x)))= 0 for everyx ∈ X. Hencer(X) ⊂ ϕ −1(0). It

follows frompoint(6) that r(X) = ϕ −1(0), hence, r is aretraction of X onto ϕ−1(0)

Point (9)followsfrom point(7)(when O = X) 2

Definition1.5.Analmostexactly(α, β)-search functional ϕ : X→ R+ will becalled:

effective if forany x ∈ X, ε > 0 and convergentseriesΣ withnonnegativetermsandthepositivesum,() thereexistsaconvergent(ϕ, Σ, x, ε)-generated sequence s such thatforitslimit r(s), ϕ(r(s))) = 0;

completely effective if for any x ∈ X, ε > 0 and convergent series Σ with nonnegative terms and thepositivesum,

()any(ϕ, Σ, x, ε)-generated sequence s is convergentandforitslimit r(s), ϕ(r(s)) = 0

Remark1.3.For analmost exactly(α, β)-search functional ϕ : X → R+ thefollowingassertionsarealent:

equiv-1 ϕ iseffective;

2 for any ε > 0 and any convergent series Σ with nonnegative terms and the positive sum, there is a

(ϕ, Σ, ε)-generated sequence of mappingsA k ∈ F (X), k ∈ N+, thatpointwise converges toamapping

r ∈ F (X) suchthat ϕ(r(x)) = 0 forevery x ∈ X (i.e. r(X) ⊂ ϕ −1(0)).

(Implication 1⇒ 2 isprovedaspoint (0) ofCorollary 1.1.Implication2⇒ 1 isobvious.)

Thefollowing definitionsweregivenbyT.N.Fomenko[7]

Thegraph G of afunctional ϕ : X→ R+ iscalled:

(a) 0-closed if (ξ,0)∈ cl(G) implies (ξ,0)∈ G;

(b) 0-complete ifforany fundamentalsequence of points xk ∈ X, k ∈ N+, suchthat lim

k→∞ ϕ(x k)= 0, the

sequence(x k , ϕ(x k)), k∈ N, convergestoapoint(ξ, η) ∈ G intheproduct X× R (i.e.,thereisalimit

ξ of thesequence xk in X, k∈ N,and ϕ(ξ) = η = 0).

Theorem1.2 Let G be the graph of an almost exactly (α, β)-search functional ϕ : X → R. If (a) G is 0-closed

and the space X is complete, or (b) G is 0-complete, then ϕ is completely effective.

Proof Takea(ϕ, Σ, x, ε)-generated sequence xk ∈ X, k∈ N+

Inthecase(a),bythecompletenessof X, thelimit ξ∈ X ofthissequenceexists.Then(ξ,0) isthelimit

of thesequence (x k , ϕ(x k)), k∈ N+,in X× R.Hence,(ξ,0)∈ cl G.So(ξ,0)∈ G,i.e., ϕ(ξ) = 0

Inthecase(b)theassertionisevident 2

Corollary 1.2 If the space X is complete and an almost exactly (α, β)-search functional ϕ : X → R is continuous, then ϕ is completely effective If X is a compactum, then for every x ∈ X there is a point

η = η(x) such that ϕ(η) = 0 and ρ(x, η)  ϕ(x)

α − β .

Proof The assertionofthefirstsentenceofthecorollaryisevident

Let X be acompactum Take x ∈ X.Foreach n ∈ N onecanfind apoint ξn such thatϕ(ξ n)= 0 and

ρ(x, ξ n) ϕ(x)

α − β +

1

n. Selectfrom the sequence ξ n, n ∈ N, aconvergent subsequenceξ n(k), k ∈ N.If η is

thelimitofthesubsequence,thenbythecontinuityof ϕ and ρ,

ϕ(η) = 0 and ρ(x, η) ϕ(x)

Corollary1.3 If an almost exactly (α, β)-search functional ϕ : X → R+is effective, then (see Remark 1.3 ) for any ε > 0 and convergent series Σ with nonnegative terms and the positive sum, there is a (ϕ, Σ, ε)-generated sequence of mappings A (k) ∈ F (X), k ∈ N+, that pointwise converges to a mapping r ∈ F (X), k ∈ N+, moreover, (see (3) from Theorem 1.1 ) ρ(x, r(x))  ϕ(x)

α − β + ε, x ∈ X, and (see (6) from Corollary 1.1 and Remark 1.3 ) r(X) = ϕ −1 (0), r | ϕ −1(0) = id X | ϕ −1(0) If all mappings A (k) are continuous and ϕ is locally bounded (for example, continuous), then (see (8) from Corollary 1.1 ) r (is continuous and) is a retraction

0 for any (α, β)-search functional ϕ, there always exists a ϕ-generated mapping.

Theorem 1.3 Let ϕ : X → R+ be an (α, β)-search functional Then for any ϕ-generated mapping A and

4 if x ∈ ϕ −1 (0), then the limit r(x) (from point 3) exists and r(x) = x;

5 if ϕ is bounded, then A k ∈ F (X, id X ), k ∈ N+, and the sequence A k , k ∈ N+, is fundamental in the metric space (F (X, idX ), d F ).

Now, suppose that for a ϕ-generated mapping A, the sequence A k , k ∈ N+, pointwise converges to a mapping r ∈ F (X) Then:

9 if ϕ is bounded, then the sequence A k , k ∈ N+, converges uniformly on X to r.

Proof Obviously, ρ(Ak (x), Ak+1 (x))  ϕ(A k (x))

k+1 (x)) β

α ϕ(A

k (x)), k ∈ N+ Points 1–3areprovedasTheorem 1.3from[7]oras Theorem 1.1for γ = 0 Point 4followsfrom thefirstinequality(1.3)

Provepoint5.Supposethat ϕ(x)  C > 0 forevery x ∈ X.Then(asinProofofTheorem 1.1for γ = 0)

onecanshowthatforevery p, q ∈ N+,p < q, and every x ∈ X,thenextrelationsare true

ρ(A p (x), A q (x))



β α

Point 6followsfrom point4

Prove point7.If ϕ(x)  C > 0 forall x ∈ O,thenforall x ∈ O,thefirstinequality(1.4)istrue,andsotheinequality ρ(Ak (x), r(x)) 



β α

k

α − β,k ∈ N+ istruetoo Itgivesusthefirstassertionofpoint 7.The continuity of A (and all A k) and the uniform convergence of A k | O, k ∈ N, to r | O on O imply thecontinuityof r| O.Point7isproved

Points 8and9areprovedaspoints(8)and (9)ofCorollary 1.1 2

Definition 1.7 An (α, β)-search functional ϕ : X → R+ will be called effective (respectively, completely effective) if there exists a ϕ-generated mapping A (respectively, if any ϕ-generated mapping A) has thefollowing property

for any x ∈ X, there exists the limit r(x) of the sequence A k (x), k ∈ N+, and ϕ(r(x)) = 0.

Thenextassertionwasobtained(in otherterms)byT.N Fomenko in[7]

Theorem 1.2 If the graph G of an (α, β)-search functional ϕ : X → R+ is (a) 0-closed (for example, ϕ is continuous), and X is complete, or (b) 0-complete, then ϕ is completely effective.

Corollary 1.4 If a functional ϕ : X → R+ is (α, β)-search and effective (completely effective), then for a (for every) ϕ-generated mapping A, the sequence A k , k ∈ N+, pointwise converges to a mapping r ∈ F (X) and points 6.–9 of Theorem 1.3 are true Moreover, in point 8., the requirement of the continuity of ϕ can

be removed without losing the property of r to be a retraction of X onto ϕ −1 (0) (since, by the effectiveness

of ϕ, the equality ϕ(r(x)) = 0 is not lost for every x ∈ X).

Definition 1.8 A functional ϕ : X → R+ is called continuously (α, β)-search, 0 < β < α, if there is acontinuous mapping A : X → X suchthat

ρ(x, Ax) ϕ(x)

α and ϕ(Ax) β

α ϕ(x).

Remark1.4.Similarly,onecandefinean almost exactly continuously (α, β)-search functional

Corollary1.5 If X is complete, and the functional ϕ : X → R+is continuous and continuously (α, β)-search,

then there is a continuous mapping A : X → X such that for every x ∈ X the sequence A k x, k ∈ N+, has the limit r(x), ρ(x, r(x)) α ϕ(x) − β and r is a retraction of X onto ϕ −1 (0).

Remark 1.5 Obviously, Banach’s fixed point theorem follows from Corollary 1.5 (if α = 1 and ϕ(x) =

ρ(x, Ax)).

2 Someapplicationsofsearchfunctionalsinthecaseofmetricspaces

Let fi bemappingsofaspace X to aspace Y , i = 1, , n, n ∈ N, and H aclosedsubsetof Y

Theorem 2.1 (Theorem on the common preimage of a closed set for a finite system of mappings.) If a functional

ϕ(x) = max {ρ(f1x, f2x), , ρ(f1x, f n x), ρ(f1x, H) }, x ∈ X,

is almost exactly (α, β)-search and effective (for example, if (a) the graph of ϕ is 0-closed and X is complete

or (b) the graph of ϕ is 0-complete), then for every ε > 0 and every convergent series ∞

k=1

σ k with nonnegative terms and the sum σ > 0, there is a mapping r ∈ F (X) such that

(ε) ρ(x, r(x))  ϕ(x)

α − β + ε, x ∈ X;

(r) r(X) = ϕ −1(0)={x ∈ X : f1(x) = = f n (x) ∈ H} and r | ϕ −1(0)= id X | ϕ −1(0).

Moreover, r(x) is the limit of a (ϕ, Σ, x, ε)-generated sequence, x ∈ X.

If the functional ϕ is (α, β)-search and effective, then there is a ϕ-generated mapping A such that the sequence A k , k ∈ N+, pointwise converges to a mapping r ∈ F (X), the equalities (r) are true and instead of

the inequalities (ε), the following more exact inequalities ρ(x, r(x))  ϕ(x)

α − β , x ∈ X, hold If A is continuous and ϕ is locally bounded, then r is a retraction of X onto ϕ −1 (0).

Proof ThefirstpartofthetheoremfollowsfromCorollary 1.3andDefinition 1.4.Thesecondpartfollowsfrom Theorem 1.3 2

Theorem 2.1mustbecompared withTheorem1.20from[7]

ConsiderpartialcasesofTheorem 2.1

4 Y = X, n > 1, f1 = id X and α = 1, then ϕ(x) = max{ρ(x, f2x), , ρ(x, f n x) }) and we obtain asufficient condition for searching common fixed points of mappings f2, , f n (This resultmustbecomparedwithTheorems 1.12–1.14from [7].)

Addthefollowing assertionto Theorem8from[6]

Proposition 2.1 Let A be a continuous mapping of a complete space X to itself Suppose that there is a number β, 0 < β < 1, such that

Proof Since ρ(x, Ax) = ϕ(x)  ϕ(x)

α and ϕ(Ax) = ρ(Ax, A

2x)  βρ(x, Ax) = β

α ϕ(x), ϕ is (1, β)-search,

and A is aϕ-generated mapping.Since A is continuous, ϕ is continuoustoo and itsgraphisclosed Since

X is complete, ϕ is completelyeffective.Therestfollowsfrom point8ofTheorem 1.3 2

Pass tocoincidencepoints

Definition2.1 AmappingΨ ofaspace X to aspace Y is called α-covering (respectively, open-α-covering)

forapositivenumber α if forany x ∈ X and r > 0,

B(Ψx, αr) ⊂ Ψ(B(x, r))

(respectively,

O(Ψx, αr) ⊂ Ψ(O(x, r)).

Thenotionof α-covering mappingiswell-known,thenotionofopen-α-coveringmappingwasintroduced

byA.V.Arutyunovin[4],wherehenotedthatevery α-covering mappingisopen-α-coveringbutthereexist

open-α-coveringmappingsthatarenot α-covering

Recall thatamappingΦ ofaspace X to aspace Y is called β-Lipschitz, β > 0, if

ρ(Φx1, Φx2) βρ(x1, x2)forany x1, x2∈ X.

In[2],Theorem 1(seealso[3]),A.V.Arutyunov obtainedthefollowingassertion

Arutyunov theorem Let a map Ψ : X → Y be α-covering, a mapping Φ : X → Y β-Lipschitz, X complete

and 0 < β < α Then for any x ∈ X and ε > 0, there exists ξ = ξ(x, ε) ∈ X such that

Ψ(ξ) = Φ(ξ) and ρ(x, ξ) ϕ(x)

α − β .

In[7],T.N Fomenkoprovedthat

if a mapping Ψ : X → Y is α-covering, a mapping Φ : X → Y is β-Lipschitz and 0 < β < α, then the functional ϕ(x) = ρ(Ψ(x), Φ(x)), x ∈ X, is (α, β)-search

Lemma 2.1 Let a mapping Ψ : X → Y be open-α-covering, a mapping Φ : X → Y be β-Lipschitz and

0 < β < α Then the functional ϕ(x) = ρ(Ψ(x), Φ(x)) on X is almost exactly (α, β)-search.

Proof Fix x ∈ X and δ > 0 Let δ = min(δ, δ

β).Considerthecase ϕ(x) > 0 only SinceΨ isopen-α-covering, B(Ψ(x), ϕ(x)) ⊂ O(Ψ(x), ϕ(x) + αδ )⊂ Ψ(O(x, ϕ(x)