Common Fixed Points of Weakly Contractive and Strongly Expansive Mappings in Topological Spaces

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Volume 2010, Article ID 746045,15pages doi:10.1155/2010/746045

Research Article

Common Fixed Points of Weakly Contractive and Strongly Expansive Mappings in Topological Spaces

M. H. Shah,

¹

N. Hussain,

²

and A. R. Khan

³

1Department of Mathematical Sciences, LUMS, DHA Lahore, Pakistan

2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

3Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

Correspondence should be addressed to A. R. Khan,arahim@kfupm.edu.sa Received 17 May 2010; Accepted 21 July 2010

Academic Editor: Yeol J. E. Cho

Copyrightq2010 M. H. Shah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Using the notion of weaklyF-contractive mappings, we prove several new common fixed point theorems for commuting as well as noncommuting mappings on a topological space X. By analogy, we obtain a common fixed point theorem of mappings which are stronglyF-expansive on X.

1. Introduction

It is well known that ifX is a compact metric space andf :X → Xis a weakly contractive mappingseeSection 2for the definition, thenf has a fixed point inX see1, p. 17. In late sixties, Furi and Vignoli2extended this result toα-condensing mappings acting on a bounded complete metric spacesee3for the definition. A generalized version of Furi- Vignoli’s theorem using the notion of weaklyF-contractive mappings acting on a topological space was proved in4 see also5.

On the other hand, in 6 while examining KKM maps, the authors introduced a new concept of lower upper semicontinuous function see Definition 2.1, Section 2 which is more general than the classical one. In 7, the authors used this definition of lower semicontinuity to redefine weaklyF-contractive mappings and stronglyF-expansive mappings see Definition 2.6, Section 2 to formulate and prove several results for fixed points.

In this article, we have used the notions of weaklyF-contractive mappingsf :X → X whereX is a topological space to prove a version of the above-mentioned fixed point theorem7, Theorem 1for common fixed pointsseeTheorem 3.1. We also prove a common

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fixed point theorem under the assumption that certain iteration of the mappings in question is weaklyF-contractive. As a corollary to this fact, we get an extension to common fixed pointsof7, Theorem 3for Banach spaces with a quasimodulus endowed with a suitable transitive binary relation. The most interesting result of this section isTheorem 3.8wherein the stronglyF-expansive condition onfwith some other conditionsimplies thatf andg have a unique common fixed point.

In Section 4, we define a new class of noncommuting self-maps and prove some common fixed point results for this new class of mappings.

2. Preliminaries

Definition 2.1see6. LetXbe a topological space. A functionf :X → Ris said to be lower semi-continuous from abovelscaatx0if for any netxλ_λ∈Λconvergent tox0with

fxλ1≤fxλ2 forλ2 ≤λ1, 2.1

we have

fx0≤lim

λ∈Λfxλ. 2.2

A functionf :X → Ris said to be lsca if it is lsca at everyx∈X.

Example 2.2. iLetXR. Definef :X → Rby

fx

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

x1, whenx >0, 1

2, whenx0,

−x1, whenx <0.

2.3

Letzn_n≥1be a sequence of nonnegative terms such thatzn_n≥1converges to 0. Then

fzn1≤fzn forλ2n≤n1λ1, f0 1

2 <1 lim

n→ ∞fzn. 2.4 Similarly, ifzn_n≥1is a sequence inXof negative terms such thatz_n_n≥1converges to 0, then

f z_n1

≤f z_n

forλ2n≤n1λ1, f0 1

2 <1 lim

n→ ∞f z_n

. 2.5

Thus,fis lsca at 0.

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ii Every lower semi-continuous function is lsca but not conversely. One can check that the functionf : X → RwithX Rdefined below is lsca at 0 but is not lower semi- continuous at 0:

fx

⎧⎨

⎩

x1, whenx≥0,

x, whenx <0. 2.6

The following lemmas state some properties of lsca mappings. The first one is an analogue of Weierstrass boundedness theorem and the second one is about the composition of a continuous function and a function lsca.

Lemma 2.3see6. LetXbe a compact topological space andf :X → Ra function lsca. Then there existsx0∈Xsuch thatfx0 inf{fx:x∈X}.

Lemma 2.4see7. LetX be a topological space andf : X → Y a continuous function. If g : X → Ris a function lsca, then the composition functionhg◦f :X → Ris also lsca.

Proof. Fixx0∈X×Xand consider a netxλ_λ∈ΛinXconvergent tox0such that

hxλ1≤hxλ2 forλ2≤λ1. 2.7

Setzλ fxλandzfx0.Then sincef is continuous, limλfxλ fx0∈X,andglsca implies that

gz g fx0

≤lim

λ g fxλ

lim

λ gzλ 2.8

withgzλ1≤gzλ2forλ2≤λ1.Thushx0≤limλhxλandhis lsca.

Remark 2.5see6. LetXbe topological space. Letf :X → Xbe a continuous function and F :X×X → Rlsca. Theng :X → Rdefined bygx Fx, fxis also lsca. For this, let xλ_λ∈Λbe a net inXconvergent tox∈X.Sincefis continuous, limλfxλ fx.Suppose that

gxλ1≤gxλ2 forλ₂≤λ₁. 2.9

Then sinceFis lsca, we have gx F

x, fx

≤lim

λ F

x_λ, fxλ lim

λ gxλ. 2.10

Definition 2.6see7. LetXbe a topological space andF:X×X → Rbe lsca. The mapping f:X → Xis said to be

iweaklyF-contractive ifFfx, fy< Fx, yfor allx, y,∈Xsuch thatx /y, iistronglyF-expansive ifFfx, fy> Fx, yfor allx, y∈Xsuch thatx /y.

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IfXis a metric space with metricdandFd, then we callf,respectively, weakly contractive and strongly expansive.

Letf, g : X → X. The set of fixed points off resp.,gis denoted byFf resp., Fg. A pointx ∈ Mis a coincidence pointcommon fixed pointoff andg iffx gx x fx gx. The set of coincidence points off andgis denoted byCf, g.Mapsf, g : X → Xare called1commuting iffgx gfxfor allx∈ X,2weakly compatible8if they commute at their coincidence points, that is, iffgx gfxwheneverfx gx, and3 occasionally weakly compatible9iffgxgfxfor somex∈Cf, g.

3. Common Fixed Point Theorems for Commuting Maps

In this section we extend some results in7to the setting of two mappings having a unique common fixed point.

Theorem 3.1. LetXbe a topological space,x₀∈X, andf, g:X → Xself-mappings such that for every countable setU⊆X,

UfU∪

gx0 ⇒Uis relatively compact 3.1

andf,gcommute onX.If

ifis continuous and weaklyF-contractive or

iigis continuous and weaklyF-contractive withgU⊆U, thenfandghave a unique common fixed point.

Proof. Letx₁ gx0and define the sequencexn_n≥1 by settingx_n1 fxnforn ≥ 1.Let A{xn:n≥1}.Then

AfA∪

gx0 , 3.2

so by hypothesisAis compact. Defineϕ:A−→R,by

ϕx

⎧⎨

⎩ F

x, fx

iff is continuous, F

x, gx

ifg is continuous. 3.3

Now ifforgis continuous and sinceFis lsca, then byRemark 2.5,ϕis lsca. So byLemma 2.3, ϕhas a minimum at, say,a∈A.

iSuppose thatfis continuous and weaklyF-contractive. Thenϕx Fx, fxasfis continuous. Now observe that ifa∈ A, f is continuous, andfA ⊆ A, thenfa ∈ A.We show thatfa a.Suppose thatfa/a; then

ϕ fa

F

fa, f fa

< F

a, fa

ϕa, 3.4

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a contradiction to the minimality ofϕata.Havingfa a,one can see thatga a.Indeed, ifga/athen we have

F

a, ga F

fa, gfa F

fa, fga

< F

a, ga

3.5 a contradiction.

ii Suppose thatg is continuous and weaklyF-contractive withgU⊆ U. Thenϕx Fx, gxasgis continuous. PutUA; thena∈A, g is continuous, andgA⊆Aimplies thatga∈A. We claim thatga a,for otherwise we will have

ϕ ga

F

ga, g

ga

< F

a, ga

ϕa 3.6

which is a contradiction. Hence the claim follows.

Now suppose thatfa/athen we have F

a, fa F

ga, fga F

ga, gfa

< F

a, fa

, 3.7

a contradiction, hencefa a.

In both cases, uniqueness follows from the contractive conditions: suppose there exists b∈Asuch thatfb bgb.Then we have

Fa, b F

fa, fb

< Fa, b, Fa, b F

ga, gb

< Fa, b 3.8

which is false. Thusfandghave a unique common fixed point.

Ifgid_X, thenTheorem 3.1ireduces to7, Theorem 1.

Corollary 3.2 see 7, Theorem 1. Let X be a topological space, x0 ∈ X, and f : X → X continuous and weakly F-contractive. If the implicationU⊆X,

UfU∪ {x0}⇒ Uis relatively compact, 3.9

holds for every countable setU⊆X,thenfhas a unique fixed point.

Example 3.3. Letc0, · _∞be the Banach space of all null real sequences. Define X

x x_n_n≥1∈c0:xn∈0,1, forn≥1 . 3.10

Letk∈Nandpn_n≥1⊆0,1a sequence such that pn_n≤k⊆ {0},

pn

n>k ⊆0,1 3.11

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withpn → 1 asn → ∞.Define the mappingsf, g:X → Xby

fx

fnxn

n≥1, gx

gnxn

n≥1, 3.12

wherex∈X, xn∈0,1andfn, gn:0,1 → 0,1are such that for 1≤n≤k, f_nxn−f_n

y_n xn−yn

2 , 3.13

g_nxn−g_n

y_n xn−yn

3 , 3.14

and forn > k

f_nxn

p_nx_n

2 , g_nxn

p_nx_n

3 . 3.15

We verify the hypothesis ofTheorem 3.1.

iObserve thatfandgare, clearly, continuous by their definition.

iiForx, y∈X,we have fx−f

ysup

n≥1

f_nxn−f_n y_n, gx−g

ysup

n≥1

g_nxn−g_n

y_n. 3.16

Since the sequencesfnxn_n≥1 andgnxn_n≥1are null sequences, there existsN ∈Nsuch that

sup

n≥1

f_nxn−f_n

y_nf_NxN−f_N y_N, sup

n≥1

gnxn−gn

yngNxN−gN

yN.

3.17

Hence fnxn−fn

ynfNxN−fN

yN<xN−yNsup

n≥1

xn−ynxn−yn, gnxn−gn

yngNxN−gN

yN<xN−yNsup

n≥1

xn−ynxn−yn. 3.18 This implies thatf andgare weakly contractive. Thusf andgare continuous and weakly contractive. Next suppose that for any countable setU⊆X,we have

UfU∪

g0c0 , 3.19

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then by the definition off, we can considerU⊆0,1.Hence closure ofUbeing closed subset of a compact set is compact. Also

fgx

pn² 2 xn

n≥N

gfx for everyx∈U. 3.20

So byTheorem 3.1,fandghave a unique common fixed point.

Corollary 3.4. Let (X, dbe a metric space,x₀ ∈X, andf, g:X → Xself-mappings such that for every countable setU⊆X,

UfU∪

gx0 ⇒Uis relatively compact, 3.21

ifis continuous and weakly contractive or

iigis continuous and weakly contractive withgU⊆U, thenfandghave a unique common fixed point.

Proof. It is immediate fromTheorem 3.1withFd.

Corollary 3.5. LetXbe a compact metric space,x₀∈X, andf, g:X → Xself-mappings such that for every countable setU⊆X,

UfU∪

gx0 ⇒Uis closed 3.22

ifis continuous and weakly contractive or

iigis continuous and weaklyF-contractive withgU⊆U, thenfandghave a unique common fixed point.

Proof. It is immediate fromTheorem 3.1.

Theorem 3.6. LetXbe a topological space,x0∈X, andf, g:X → Xself-mappings such that for every countable setU⊆X,

(1)UfU∪ {gx0}⇒Uis relatively compact;

(2)Uf^kU∪ {gx0}⇒Uis relatively compact for somek∈N;

(3)Uf^kU∪ {g^kx0}⇒Uis relatively compact for somek∈N.

Andf,gcommute onX.Further, if

i f is continuous andf^k weaklyF-contractive or

ii g is continuous andg^k weaklyF-contractive with gU⊆U,

3.23

thenfandghave a unique common fixed point.

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Proof. Part3: we proceed as inTheorem 3.1. Letx1 g^kx0for somek∈Nand define the sequencexn_n≥1 by settingx_n1f^kxnforn≥1.LetA{xn:n≥1}.Then

Af^kA∪

g^kx0

, 3.24

so by hypothesis3,Ais compact. Defineϕ:A → R by

ϕx

⎧⎨

⎩ F

x, f^kx

iff is continuous, F

x, g^kx

ifg is continuous. 3.25

Now sinceFis lsca and ifforgis continuous, then byRemark 2.5ϕwould be lsca and hence byLemma 2.3,ϕwould have a minimum, say, ata∈A.

iSuppose thatfis continuous andf^kweaklyF-contractive. Thenϕx Fx, f^kxasf is continuous. Now observe thata∈A, fis continuous, andfA⊆Aimplies that f^k is continuous andf^kA ⊆ Aand sof^ka∈ Afor somek ∈N. We show that f^ka a. Suppose thatf^ka/afor anyk∈N, then

ϕ f^ka

F

f^ka, f^k

f^ka

< F

a, f^ka

ϕa, 3.26

a contradiction to the minimality ofϕata.Therefore, f^ka a,for somek ∈ N. One can check thatga a. Suppose thatg^ka/a, then we have

F

a, g^ka F

f^ka, g^k

f^ka F

f^ka, f^k

g^ka

< F

a, g^ka 3.27

a contradiction. Thusais a common fixed point off^kandg^kand hence offandg.

iiSuppose thatg is continuous andg^k weaklyF-contractive withgU⊆U. Thenϕx Fx, g^kxasg is continuous. PutU A.Thena ∈ A, g continuous andgA ⊆ Aimply thatg^ka∈A. We claim thatg^ka a,for otherwise we will have

ϕ g^ka

F

g^ka, g^k

g^ka

< F

a, g^ka

ϕa 3.28

which is a contradiction. Hence the claim follows.

Now suppose thatf^ka/athen we have F

a, f^ka F

g^ka, f^k

g^ka F

g^ka, g^k

f^ka

< F

a, f^ka 3.29

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a contradiction, hencef^ka a.Thusais a common fixed point off^kandg^kand hence off andg.

Now we establish the uniqueness ofa.Suppose there existsb∈ Asuch thatf^kb b g^kbfor somek ∈ N. Now if fis continuous andf^kis weaklyF-contractive, then we have

Fa, b F

f^ka, f^kb

< Fa, b 3.30

and ifgis continuous andg^kis weaklyF-contractive, then we have Fa, b F

g^ka, g^kb

< Fa, b 3.31

which is false. Thusf^kandg^khave a unique common fixed point which obviously is a unique common fixed point offandg.

Part2. The conclusion follows if we sethg^kin part3.

Part1. The conclusion follows if we setSf^kandT g^kin part3.

A nice consequence ofTheorem 3.6is the following theorem whereX is taken as a Banach space equipped with a transitive binary relation.

Theorem 3.7. LetX X, · be a Banach space with a transitive binary relationsuch that x ≤ yforx, y ∈X withxy.Suppose, further, that the mappingsA, m :X → X are such that the following conditions are satisfied:

i0mxandmxxfor allx∈X;

ii0xy,thenAxAy;

iiiA is bounded linear operator andA^kx<xfor somek∈Nand for allx∈Xsuch that x /0 with 0x.

If either

am

fx−f y

Am

gx−g y

andg is contractive, bm

gx−g y

Am

fx−f y

andf is contractive, 3.32

for allx, y∈Xwithf, gcommuting onXand if one of the conditions, (1)–(3), ofTheorem 3.6holds, thenfandghave a unique common fixed point.

Proof. aSuppose thatmfx−fyAmgx−gyfor allx, y∈Xwithf, gcommuting onXandgis contractive.Then we have

0m

fx−f y

Am

gx−g y

. 3.33

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f²x−f² y

Am

gfx−gf y Am

fgx−fg y A²m

gx−g y

.

3.34

Therefore, afterk-steps,k∈N, we get 0m

f^kx−f^k y

A^km

gx−g y

.

3.35

Hence,

f^kx−f^k

ym

f^kx−f^k y

≤A^km

gx−g y

<m

gx−g

y

gx−g y

≤x−y.

3.36

Sof^k is weakly contractive. Sincef is continuousasAis bounded andg contractiveby Theorem 3.6,fandghave a unique common fixed point.

bSuppose thatmgx−gyAmfx−fyandfis contractive for allx, y∈X withf, g commuting onX andf being contractive.The proof now follows if we mutually interchangef, ginaabove.

Theorem 3.8. LetXbe a topological space,Y ⊂Z⊂XwithY closed andx₀ ∈Y.Letf, g:Y → Z be mappings such that for every countable setU⊆Y,

fU U∪

gx0 ⇒Uis relatively compact 3.37

andf,g commute onX.If f is a homeomorphism and stronglyF-expansive, thenf and g have a unique common fixed point.

Proof. Suppose that f is a homeomorphism and stronglyF-expansive. Let z, w ∈ Z with z /w.Then there existsx, y∈Y such thatzfxandwfyorf⁻¹z xandf⁻¹w y.Sincefis stronglyF-expansive, we have

Fz, w F

fx, f y

> F x, y

F

f⁻¹z, f⁻¹w

, 3.38

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or

F

f⁻¹z, f⁻¹w

< Fz, w. 3.39

Sof⁻¹is a weaklyF-contractive mapping. Choose any countable subsetV ofZand setB V∩Y.Suppose that

Bf⁻¹B∪

gx0 . 3.40

Thenf⁻¹B Ufor someU⊆Yand we get fU U∪

gx0 . 3.41

So by hypothesisUis compact and sincefis a homeomorphism,fU Bis compact. Since fgx gfxfor everyx∈Uandf⁻¹B U, we have

f⁻¹gx f⁻¹g

ff⁻¹x f⁻¹

gf

f⁻¹x f⁻¹

fg

f⁻¹x

gf⁻¹x 3.42

for everyx∈B.Thus

Bf⁻¹B∪

gx0 ⇒Bis relatively compact 3.43

andf⁻¹gx gf⁻¹xfor everyx∈B.Sincef⁻¹is continuous and weaklyF-contractive, by Theorem 3.1, the mappingsf⁻¹andg have a unique common fixed point, say,a∈ B. Since f⁻¹a aimplies thatafa,soais a unique common fixed point offandg.

The following example illustratesTheorem 3.8.

Example 3.9. LetXR²with the River metricd:X×X → Rdefined by

d x, y

⎧⎨

⎩ δ

x, y

ifx, y are collinear, δx,0 δ

0, y

, otherwise, 3.44

where x x1, y1, y x2, y2, and δ denotes the Euclidean metric on X. ThenX is a topological space with a topology induced by the metricd. Consider the sets Y, Zdefined by

Y

u, v∈R²:uv∈0,1 , Z

u, v∈R²:uv∈

0,3 2

.

3.45

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Let the mappings f, g : Y → Z be defined by fu, v 3/2u,3/2vand gu, v 2/3u,2/3v for u, v ∈ Y. Then f is clearly a homeomorphism and for an arbitrary countable subsetAofYandx₀ 0,0∈Y,

fA A∪

gx0 . 3.46

If and only ifA{0,0}. Indeed, ifu, v∈Asuch thatu, v/ 0,0, then fA 3

2A /A∪ {0,0}A∪

gx0 . 3.47

Further,fgu, v gfu, vfor everyu, v∈Y.SetFu, v ρu, vwhereρ:X×X → R is the Radial metric defined by

ρ x, y

⎧⎨

⎩

y₁−y2 ifx1x2,

y1y2|x1−x2| ifx1/x2, 3.48

andx x1, y1;y x2, y2.Now forx, y∈Y,since F

fx, f y

ρ

fx, f y

3 2ρ

x, y

> ρ x, y

F x, y

, 3.49

f is stronglyF-expansive. AlsoF ρ :X, d×X, d → Ris lower semi-continuous and hence lsca. Thus all the conditions ofTheorem 3.8are satisfied and f and g have a unique common fixed point.

4. Occasionally Banach Operator Pair and Weak F-Contractions

In this section, we define a new class of noncommuting self-maps and prove some common fixed point results for this new class of maps.

The pairT, Iis called a Banach operator pair10if the setFIisT-invariant, namely, TFI ⊆ FI. Obviously, commuting pairT, Iis a Banach operator pair but converse is not true, in general; see10–13. IfT, Iis a Banach operator pair, thenI, Tneed not be a Banach operator pair.

Definition 4.1. The pairT, Iis called occasionally Banach operator pair if

du, Tu≤diam FIfor someu∈FI. 4.1

Clearly, Banach operator pairBOP T, Iis occasionally Banach operator pairOBOPbut not conversely, in general.

Example 4.2. Let X R M with usual norm. Define I, T : M → M by Ix x² and Tx2−x², forx /−1 andI−1 T−1 1/2.FI {0,1}andCI, T {−1,1}. Obviously T, Iis OBOP but not BOP asT02/∈FI. Further,T, Iis not weakly compatible and hence not commuting.

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Example 4.3. LetXRwith usual norm andM 0,1. DefineT, I:M → Mby

Tx

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩ 1

2, ifx∈

0,1 4

, 1−2x, ifx∈

1 4,1

2

,

0, ifx∈

1 2,1

,

4.2

Ix

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

2x, ifx∈

0,1 2

, 1, ifx∈

1 2,1

.

4.3

Here FI {0,1} and T0 1/2/∈FI implies that T, I is not Banach operator pair.

Similarly,I, Tis not Banach operator pair. Further,

|0−T0|

0−1 2

1

2 ≤1diamFI 4.4

imply thatT, Iis OBOP. Further, note thatCT, I {1/4} andTI1/4 /IT1/4. Hence {T, I}is not occasionally weakly compatible pair.

Definition 4.4. LetXbe a nonempty set andd:X×X → 0,∞be a mapping such that d

x, y

0 if and only ifxy. 4.5

For a spaceX, dsatisfying4.5andA⊆X,the diameter ofAis defined by diamA sup

d x, y

:x, y∈A . 4.6

Here we extend this concept to the spaceX, dsatisfying condition4.5.

Definition 4.5. LetX, dbe a space satisfying4.5. The pairT, Iis called occasionally Banach operator pair onXiﬀthere is a pointuinXsuch thatu∈FIand

du, Tu≤diamFI, dTu, u≤diamFI. 4.7

UfU∪ {x0}⇒Uis relatively compact. 4.8

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If f is continuous and weakly F-contractive, F satisfies condition 4.5, and the pair g, f is occasionally Banach operator pair, thenfandghave a unique common fixed point.

Proof. ByCorollary 3.2,Ffis a singleton. Letu∈Ff. Then, by our hypothesis, d

u, gu

≤diam F f

0. 4.9

Therefore,ugufu. That is,uis unique common fixed point offandg.

UfU∪ {x₀}⇒Uis relatively compact. 4.10 Iffis continuous and weakly contractive and the pairg, fis occasionally Banach operator pair, then fandghave a unique common fixed point.

Proof. It is immediate fromTheorem 4.6withFd.

Corollary 4.8. LetXbe a compact metric space,x0∈X, andf, g:X → Xself-mappings such that for every countable setU⊆X,

UfU∪ {x₀}⇒Uis closed. 4.11

Iffis continuous and weakly contractive and the pairg, fis occasionally Banach operator pair, then fandghave a unique common fixed point.

Proof. It is immediate fromTheorem 4.6.

Theorem 4.6holds for a Banach operator pair without condition4.5as follows.

UfU∪ {x₀}⇒Uis relatively compact. 4.12 Iffis continuous and weaklyF-contractive and the pairg, fis a Banach operator pair, thenfand ghave a unique common fixed point.

Proof. ByCorollary 3.2,Ffis a singleton. Letu∈Ff. Asg, fis a Banach operator pair, by definitiongFf ⊂ Ff. Thusgu ∈Ffand henceu gu fu. That is,uis unique common fixed point offandg.

UfU∪ {x0}⇒Uis relatively compact. 4.13

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Iffis continuous and weakly contractive and the pairg, fis a Banach operator pair, thenfandg have a unique common fixed point.

Acknowledgments

N. Hussain thanks the Deanship of Scientific Research, King Abdulaziz University for the support of the Research Project no. 3-74/430. A. R. Khan is grateful to the King Fahd University of Petroleum & Minerals and SABIC for the support of the Research Project no.

SB100012.

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