Common Fixed Point Theorems for Weakly Compatible Pairs on Cone Metric Spaces

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Volume 2009, Article ID 643840,13pages doi:10.1155/2009/643840

Research Article

Common Fixed Point Theorems for Weakly

Compatible Pairs on Cone Metric Spaces

G. Jungck,

1

_{S. Radenovi ´c,}

2

_{S. Radojevi ´c,}

2

_{and V. Rako ˇcevi´c}

3

1_{Department of Mathematics, Bradley University, Peoria, IL 61625, USA}

2_{Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11 120 Beograd, Serbia} 3_{Department of Mathematics, Faculty of Science and Mathematics, University of Niˇs, Viˇsegradska 33,}

18 000 Niˇs, Serbia

Correspondence should be addressed to S. Radenovi´c,[email protected]

Received 17 December 2008; Accepted 4 February 2009

Recommended by Mohamed Khamsi

We prove several fixed point theorems on cone metric spaces in which the cone does not need to be normal. These theorems generalize the recent results of Huang and Zhang2007, Abbas and Jungck2008, and Vetro2007. Furthermore as corollaries, we obtain recent results of Rezapour and Hamlborani2008.

Copyrightq2009 G. Jungck 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.

1. Introduction and Preliminaries

Recently, Abbas and Jungck1, have studied common fixed point results for noncommuting mappings without continuity in cone metric space with normal cone. In this paper, our results are related to the results of Abbas and Jungck, but our assumptions are more general, and also we generalize some results of1–3, and4by omitting the assumption of normality in the results.

Let us mention that nonconvex analysis, especially ordered normed spaces, normal cones, and topical functions2,4–9have some applications in optimization theory. In these cases, an order is introduced by using vector space cones. Huang and Zhang2used this approach, and they have replaced the real numbers by ordering Banach space and defining cone metric space. Consistent with Huang and Zhang 2, the following definitions and results will be needed in the sequel.

LetEbe a real Banach space. A subsetPofEis called a cone if and only if:

iPis closed, nonempty, andP /{0};

iia, b∈R, a, b≥0,andx, y∈Pimplyaxby∈P;

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Given a coneP ⊂E,we define a partial ordering≤onEwith respect toPbyx≤yif and only ify−x∈P.We will writex < yto indicate thatx≤ybutx /y,whilex ywill stand fory−x ∈ intP interior ofP. A coneP ⊂ Eis called normal if there are a number

K >0 such that for allx, y∈E,

0≤x≤yimpliesx ≤Ky. 1.1

The least positive number satisfying the above inequality is called the normal constant ofP.

It is clear thatK≥1.From4we know that there exists ordered Banach spaceEwith coneP

which is not normal but with intP /∅.

Definition 1.1see2. LetXbe a nonempty set. Suppose that the mappingd :X×X → E

satisfies

d10≤dx, yfor allx, y∈Xanddx, y 0 if and only ifxy;

d2dx, y dy, xfor allx, y∈X;

d3dx, y≤dx, z dz, yfor allx, y, z∈X.

Thend is called a cone metric on X, and X, d is called a cone metric space. The concept of a cone metric space is more general than of a metric space.

Definition 1.2see2. LetX, dbe a cone metric space. We say that{xn}is

eCauchy sequence if for everyc in Ewith 0 c,there is an N such that for all

n, m > N, dxn, xmc;

fconvergent sequence if for everycinEwith 0 c,there is anN such that for all

n > N, dxn, xcfor some fixedxinX.

A cone metric space X is said to be complete if every Cauchy sequence in X is convergent inX.The sequence{xn}converges tox∈Xif and only ifdxn, x → 0 asn → ∞.

It is a Cauchy if and only ifdxn, xm → 0 asn, m → ∞.

Remark 1.3. see10LetEbe an ordered Banachnormedspace. Thencis an interior point ofP,if and only if−c, cis a neighborhood of 0.

Corollary 1.4see, e.g.,11without proof. (1) Ifa≤bandbc,thenac.

Indeed,c−a c−b b−a≥c−bimplies−c−a, c−a⊇−c−b, c−b.

(2) Ifabandbc,thenac.

Indeed,c−a c−b b−a> c−bimplies−c−a, c−a⊃−c−b, c−b.

(3) If0≤ucfor eachc∈intP, thenu0.

Remark 1.5. Ifc∈ intP, 0≤anandan → 0,then there existsn0such that for alln > n0we

haveanc.

Proof. Let 0 cbe given. Choose a symmetric neighborhoodV such thatcV ⊂ P.Since

an → 0, there isn0such thatan ∈V −V forn > n0.This means thatc±an∈cV ⊂P for n > n0,that is,anc.

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cone, we have only half of the lemmas 1 and 4 from2. Also, the fact thatdxn, yn → dx, y

ifxn → xandyn → yis not applicable.

Remark 1.6. Let 0c.If 0≤dxn, x≤bnandbn → 0,then eventuallydxn, x c,where xn, xare sequence and given point inX.

Proof. It follows fromRemark 1.5,Corollary 1.41, andDefinition 1.2f.

Remark 1.7. If 0≤an ≤bnandan → a, bn → b,thena≤b,for each coneP.

Remark 1.8. IfEis a real Banach space with conePand ifa≤λawherea∈Pand 0< λ <1,

thena0.

Proof. The conditiona ≤ λameans thatλa−a ∈P,that is,−1−λa∈ P.Sincea∈ P and 1−λ >0,then also1−λa∈P.Thus we have1−λa∈P∩−P {0}anda0.

Remark 1.9. Let X, dbe a cone metric space. Let us remark that the family{Nx, e : x ∈ X,0e}, whereNx, e {y∈X:dy, xe}, is a subbasis for topology onX. We denote this cone topology by τc, and note that τc is a Hausdorﬀtopology see, e.g.,11without

proof.

For the proof of the last statement, we suppose that for eachc, 0cwe haveNx, c∩ Ny, c/∅. Thus, there existsz∈X such thatdz, xcanddz, yc. Hence,dx, y≤ dx, z dz, yc/2c/2c. Clearly, for eachn, we havec/n∈intP, soc/n−dx, y∈

intP ⊂P. Now, 0−dx, y∈P, that is,dx, y∈ −P∩P, and we havedx, y 0. We find it convenient to introduce the following definition.

Definition 1.10. Let X, d be a cone metric space and P a cone with nonempty interior. Suppose that the mappingsf, g : X → X are such that the range of g contains the range off, andfXor gXis a complete subspace ofX. In this case we will say that the pair

f, gis Abbas and Jungck’s pair, or shortly AJ’s pair.

Definition 1.11see1. Letfandgbe self-maps of a setXi.e.,f, g:X → X.Ifwfx gxfor somexinX,thenxis called a coincidence point offandg,andwis called a point of coincidence offandg.Self-mapsfandgare said to be weakly compatible if they commute at their coincidence point, that is, iffxgxfor somex∈X,thenfgxgfx.

Proposition 1.12see1. Letfandgbe weakly compatible self-maps of a setX.Iffandghave a unique point of coincidencewfxgx,thenwis the unique common fixed point offandg.

2. Main Results

In this section we will prove some fixed point theorems of contractive mappings for cone metric space. We generalize some results of1–4by omitting the assumption of normality in the results.

Theorem 2.1. Suppose thatf, gis AJ’s pair, and that for some constantλ ∈0,1and for every

x, y∈X,there exists

u≡ux, y∈

dgx, gy, dfx, gx, dfy, gy,dfx, gy dfy, gx

2

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such that

dfx, fy≤λ·u. 2.2

Thenfandghave a unique coincidence point inX. Moreover iffandgare weakly compatible,fand

ghave a unique common fixed point.

Proof. Letx0 ∈X,and let x1 ∈ X be such thatgx1 fx0 y0. Having definedxn ∈X,let xn1∈Xbe such thatgxn1fxnyn.

We first show that

dyn, yn1

≤λdyn−1, yn

, forn≥1. 2.3

We have that

dyn, yn1

dfxn, fxn1

≤λ·u, 2.4

where

u∈

dgxn, gxn1

, dfxn, gxn

, dfxn1, gxn1

,d

fxn, gxn1

dfxn1, gxn

2

dyn−1, yn

, dyn, yn1

,d

yn−1, yn1

2

.

2.5

Now we have to consider the following three cases.

If u dyn−1, yn then clearly 2.3 holds. If u dyn, yn1 then according

to Remark 1.8dgxn, gxn1 0, and 2.3 is immediate. Finally, suppose that u

1/2dyn−1, yn1.Now,

dyn, yn1

≤λd

yn−1, yn1

2 ≤

λ

2d

yn−1, yn

1

2

yn, yn1

. 2.6

Hence,dyn, yn1≤λdyn−1, yn, and we proved2.3.

Now, we have

dyn, yn1

≤λndy0, y1

. 2.7

We will show that{yn}is a Cauchy sequence. Forn > m, we have

dyn, ym

≤dyn, yn−1

dyn−1, yn−2

· · ·dym1, ym

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and we obtain

dyn, ym

≤λn−1λn−2· · ·λmdy1, y0

≤ λm

1−λd

y1, y0

−→0 asm−→ ∞.

2.9

From Remark 1.5 it follows that for 0 c and largem : λm1−λ−1dy1, y0 c; thus,

according toCorollary 1.41,dyn, ym c.Hence, by Definition 1.2e,{yn}is a Cauchy

sequence. Since fX ⊆ gX and fX or gX is complete, there exists a q ∈ gX

such that gxn → q ∈ gX as n → ∞. Consequently, we can find p ∈ X such that gpq.

Let us show thatfpq.For this we have

dfp, q≤dfp, fxn

dfxn, q

≤λ·und

fxn, q

, 2.10

where

un ∈

dgxn, gp

, dfxn, gxn

, dfp, gp,d

fxn, gp

dfp, gp

2

. 2.11

Let 0c.Clearly at least one of the following four cases holds for infinitely manyn.

Case 1

0

dfp, q≤λ·dgxn, gp

dfxn, q

λ· c

2λ c

2 c. 2.12

Case 2

0

dfp, q≤λ·dfxn, gxn

dfxn, q

≤λ·dfxn, q

λ·dq, gxn

dfxn, q

λ1·dfxn, q

λ·dq, gxn

λ1· c

2λ1λ·

c

2λ c.

2.13

Case 3

0

dfp, q≤λ·dfp, gp dfxn, q

, that is,

dfp, q 1

1−λ· c

1/1−λ c.

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Case 4

0

dfp, q≤λ·d

fxn, gp

dfp, gp

2 d

fxn, q

≤ λd

fxn, gp

2

1

2dfp, gp d

fxn, q

, that is,

dfp, q≤λ2dfxn, q

λ2 c

λ2 c.

2.15

In all cases, we obtaindfp, q cfor eachc∈intP.UsingCorollary 1.43, it follows that

dfp, q 0,orfpq.

Hence, we proved that f and g have a coincidence point p ∈ X and a point of coincidenceq ∈ X such that q fp gp.If q1 is another point of coincidence, then

there isp1∈Xwithq1fp1gp1.Now,

dq, q1 dfp, fp1≤λ·u, 2.16

where

u∈

dgp, gp1

, dfp, gp, dfp1, gp1

,d

fp, gp1

dfp1, gp

2

dq, q1

,0,d

q, q1

dq1, q

2

0, dq, q1

.

2.17

Hence,dq, q1 0,that is,qq1.

Sinceq fp gp is the unique point of coincidence off and g,and f and g

are weakly compatible,qis the unique common fixed point off andg byProposition 1.12 1.

In the next theorem, among other things, we generalize the well-known Zamfirescu result12,21.

Theorem 2.2. Suppose thatf, gis AJ’s pair, and that for some constantλ ∈0,1and for every

u≡ux.y∈

dgx, gy,dfx, gx dfy, gy

2 ,

dfx, gy dfy, gx

2

, 2.18

such that

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Thenf andg have a unique coincidence point inX. Moreover, iff andg are weakly compatible,f

andghave a unique common fixed point.

Proof. Letx0 ∈X,and let x1 ∈ X be such thatgx1 fx0 y0. Having definedxn ∈X,let xn1∈Xbe such thatgxn1fxnyn.

We first show that

dyn, yn1

≤λdyn−1, yn

forn≥1. 2.20

Notice that

dyn, yn1

dfxn, fxn1

≤λ·un, 2.21

where

un∈

dgxn, gxn1

,d

fxn, gxn

dfxn1, gxn1

2 ,

dfxn, gxn1

dfxn1, gxn

2

dyn−1, yn

,d

yn−1, yn

dyn, yn1

2 ,

dyn−1, yn1

2

.

2.22

As inTheorem 2.1, we have to consider three cases.

Ifu dyn−1, yn, then clearly2.20holds. Ifu dyn−1, yn dyn, yn1/2,then

from2.19withxxnandyxn1,asλ∈0,1,we have

dyn, yn1

≤λd

yn−1, yn

dyn, yn1

2 ≤λ

dyn−1, yn

2

dyn, yn1

2 . 2.23

Hence, dyn, yn1 ≤ λdyn−1, yn, and in this case 2.20 holds. Finally, if u

dyn−1, yn1/2,then

dyn, yn1

≤λd

yn−1, yn1

2 ≤λ

dyn−1, yn

dyn, yn1

2

≤λd

yn−1, yn

2

dyn, yn1

2 ,

2.24

and2.20holds. Thus, we proved that in all three cases2.20holds.

Now, from the proof ofTheorem 2.1, we know that{gxn1}{fxn}{yn}is a Cauchy

sequence. Hence, there existqingXandp∈Xsuch thatgxn → q,n → ∞,andgp q.

Now we have to show thatfpq.For this we have

dfxn, q

≤λ·und

fxn, q

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where

un∈

dgxn, gp

,d

fxn, gxn

dfp, gp

2 ,

dfxn, gp

dfp, gxn

2

. 2.26

Let 0c.Clearly at least one of the following three cases holds for infinitely manyn.

Case 1

0

dfp, q≤λ·dgxn, gp

dfxn, q

λ· c

2λ c

2 c. 2.27

Case 2

0

dfp, q≤λ·d

fxn, gxn

dfp, gp

2 dfxn, q

≤ λd

fxn, gxn

2

dfp, gp

2 d

fxn, q

, that is,

λdgxn, q

λ2 c

2λ2λ

c

2λ c.

2.28

Case 3

0

dfp, q≤λ·d

fxn, gp

dfp, gxn

2 d

fxn, q

≤ λd

fxn, gp

2

1 2dfp, q

λ

2d

q, gxn

dfxn, q

, that is,

λdgxn, q

λ2 c

2λ2λ

c

2λ c.

2.29

In all cases we obtaindfp, q cfor eachc ∈intP.UsingCorollary 1.43, it follows that

dfp, q 0,orfpq.

Thus we showed that f and g have a coincidence point p ∈ X, that is, point of coincidenceq ∈ X such thatq fp gp.Ifq1 is another point of coincidence then there

isp1 ∈Xwithq1fp1gp1.Now from2.19, it follows that

dq, q1

dfp, fp1

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where

u∈

dgp, gp1

,dfp, gp d

fp1, gp1

2 ,

dfp, gp1

dfp1, gp

2

dq, q,0,d

q, q1

dq1, q

2

0, dq, q1

.

2.31

Hence,dq, q1 0,that is,q q1.Iffandgare weakly compatible, then as in the proof of Theorem 2.1, we have thatqis a unique common fixed point offandg.The assertion of the theorem follows.

Now as corollaries, we recover and generalize the recent results of Huang and Zhang

2, Abbas and Jungck1, and Vetro3. Furthermore as corollaries, we obtain recent results of Rezapour and Hamlborani4.

Corollary 2.3. Suppose thatf, gis AJ’s pair, and that for some constantλ ∈ 0,1and for every

x, y∈X,

dfx, fy≤λ·dgx, gy. 2.32

x, y∈X,

dfx, fy≤λ·dfx, gx dfy, gy

2 . 2.33

x, y∈X,

dfx, fy≤λ·dfx, gy dfy, gx

2 . 2.34

In the next corollary, among other things, we generalize the well-known result12,

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uux, y∈ {dgx, gy, dfx, gx, dfy, gy}, 2.35

such that

dfx, fy≤λ·u. 2.36

Now, we generalize the well-known Bianchini result12,5.

uux, y∈ {dfx, gx, dfy, gy}, 2.37

such that

dfx, fy≤λ·u. 2.38

When in the next theorem we setg IX,the identity map onX, E −∞,∞and P 0,∞, we get the theorem of Hardy and Rogers12,18.

Theorem 2.8. Suppose thatf, gis AJ’s pair, and that there exist nonnegative constantsaisatisfying ₅

i1ai<1such that, for eachx, y∈X,

dfx, fy≤a1dgx, gy a2dgx, fx a3dgy, fy a4dgx, fy a5dgy, fx. 2.39

Proof. Let us define the sequencesxnandynas in the proof ofTheorem 2.1We have to show

that

dyn, yn1

≤λdyn−1, yn

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From

dyn1, yn

dfxn1, fxn

≤a1d

gxn1, gxn

a2d

gxn1, fxn1

a3d

gxn, fxn

a4d

gxn1, fxn

a5d

gxn, fxn1

a1d

yn, yn−1

a2d

yn, yn1

a3d

yn−1, yn

a5d

yn−1, yn1

,

dyn, yn1

dfxn, fxn1

≤a1d

gxn, gxn1

a2d

gxn, fxn

a3d

gxn1, fxn1

a4d

gxn, fxn1

a5d

gxn1, fxn

a1d

yn−1, yn

a2d

yn−1, yn

a3d

yn, yn1

a4d

yn−1, yn1

,

2.41

we obtain

2dyn, yn1

≤2a1d

yn−1, yn

2a2d

yn−1, yn

a3d

yn, yn1

a4d

yn−1, yn1

a3a4a5

dyn, yn1

. 2.42

Thus,

dyn, yn1

≤ 2a1a2a3a4

2−a3−a4−a5 ·

dyn−1, yn

λ·dyn−1, yn

, 2.43

whereλ 2a1a2a3a42−a3−a4−a5−1∈0,1,and we proved2.40.

Now, from the proof ofTheorem 2.1, we know that{gxn1}{fxn}{yn}is a Cauchy

sequence. Hence, there existqingXandp∈Xsuch thatgxn → q,n → ∞,andgpq.

We have to show thatfpq.For this we have

dfxn, q

≤a1d

gxn, gp

a2d

gxn, fxn

a3d

gp, fpa4d

gxn, fp

a5d

gp, fxn

dfxn, q

≤a1d

gxn, q

a2d

gxn, q

a2d

q, fxn

a3dq, fp a4d

gxn, q

a4dq, fp

a51

dq, fxn

,that is,

dfp, q≤ a1a2a4

1−a3−a4

dgxn, q

1a2a5

1−a3−a4

dfxn, q

a1a2a4

1−a3−a4

c

2a1a2a4/1−a3−a4

1a2a5

1−a3−a4

c

21a2a5/1−a3−a4

c.

2.44

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Thus we showed that f and g have a coincidence point p ∈ X, that is, point of coincidenceq ∈ X such thatq fp gp.Ifq1 is another point of coincidence then there

isp1 ∈Xwithq1fp1gp1.Now,

dq, q1

dfp, fp1

≤a1d

gp, gp1

a2d

gp, fpa3d

gp1, fp1

a4d

gp, fp1

a5d

gp1, fp

a1a4a5

dq, q1

.

2.45

According toRemark 1.8, and because 0≤a1a4a5 ≤

5

i1ai <1,we getdq, q1 0,that

is,qq1.Iffandgare weakly compatible, then as in the proof ofTheorem 2.1, we have that

qis a unique common fixed point offandg.The assertion of the theorem follows.

It is clear that, for the special choice ofaiinTheorem 2.8, all the results from Corollaries

2.3,2.4, and2.5, could be obtained.

Finally, we add an example with Banach type contraction on non-normal cone metric spaceseeCorollary 2.3.

Example 2.9. LetX R, E C1

R0,1, andP {ϕ ∈ E : ϕ ≥ 0}.Defined : X×X → Eby

dx, y |x−y|ϕwhereϕ : 0,1 → Rsuch thatϕt et.It is easy to see thatdis a cone metric onX.Consider the mappingsf, g:X → Xin the following manner:

fx

⎧ ⎨ ⎩

1

1αxβ, x /0,

0, x0,

gx

x 1αβ, x /0,

0, x0, 2.46

whereα >1, β∈R.One can see that

dfx, fy≤kdgx, gy, 2.47

for allx, y ∈ X,wherek 1/α∈ 0,1.The mappingsf andg commute atx 0,the only coincidence point. Sofandgare weakly compatible. All the conditions of theCorollary 2.3

hold, thenfandghave a common fixed point.

Acknowledgments

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