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

1Department of Mathematics, Bradley University, Peoria, IL 61625, USA

2Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11 120 Beograd, Serbia

3Department 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;

iiiP∩−P {0}.

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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, x_m → 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) If 0≤ucfor eachc∈intP, thenu0.

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

Proof. Let 0 cbe given. Choose a symmetric neighborhoodV such thatcV ⊂ P.Since a_n → 0, there isn₀such thata_n ∈V −V forn > n₀.This means thatc±a_n∈cV ⊂P for n > n0,that is,anc.

From this it follows that: the sequence{xn}converges tox∈Xifdxn, x → 0 asn →

∞,and{xn}is a Cauchy ifdxn, x_m → 0 asn, m → ∞.In the situation with non-normal

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cone, we have only half of the lemmas 1 and 4 from2. Also, the fact thatdxn, yn → dx, y ifx_n → xandy_n → yis not applicable.

Remark 1.6. Let 0c.If 0≤dxn, x≤b_nandb_n → 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≤a_n ≤b_nanda_n → a, b_n → 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. LetX, 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

, 2.1

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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 x_n1∈Xbe such thatgx_n1fx_ny_n.

We first show that d

y_n, y_n1

≤λd

y_n−1, y_n

, forn≥1. 2.3

We have that

d

yn, y_n1 d

fxn, fx_n1

≤λ·u, 2.4

where

u∈

d

gx_n, gx_n1 , d

fx_n, gx_n , d

fx_n1, gx_n1 ,d

fxn, gx_n1 d

fx_n1, gxn

2

d y_n−1, yn

, d

yn, y_n1 ,d

y_n−1, y_n1 2

.

2.5

Now we have to consider the following three cases.

If u dy_n−1, y_n then clearly 2.3 holds. If u dyn, y_n1 then according to Remark 1.8dgxn, gx_n1 0, and 2.3 is immediate. Finally, suppose that u 1/2dyn−1, y_n1.Now,

d

yn, y_n1

≤λd

y_n−1, y_n1

2 ≤ λ

2d y_n−1, yn

1 2

yn, y_n1

. 2.6

Hence,dyn, y_n1≤λdy_n−1, y_n, and we proved2.3.

Now, we have

d

yn, y_n1

≤λⁿd y0, y1

. 2.7

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

y_n, y_m

≤d

y_n, y_n−1 d

y_n−1, y_n−2

· · ·d

y_m1, y_m

, 2.8

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and we obtain d

yn, ym

≤

λⁿ⁻¹λⁿ⁻²· · ·λ^m d

y1, y0

≤ λ^m 1−λd

y₁, y₀

−→0 asm−→ ∞.

2.9

From Remark 1.5 it follows that for 0 c and largem : λ^m1−λ⁻¹dy1, y0 c; thus, according toCorollary 1.41,dyn, y_m 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≤d

fp, fx_n d

fx_n, q

≤λ·u_nd fx_n, q

, 2.10

where

u_n ∈

d

gx_n, gp , d

fx_n, gx_n , d

fp, gp ,d

fxn, gp d

fp, gp 2

. 2.11

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

Case 1⁰

dfp, q≤λ·d

gxn, gp d

fxn, q

λ· c 2λ c

2 c. 2.12

Case 2⁰

dfp, q≤λ·d

fx_n, gx_n d

fx_n, q

≤λ·d fx_n, q

λ·d q, gx_n

d fx_n, q λ1·d

fxn, q

λ·d q, gxn

λ1· c

2λ1λ· c 2λ c.

2.13

Case 3⁰

dfp, q≤λ·dfp, gp d fx_n, q

, that is, dfp, q 1

1−λ· c

1/1−λ c.

2.14

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Case 4⁰

dfp, q≤λ·d

fx_n, gp d

fp, gp

2 d

fxn, q

≤ λd

fx_n, gp

2 1

2dfp, gp d fxn, q

, that is, dfp, q≤λ2d

fx_n, 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∈

d

gp, gp₁

, dfp, gp, d

fp₁, gp₁ ,d

fp, gp1

d

fp1, gp 2

d q, q₁

,0,d q, q1

d q1, q 2

0, d q, 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 x, y∈X,there exists

u≡ux.y∈

dgx, gy,dfx, gx dfy, gy

2 ,dfx, gy dfy, gx

2

, 2.18

such that

dfx, fy≤λ·u. 2.19

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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 definedx_n ∈X,let x_n1∈Xbe such thatgx_n1fxnyn.

We first show that d

yn, y_n1

≤λd y_n−1, yn

forn≥1. 2.20

Notice that

d

yn, y_n1 d

fxn, fx_n1

≤λ·un, 2.21

where

un∈

d

gxn, gx_n1 ,d

fx_n, gx_n d

fx_n1, gx_n1

2 ,d

fx_n, gx_n1 d

fx_n1, gx_n 2

d

y_n−1, y_n ,d

y_n−1, yn

d

yn, y_n1

2 ,d

y_n−1, y_n1 2

.

2.22

As inTheorem 2.1, we have to consider three cases.

Ifu dy_n−1, y_n, then clearly2.20holds. Ifu dy_n−1, y_n dyn, y_n1/2,then from2.19withxxnandyx_n1,asλ∈0,1,we have

d

y_n, y_n1

≤λd

y_n−1, y_n d

y_n, y_n1

2 ≤λd

y_n−1, y_n

2 d

y_n, y_n1

2 . 2.23

Hence, dyn, y_n1 ≤ λdyn−1, yn, and in this case 2.20 holds. Finally, if u dy_n−1, y_n1/2,then

d

y_n, y_n1

≤λd

y_n−1, y_n1

2 ≤λd

y_n−1, y_n d

y_n, y_n1 2

≤λd

y_n−1, y_n

2 d

y_n, y_n1

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 d

fp, q

≤d

fp, fx_n d

fx_n, q

≤λ·u_nd fx_n, q

, 2.25

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where

un∈

d

gxn, gp ,d

fx_n, gx_n

dfp, gp

2 ,d

fx_n, gp d

fp, gx_n 2

. 2.26

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

Case 1⁰

dfp, q≤λ·d

gx_n, gp d

fx_n, q

λ· c 2λ c

2 c. 2.27

Case 2⁰

dfp, q≤λ·d

fxn, gxn

dfp, gp

2 dfxn, q

≤ λd

fxn, gxn

2 dfp, gp

2 d

fx_n, q

fxn, q λd

gxn, q

λ2 c

2λ2λ c 2λ c.

2.28

Case 3⁰

dfp, q≤λ·d

fx_n, gp d

fp, gx_n

2 d

fxn, q

≤ λd

fx_n, gp

2 1

2dfp, q λ 2d

q, gxn

d fxn, q

fx_n, q λd

gx_n, 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

d q, q1

d fp, fp1

≤λ·u, 2.30

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where

u∈

d gp, gp1

,dfp, gp d

fp₁, gp₁

2 ,d

fp, gp₁ d

fp₁, gp 2

dq, q,0,d q, q1

d q1, q 2

0, d q, 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

dfx, fy≤λ·dfx, gx dfy, gy

2 . 2.33

dfx, fy≤λ·dfx, gy dfy, gx

2 . 2.34

In the next corollary, among other things, we generalize the well-known result12, 9.

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Corollary 2.6. Suppose thatf, gis AJ’s pair, and that for some constantλ ∈ 0,1and for every x, y∈X,there exists

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.

Corollary 2.7. Suppose thatf, gis AJ’s pair, and that for some constantλ ∈ 0,1and for every x, y∈X,there exists

uux, y∈ {dfx, gx, dfy, gy}, 2.37

such that

dfx, fy≤λ·u. 2.38

When in the next theorem we setg I_X,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

i1a_i<1 such that, for eachx, y∈X,

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

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

d

yn, y_n1

≤λd y_n−1, yn

, for someλ∈0,1, n≥1. 2.40

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From d

y_n1, y_n d

fx_n1, fx_n

≤a₁d

gx_n1, gx_n a₂d

gx_n1, fx_n1 a3d

gxn, fxn

a4d

gx_n1, fxn

a5d

gxn, fx_n1 a₁d

y_n, y_n−1 a₂d

y_n, y_n1 a₃d

y_n−1, y_n a₅d

y_n−1, y_n1 , d

y_n, y_n1 d

fx_n, fx_n1

≤a₁d

gx_n, gx_n1 a₂d

gx_n, fx_n a3d

gx_n1, fx_n1 a4d

gxn, fx_n1 a5d

gx_n1, fxn

a₁d

y_n−1, y_n a₂d

y_n−1, y_n a₃d

y_n, y_n1 a₄d

y_n−1, y_n1 ,

2.41

we obtain

2d

y_n, y_n1

≤2a₁d

y_n−1, y_n

2a₂d

y_n−1, y_n a₃d

y_n, y_n1 a₄d

y_n−1, y_n1

a₃a₄a₅ d

y_n, y_n1

. 2.42

Thus,

d

y_n, y_n1

≤ 2a₁a₂a₃a₄ 2−a3−a4−a5 ·d

y_n−1, y_n

λ·d

y_n−1, y_n

, 2.43

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

Now, from the proof ofTheorem 2.1, we know that{gx_n1}{fxn}{yn}is a Cauchy sequence. Hence, there existqingXandp∈Xsuch thatgx_n → q,n → ∞,andgpq.

We have to show thatfpq.For this we have dfp, q≤d

fp, fxn

d fxn, q

≤a1d

gxn, gp a2d

gxn, fxn

a₃d

gp, fp a₄d

gx_n, fp a₅d

gp, fx_n d

fx_n, q

≤a₁d gx_n, q

a₂d gx_n, q

a₂d q, fx_n

a₃dq, fp a₄d

gx_n, q

a₄dq, fp a₅1

d q, fx_n

,that is, dfp, q≤ a₁a₂a₄

1−a3−a4

d gx_n, q

1a₂a₅ 1−a3−a4

d fx_n, q a1a2a4

1−a₃−a₄

c

2a1a₂a₄/1−a₃−a₄ 1a2a5

1−a₃−a₄

c

21a₂a₅/1−a₃−a₄ c.

2.44

Then according toCorollary 1.43,dfp, q 0, that is,fpq.

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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,

d q, q₁

d

fp, fp₁

≤a₁d

gp, gp₁ a₂d

gp, fp a₃d

gp₁, fp₁ a₄d

gp, fp₁ a₅d

gp₁, fp

a₁a₄a₅ d

q, q₁ .

2.45

According toRemark 1.8, and because 0≤a₁a₄a₅ ≤₅

i1a_i <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 ofa_iinTheorem 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 e^t.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

The fourth author would like to express his gratitude to Professor Sh. Rezapour and to Professor S. M. Veazpour for the valuable comments. The second, third, and fourth authors thank the Ministry of Science and the Ministry of Environmental Protection of Serbia for their support.

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