Some Common Fixed Point Results in Cone Metric Spaces

Volume 2009, Article ID 493965,11pages doi:10.1155/2009/493965

Research Article

Some Common Fixed Point Results in Cone Metric Spaces

Muhammad Arshad,

Akbar Azam,

and Pasquale Vetro

1Department of Mathematics, Faculty of Basic and Applied Sciences, International Islamic University, H-10, 44000 Islamabad, Pakistan

2Department of Mathematics, F.G. Postgraduate College, H-8, 44000 Islamabad, Pakistan

3Dipartimento di Matematica ed Applicazioni, Universit`a degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy

Correspondence should be addressed to Pasquale Vetro,[email protected] Received 5 September 2008; Revised 26 December 2008; Accepted 5 February 2009 Recommended by Lech G ´orniewicz

We prove a result on points of coincidence and common fixed points for three self-mappings satisfying generalized contractive type conditions in cone metric spaces. We deduce some results on common fixed points for two self-mappings satisfying contractive type conditions in cone metric spaces. These results generalize some well-known recent results.

Copyrightq2009 Muhammad Arshad 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

Huang and Zhang 1 recently have introduced the concept of cone metric space, where the set of real numbers is replaced by an ordered Banach space, and they have established some fixed point theorems for contractive type mappings in a normal cone metric space.

Subsequently, some other authors2–5have generalized the results of Huang and Zhang1 and have studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces.

Vetro 5 extends the results of Abbas and Jungck 2 and obtains common fixed point of two mappings satisfying a more general contractive type condition. Rezapour and Hamlbarani6prove that there aren’t normal cones with normal constant c < 1 and for each k > 1 there are cones with normal constant c > k. Also, omitting the assumption of normality they obtain generalizations of some results of 1. In 7 Di Bari and Vetro obtain results on points of coincidence and common fixed points in nonnormal cone metric spaces. In this paper, we obtain points of coincidence and common fixed points for three self- mappings satisfying generalized contractive type conditions in a complete cone metric space.

Our results improve and generalize the results in1,2,5,6,8.

2. Preliminaries

We recall the definition of cone metric spaces and the notion of convergence1. LetEbe a real Banach space andP be a subset ofE. The subsetP is called an order cone if it has the following properties:

iPis nonempty, closed, andP /{0};

ii0a, b∈Randx, y∈P ⇒axby∈P; iiiP∩−P {0}.

For a given coneP ⊆E, we can define a partial orderingonEwith respect toP by xyif and only ify−x∈P. We will writex < yifxyandx /y, whilexywill stands fory−x∈IntP, where IntP denotes the interior ofP.The conePis called normal if there is a numberκ1 such that for allx, y∈E:

0xy⇒ xκy. 2.1

The least numberκ1 satisfying2.1is called the normal constant ofP.

In the following we always suppose thatEis a real Banach space andP is an order cone inE with IntP /∅andis the partial ordering with respect toP.

Definition 2.1. LetXbe a nonempty set. Suppose that the mappingd:X×X → Esatisfies i0dx, y,for allx, y∈X,anddx, y 0 if and only ifxy;

iidx, y dy, xfor allx, y∈X;

iiidx, ydx, z dz, y, for allx, y, z∈X.

Thendis called a cone metric onX, andX, dis called a cone metric space.

Let{xn} be a sequence inX, andx∈X. If for everyc∈E,with0cthere isn₀ ∈N such that for alln≥n0, dxn, xc,then{xn}is said to be convergent,{xn} converges tox andx is the limit of {xn}.We denote this by lim_nx_n x,orx_n → x,asn → ∞.If for every c∈Ewith0cthere isn₀∈Nsuch that for alln, m≥n₀, dxn, x_mc,then{xn}is called a Cauchy sequence inX. If every Cauchy sequence is convergent inX, thenXis called a complete cone metric space.

3. Main Results

First, we establish the result on points of coincidence and common fixed points for three self- mappings and then show that this result generalizes some of recent results of fixed point.

A pairf, Tof self-mappings onXis said to be weakly compatible if they commute at their coincidence pointi.e.,fTxTfxwheneverfxTx. A pointy∈Xis called point of coincidence of a familyTj,j ∈J, of self-mappings onXif there exists a pointx∈Xsuch thatyTjxfor allj∈J.

Lemma 3.1. Let X be a nonempty set and the mappings S, T, f : X → X have a unique point of coincidencev in X. IfS, fand T, fare weakly compatibles, thenS, T, andf have a unique common fixed point.

Proof. Sincevis a point of coincidence ofS, T, and f. Therefore,vfuSuTufor some u∈X.By weakly compatibility ofS, fandT, fwe have

SvSfufSufv, TvTfufTufv. 3.1

It implies thatSv Tv fv w say. Thenw is a point of coincidence of S, T, and f.

Therefore,vwby uniqueness. Thusvis a unique common fixed point ofS, T, andf.

Let X, d be a cone metric space, S, T, f be self-mappings on X such thatSX ∪ TX ⊆ fXandx0 ∈ X. Choose a pointx1 inX such thatfx1 Sx0. This can be done sinceSX⊆fX. Successively, choose a pointx2 inXsuch thatfx2Tx1.Continuing this process having chosenx₁, . . . , x_2k, we choosex_2k1andx_2k2inXsuch that

fx_2k1Sx_2k,

fx_2k2Tx_2k1, k0,1,2, . . . . 3.2

The sequence{fxn}is called anS-T-sequence with initial pointx₀.

Proposition 3.2. LetX, dbe a cone metric space andP be an order cone. LetS, T, f :X → Xbe such thatSX∪TX⊆fX. Assume that the following conditions hold:

idSx, Tyαdfx, Sx βdfy, Ty γdfx, fy, for allx, y∈X, withx /y, where α, β, γare nonnegative real numbers withαβγ <1;

iidSx, Tx< dfx, Sx dfx, Tx, for allx∈X, wheneverSx /Tx.

Then everyS-T-sequence with initial pointx₀∈Xis a Cauchy sequence.

Proof. Let x0 be an arbitrary point inXand{fxn}be anS-T-sequence with initial pointx0. First, we assume thatfx_n/fx_n1for alln∈N. It implies thatx_n/x_n1for alln. Then,

d

fx_2k1, fx_2k2 d

Sx_2k, Tx_2k1

αd

fx_2k, Sx_2k βd

fx_2k1, Tx_2k1 γd

fx_2k, fx_2k1

αγd

fx2k, fx_2k1 βd

fx_2k1, fx_2k2 .

3.3

It implies that

1−βd

fx_2k1, fx_2k2

αγd

fx_2k, fx_2k1

, 3.4

so

d

fx_2k1, fx_2k2

αγ 1−β

d

fx_2k, fx_2k1

. 3.5

Similarly, we obtain

d

fx_2k2, fx_2k3

βγ 1−α

d

fx_2k1, fx_2k2

. 3.6

Now, by induction, for eachk0,1,2, . . . ,we deduce

d

fx_2k1, fx_2k2

αγ 1−β

d

fx2k, fx_2k1

αγ

1−β

βγ 1−α

d

fx_2k−1, fx_2k

· · · αγ

1−β

βγ 1−α

αγ 1−β

k

d

fx0, fx1

,

d

fx_2k2, fx_2k3

βγ 1−α

d

fx_2k1, fx_2k2

· · ·

βγ 1−α

αγ 1−β

_k1 d

fx₀, fx₁ .

3.7

Let

λ αγ

1−β

, μ

βγ 1−α

. 3.8

Thenλμ <1.Now, forp < q, we have d

fx_2p1, fx_2q1

d

fx_2p1, fx_2p2 d

fx_2p2, fx_2p3 d

fx_2p3, fx_2p4 · · ·d

fx2q, fx_2q1

λ

q−1 ip

λμⁱ

q ip1

λμⁱ

d

fx0, fx1

λλμ^p

1−λμ λμ^p1 1−λμ

d

fx0, fx1

1μλλμ^p

1−λμd

fx₀, fx₁

2λμ^p

1−λμd

fx₀, fx₁ .

3.9

In analogous way, we deduce

d

fx2p, fx_2q1

1λλμ^p

1−λμd

fx0, fx1

≤ 2λμ^p 1−λμd

fx0, fx1

,

d

fx2p, fx2q

1λλμ^p 1−λμd

fx0, fx1

≤ 2λμ^p 1−λμd

fx0, fx1

,

d

fx_2p1, fx2q

1μλλμ^p 1−λμd

fx0, fx1

≤ 2λμ^p 1−λμd

fx0, fx1

.

3.10

Hence, for 0< n < m

d

fxn, fxm

2λμ^p

1−λμ, 3.11

wherepis the integer part ofn/2.

Fix0cand chooseI0, δ {x∈E:x< δ}such thatcI0, δ⊂IntP.Since

p→ ∞lim 2λμ^p 1−λμd

fx₀, fx₁

0, 3.12

there existsn₀∈Nbe such that

2λμ^p 1−λμd

fx₀, fx₁

∈I0, δ 3.13

for allp≥n₀. The choice ofI0, δassures

c−2λμ^p 1−λμd

fx0, fx1

∈IntP, 3.14

so

2λμ^p 1−λμd

fx0, fx1

c. 3.15

Consequently, for alln, m∈N, with 2n0< n < m, we have d

fxn, fxm

c, 3.16

and hence{fxn}is a Cauchy sequence.

Now, we suppose thatfxmfx_m1for somem∈N. Ifxm x_m1andm2k, byii we have

d

fx_2k1, fx_2k2 d

Sx_2k, Tx_2k1

< d

fx_2k, Sx_2k d

fx_2k1, Tx_2k1 d

fx_2k1, fx_2k2 ,

3.17

which impliesfx_2k1 fx_2k2. Ifxm/x_m1we useito obtainfx_2k1fx_2k2. Similarly, we deduce thatfx_2k2 fx_2k3 and sofxn fxmfor everyn ≥ m. Hence{fxn}is a Cauchy sequence.

Theorem 3.3. LetX, dbe a cone metric space andPbe an order cone. LetS, T, f :X → Xbe such thatSX∪TX⊆fX. Assume that the following conditions hold:

idSx, Tyαdfx, Sx βdfy, Ty γdfx, fy, for allx, y∈X, withx /y, where α, β, γare nonnegative real numbers withαβγ <1;

iidSx, Tx< dfx, Sx dfx, Tx, for allx∈X, wheneverSx /Tx.

If fX or SX ∪TX is a complete subspace of X, then S, T, and f have a unique point of coincidence. Moreover, ifS, fand T, fare weakly compatibles, thenS, T, and f have a unique common fixed point.

Proof. Let x0 be an arbitrary point inX. ByProposition 3.2everyS-T-sequence{fxn}with initial pointx0is a Cauchy sequence. IffXis a complete subspace ofX, there existu, v∈X such thatfx_n → vfuthis holds also ifSX∪TXis complete withv∈SX∪TX.

From

dfu, Sud

fu, fx2n

d

fx2n, Su

d

v, fx2n

d

Tx_2n−1, Su

d

v, fx2n

αdfu, Su βd

fx_2n−1, Tx_2n−1 γd

fu, fx_2n−1 ,

3.18

we obtain

dfu, Su 1 1−α

d v, fx2n

βd

fx_2n−1, fx2n

γd

v, fx_2n−1

. 3.19

Fix0cand choosen0∈Nbe such that d

v, fx2n

kc, d

fx_2n−1, fx2n

kc, d

v, fx_2n−1

kc 3.20

for alln≥n0, wherek 1−α/1βγ. Consequentlydfu, Sucand hencedfu, Su c/mfor everym∈N. From

c

m−dfu, Su∈IntP, 3.21

beingP closed, asm → ∞, we deduce−dfu, Su ∈ P and sodfu, Su 0. This implies thatfuSu.

Similarly, by using the inequality, dfu, Tud

fu, fx_2n1 d

fx_2n1, Tu

, 3.22

we can show thatfuTu.It implies thatvis a point of coincidence ofS, T, andf, that is

vfuSuTu. 3.23

Now, we show thatS, T, andfhave a unique point of coincidence. For this, assume that there exists another pointv^∗inXsuch thatv^∗fu^∗Su^∗Tu^∗, for some u^∗inX.From

d v, v^∗

d

Su, Tu^∗ αdfu, Su βd

fu^∗, Tu^∗ γd

fu, fu^∗ αdv, v βd

v^∗, v^∗ γd

v, v^∗

γd

v, v^∗

3.24

we deducevv^∗.Moreover, ifS, fandT, fare weakly compatibles, then

SvSfufSufv, TvTfufTufv, 3.25

which impliesSv Tv fv w say. Thenw is a point of coincidence ofS, T, andf therefore,vw,by uniqueness. Thusv is a unique common fixed point ofS, T, andf.

FromTheorem 3.3, if we chooseST, we deduce the following theorem.

Theorem 3.4. LetX, dbe a cone metric space,Pbe an order cone andT, f:X → Xbe such that TX⊆fX. Assume that the following condition holds:

dTx, Tyαdfx, Tx βdfy, Ty γdfx, fy 3.26

for allx, y∈Xwhereα, β, γ ∈0,1withαβγ <1.

IffXor TXis a complete subspace ofX, thenTandfhave a unique point of coincidence.

Moreover, if the pairT, fis weakly compatible, thenTandfhave a unique common fixed point.

Theorem 3.4generalizes Theorem 1 of5.

Remark 3.5. InTheorem 3.4the condition3.26can be replaced by

dTx, Tyαdfx, Tx dfy, Ty γdfx, fy 3.27

for allx, y∈X, whereα, γ ∈0,1with 2αγ <1.

3.27⇒3.26is obivious.3.26⇒3.27. If in3.26interchanging the roles ofxandy and adding the resultant inequality to3.26, we obtain

dTx, Tyαβ

2 dfx, Tx dfy, Ty γdfx, fy. 3.28 FromTheorem 3.4, we deduce the followings corollaries.

Corollary 3.6. LetX, dbe a cone metric space,Pbe an order cone and the mappingsT, f:X → X satisfy

dTx, Tyγdfx, fy 3.29

for allx, y∈Xwhere, 0γ <1.IfTX⊆fXandfXis a complete subspace ofX, thenTand fhave a unique point of coincidence. Moreover, if the pairT, fis weakly compatible, thenT andf have a unique common fixed point.

Corollary 3.6generalizes Theorem 2.1 of2, Theorem 1 of1, and Theorem 2.3 of6.

Corollary 3.7. LetX, dbe a cone metric space,Pbe an order cone and the mappingsT, f:X → X satisfy

dTx, Tyαdfx, Tx dfy, Ty 3.30

for allx, y∈X, where 0α <1/2.IfTX⊆fXandfXis a complete subspace ofX, thenT andfhave a unique point of coincidence. Moreover, if the pairT, fis weakly compatible, thenTand fhave a unique common fixed point.

Corollary 3.7generalizes Theorem 2.3 of2, Theorem 3 of1, and Theorem 2.6 of6.

Example 3.8. LetX {a, b, c},ER²andP {x, y∈E|x, y0}.Defined:X×X → E as follows:

dx, y

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

0,0 ifxy, 5

7,5

ifx /y, x, y∈X− {b}, 1,7 ifx /y, x, y∈X− {c}, 4

7,4

ifx /y, x, y∈X− {a}.

3.31

Define mappingsf, T :X → Xas follow:

fx x,

Tx

⎧⎨

⎩

c, if x /b, a, ifxb.

3.32

Then, if 2αγ <1

7α4γ

7 ,7α4γ

8α4γ

7 ,8α4γ

42αγ

7 ,42αγ

<

4 7,4

<

5 7,5

,

3.33

which implies

αdfb, Tb dfc, Tc γdfb, fc< dTb, Tc, 3.34

for allα, γ ∈0,1with 2αγ <1.

Therefore,Theorem 3.4is not applicable to obtain fixed point ofT or common fixed points offand T.

Now define a constant mappingS:X → XbySxc, then forα0γ, β5/7.

dSx, Ty

⎧⎪

⎨

⎪⎩

0,0, ify /b, 5

7,5

, ifyb, αdfx, Sx βdfy, Ty γdfx, fy

5 7,5

ifyb.

3.35

It follows that all conditions ofTheorem 3.3are satisfied forα 0 γ, β 5/7 and soS, T, andfhave a unique point of coincidence and a unique common fixed pointc.

4. Applications

In this section, we prove an existence theorem for the common solutions for two Urysohn integral equations. Throughout this section letX Ca, b,Rⁿ,P {u, v∈R² :u, v≥0}, anddx, y x−y∞, px−y∞for everyx, y∈X, wherep≥0 is a constant. It is easily seen thatX, dis a complete cone metric space.

Theorem 4.1. Consider the Urysohn integral equations xt

_b

a

K₁t, s, xsdsgt,

xt _b

a

K2t, s, xsdsht,

4.1

wheret∈a, b⊂R,x, g, h∈X. Assume thatK₁, K₂:a, b×a, b×Rⁿ → Rⁿare such that

iFx, Gx∈Xfor eachx∈X,where

F_xt _b

a

K₁t, s, xsds, G_xt _b

a

K₂t, s, xsds ∀t∈a, b, 4.2

iithere existα, β, γ≥0 such that

F_xt−G_yt gt−ht, pF_xt−G_yt gt−ht

≤αF_xt gt−xt, pF_xt gt−xt βG_yt ht−yt, pG_yt ht−yt γ|xt−yt|, p|xt−yt|,

4.3

whereαβγ <1, for everyx, y∈Xwithx /yandt∈a, b.

iiiwheneverFxg /Gxh

sup

t∈a,b

Fxt−Gxt gt−ht, pFxt−Gxt gt−ht

< sup

t∈a,b

F_xt gt−xt, pF_xt gt−xt sup

t∈a,b

G_xt ht−xt, pG_xt ht−xt,

4.4

for everyx∈X.

Then the system of integral equations4.1have a unique common solution.

Proof. DefineS, T :X → XbySx Fxg, Tx Gxh. It is easily seen that S−T_∞, pS−T_∞

≤αSx−x

∞, pSx−x

∞

βTy−y

∞, pTy−y

∞

γ

x−y_∞, px−y_∞ ,

4.5

for everyx, y∈X, withx /yand ifSx/Tx S−T_∞, pS−T_∞

<Sx−x

∞, pSx−x

∞

Tx−x_∞, pTx−x_∞ 4.6 for everyx∈X. ByTheorem 3.3, iffis the identity map onX, the Urysohn integral equations 4.1have a unique common solution.

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