Some Coupled Fixed Point Results on Partial Metric Spaces

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Volume 2011, Article ID 647091,11pages doi:10.1155/2011/647091

Research Article

Some Coupled Fixed Point Results on

Partial Metric Spaces

Hassen Aydi

Institut Supérieur d’Informatique de Mahdia, Université de Monastir, route de Réjiche, Km 4, BP 35, Mahdia 5121, Tunisia

Correspondence should be addressed to Hassen Aydi,[email protected]

Received 20 December 2010; Revised 19 February 2011; Accepted 19 March 2011

Academic Editor: Enrique Llorens-Fuster

Copyrightq2011 Hassen Aydi. 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.

We give some coupled fixed point results for mappings satisfying diﬀerent contractive conditions on complete partial metric spaces.

1. Introduction and Preliminaries

For a given partially ordered set X, Bhaskar and Lakshmikantham in 1 introduced the concept of coupled fixed point of a mapping F : X ×X → X. Later in 2, C´ır´ıc and Lakshmikantham investigated some more coupled fixed point theorems in partially ordered sets. The following is the corresponding definition of a coupled fixed point.

Definition 1.1see3. An elementx, y∈ X×Xis said to be a coupled fixed point of the mappingF:X×X → XifFx, y xandFy, x y.

Sabetghadam et al.4obtained the following.

Theorem 1.2. LetX, dbe a complete cone metric space. Suppose that the mappingF:X×X → X

satisfies the following contractive condition for allx, y, u, v∈X

dFx, y, Fu, v≤kdx, u ldy, v, 1.1

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In this paper, we give the analogous of this resultand some others in4on partial metric spaces, and we establish some coupled fixed point results.

The concept of partial metric spaceX, p was introduced by Matthews in 1994. In such spaces, the distance of a point in the self may not be zero. First, we start with some preliminaries definitions on the partial metric spaces3,5–13.

Definition 1.3see6–8. A partial metric on a nonempty setXis a functionp:X×X−→Ê

such that for allx, y, z∈X:

p1xy⇐⇒px, x px, y py, y,

p2px, x≤px, y,

p3px, y py, x,

p4px, y≤px, z pz, y−pz, z.

A partial metric space is a pairX, psuch thatXis a nonempty set andpis a partial metric onX.

Remark 1.4. It is clear that ifpx, y 0, then fromp1,p2, andp3,x y. But ifx y,

px, ymay not be 0.

Ifpis a partial metric onX, then the functionps_:_X_×_X_−→_R

given by

psx, y2px, y−px, x−py, y, 1.2

is a metric onX.

Definition 1.5see6–8. LetX, pbe a partial metric space. Then,

ia sequence{xn}in a partial metric spaceX, pconverges to a pointx∈ X if and

only ifpx, x limn→∞px, xn;

iia sequence{xn}in a partial metric spaceX, pis called a Cauchy sequence if there

existsand is finitelimn,m→∞pxn, xm;

iiia partial metric spaceX, pis said to be complete if every Cauchy sequence{xn}

inXconverges to a pointx∈X, that is,px, x limn,m→∞pxn, xm.

Lemma 1.6see6,7,9. LetX, pbe a partial metric space;

a{xn}is a Cauchy sequence inX, pif and only if it is a Cauchy sequence in the metric

spaceX, ps_,

ba partial metric spaceX, pis complete if and only if the metric spaceX, ps_{is complete;}

furthermore, limn→∞psxn, x 0 if and only if

px, x lim

n→∞pxn, x n,m→lim∞pxn, xm. 1.3

2. Main Results

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Theorem 2.1. LetX, pbe a complete partial metric space. Suppose that the mappingF:X×X → Xsatisfies the following contractive condition for allx, y, u, v∈X

pFx, y, Fu, v≤kpx, u lpy, v, 2.1

wherek,lare nonnegative constants withkl <1. Then,Fhas a unique coupled fixed point.

Proof. Choosex0, y0 ∈Xand setx1Fx0, y0andy1Fy0, x0. Repeating this process, set

xn1Fxn, ynandyn1Fyn, xn. Then, by2.1, we have

pxn, xn1 p

Fxn−1, yn−1, Fxn, yn

≤kpxn−1, xn lp

yn−1, yn

, 2.2

and similarly

pyn, yn1pFyn−1, xn−1, Fyn, xn

≤kpyn−1, yn

lpxn−1, xn.

2.3

Therefore, by letting

dnpxn, xn1 pyn, yn1, 2.4

we have

dnpxn, xn1 p

yn, yn1

≤kpxn−1, xn lp

yn−1, yn

kpyn−1, yn

lpxn−1, xn

klpyn−1, yn

pxn−1, xn

kldn−1.

2.5

Consequently, if we setδkl, then, for eachn∈Æ, we have

dn≤δdn−1≤δ2dn−2 ≤ · · · ≤δnd0. 2.6

Ifd00 thenpx0, x1 py0, y1 0. Hence, from Remark1.4, we getx0x1Fx0, y0and

y0 y1 Fy0, x0, meaning thatx0, y0is a coupled fixed point ofF. Now, letd0 > 0. For

eachn≥m, we have, in view of the conditionp4

pxn, xm≤pxn, xn−1 pxn−1, xn−2−pxn−1, xn−1

pxn−2, xn−3 pxn−3, xn−4−pxn−3, xn−3

· · ·pxm2, xm1 pxm1, xm−pxm1, xm1

≤pxn, xn−1 pxn−1, xn−2 · · ·pxm1, xm.

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Similarly, we have

pyn, ym

≤pyn, yn−1pyn−1, yn−2· · ·pym1, ym

. 2.8

Thus,

pxn, xm p

yn, ym

≤dn−1dn−2· · ·dm

≤δn−1δn−2· · ·δmd0

≤ δm 1−δd0.

2.9

By definition ofps_{, we have}_ps_{x, y}_≤₂_p_{x, y}_{, so, for any}_n_≥_m

psxn, xm ps

yn, ym

≤2pxn, xm 2p

yn, ym

≤2 δ

m

1−δd0, 2.10

which implies that{xn}and{yn}are Cauchy sequences inX, psbecause of 0≤δkl <1.

Since the partial metric spaceX, pis complete, hence thanks to Lemma1.6, the metric space

X, psis complete, so there existu∗, v∗∈Xsuch that

lim

n→∞p

s_x

n, u∗ lim

n→∞p

s_y

n, v∗

0. 2.11

Again, from Lemma1.6, we get

pu∗, u∗ lim

n→∞pxn, u

∗ _lim

n→∞pxn, xn,

pv∗, v∗ lim

n→∞p

yn, v∗

lim

n→∞p

yn, yn

.

2.12

But, from conditionp2and2.6,

pxn, xn≤pxn, xn1≤dn≤δnd0, 2.13

so sinceδ∈0,1, hence lettingn → ∞, we get limn→∞pxn, xn 0. It follows that

pu∗, u∗ lim

n→∞pxn, u

∗ _lim

n→∞pxn, xn 0. 2.14

Similarly, we get

pv∗, v∗ lim

n→∞p

yn, v∗

lim

n→∞p

yn, yn

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Therefore, we have, using2.1,

pFu∗, v∗, u∗≤pFu∗, v∗, xn1 pxn1, u∗−pxn1, xn1, byp4

≤pFu∗, v∗, Fxn, yn

pxn1, u∗

≤kpxn, u∗ lp

yn, v∗

pxn1, u∗,

2.16

and letting n → ∞, then from 2.14 and 2.15, we obtain pFu∗, v∗, u∗ 0, so

Fu∗, v∗ u∗. Similarly, we haveFv∗, u∗ v∗, meaning that u∗, v∗ is a coupled fixed point ofF.

Now, ifu, vis another coupled fixed point ofF, then

pu, u∗pFu, v, Fu∗, v∗≤kpu, u∗lpv, v∗,

pv, v∗pFv, u, Fv∗, u∗≤kpv, v∗lpu, u∗. 2.17

It follows that

pu, u∗pv, v∗≤klpu, u∗pv, v∗. 2.18

In view ofkl <1, this implies thatpu, u∗ pv, v∗ 0, sou∗uandv∗v. The proof of Theorem2.1is completed.

It is worth noting that when the constants in Theorem 2.1 are equal, we have the following corollary

Corollary 2.2. LetX, pbe a complete partial metric space. Suppose that the mappingF :X×X → Xsatisfies the following contractive condition for allx,y, u, v∈X

pFx, y, Fu, v≤ k

2

px, u py, v, 2.19

where 0≤k <1. Then,Fhas a unique coupled fixed point.

Example 2.3. LetX 0,∞endowed with the usual partial metricpdefined byp:X×X → 0,∞withpx, y max{x, y}. The partial metric spaceX, pis complete becauseX, ps

is complete. Indeed, for anyx, y∈X,

psx, y2px, y−px, x−py, y2 maxx, y −xyx−y, 2.20

Thus, X, ps _{is the Euclidean metric space which is complete. Consider the mapping} _F _:

X×X → Xdefined byFx, y xy/6. For anyx, y, u, v∈X, we have

pFx, y, Fu, v 1

6max

xy, uv ≤ 1

6

max{x, u}maxy, v

1

6

px, u py, v.

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which is the contractive condition2.19 fork 1/3. Therefore, by Corollary2.2,F has a unique coupled fixed point, which is0,0. Note that if the mappingF:X×X → Xis given byFx, y xy/2, thenFsatisfies the contractive condition2.19fork1, that is,

pFx, y, Fu, v 1

2max

xy, uv ≤ 1

2

max{x, u}maxy, v

1

2

px, u py, v.

2.22

In this case,0,0and1,1are both coupled fixed points ofF, and, hence, the coupled fixed point ofF is not unique. This shows that the condition k < 1 in Corollary2.2, and hence

kl <1 in Theorem2.1cannot be omitted in the statement of the aforesaid results.

Theorem 2.4. LetX, pbe a complete partial metric space. Suppose that the mappingF:X×X → Xsatisfies the following contractive condition for allx, y, u, v∈X

pFx, y, Fu, v≤kpFx, y, xlpFu, v, u, 2.23

wherek,lare nonnegative constants withkl <1. Then,Fhas a unique coupled fixed point.

Proof. We take the same sequences{xn}and{yn}given in the proof of Theorem2.1by

xn1Fxn, yn

, yn1Fyn, xn

for any n∈Æ. 2.24

Applying2.23, we get

pxn, xn1≤δpxn−1, xn 2.25

pyn, yn1≤δpyn−1, yn

2.26

whereδk/1−l. By definition ofps_{, we have}

psxn, xn1≤2pxn, xn1≤2δnpx1, x0, 2.27

psyn, yn1≤2pyn, yn1≤2δnpy1, y0

. 2.28

Sincekl <1, henceδ <1, so the sequences{xn}and{yn}are Cauchy sequences in the metric

spaceX, ps_{. The partial metric space}_{X, p}_{is complete, hence from Lemma}_1.6,_{X, p}s_is

complete, so there existu∗, v∗∈Xsuch that

lim

n→∞p

s_x

n, u∗ lim

n→∞p

s_y

n, v∗

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From Lemma1.6, we get

pu∗, u∗ lim

n→∞pxn, u

∗ _lim

n→∞pxn, xn,

pv∗, v∗ lim

n→∞p

yn, v∗

lim

n→∞p

yn, yn

.

2.30

By the conditionp2and2.25, we have

pxn, xn≤pxn, xn1≤δnpx1, x0, 2.31

so limn→∞pxn, xn 0. It follows that

pu∗, u∗ lim

n→∞pxn, u

∗ _lim

n→∞pxn, xn 0. 2.32

Similarly, we find

pv∗, v∗ lim

n→∞p

yn, v∗

lim

n→∞p

yn, yn

0. 2.33

Therefore, by2.23,

pFu∗, v∗, u∗≤pFu∗, v∗, xn1 pxn1, u∗

pFu∗, v∗, Fxn, yn

pxn1, u∗

≤kpFu∗, v∗, u∗ lpFxn, yn

, xn

pxn1, u∗

kpFu∗, v∗, u∗ lpxn1, xn pxn1, u∗,

2.34

and lettingn → ∞, then from2.27–2.32, we obtain

pFu∗, v∗, u∗≤kpFu∗, v∗, u∗. 2.35

From the preceding inequality, we can deduce a contradiction if we assume that

pFu∗, v∗, u∗/0, because in that case we conclude that 1 ≤ k and now this inequality is, in fact, a contradiction, so pFu∗, v∗, u∗ 0, that is, Fu∗, v∗ u∗. Similarly, we have

Fv∗, u∗ v∗, meaning thatu∗, v∗is a coupled fixed point ofF. Now, ifu, vis another coupled fixed point ofF, then, in view of2.23,

pu, u∗pFu, v, Fu∗, v∗

≤kpFu, v, ulpFu∗, v∗, u∗ kpu, ulpu∗, u∗

≤kpu, u∗lpu, u∗ klpu, u∗, usingp2,

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that is,pu, u∗ 0 sincekl<1. It follows thatu∗u. Similarly, we can havev∗v, and the proof of Theorem2.4is completed.

Theorem 2.5. LetX, pbe a complete partial metric space. Suppose that the mappingF:X×X → Xsatisfies the following contractive condition for allx,y, u, v∈X

pFx, y, Fu, v≤kpFx, y, ulpFu, v, x, 2.37

wherek, lare nonnegative constants withk2l <1. Then,Fhas a unique coupled fixed point.

Proof. Since,k2l < 1, hencekl < 1, and as a consequence the proof of the uniqueness in this theorem is as trivial as in the other results. To prove the existence of the fixed point, choose the sequences{xn}and{yn}like in the proof of Theorem2.1, that is

xn1Fxn, yn

, yn1Fyn, xn

, for anyn∈Æ. 2.38

Applying again2.37, we have

pxn, xn1 pFxn−1, yn−1, Fxn, yn

≤kpFxn−1, yn−1, xn

lpFxn, yn

, xn−1

kpxn, xn lpxn1, xn−1

≤ kpxn1, xn lpxn1, xn−1

, byp2

≤kpxn1, xn lpxn1, xn lpxn, xn−1−lpxn, xn, usingp4

≤klpxn, xn1 lpxn−1, xn.

2.39

It follows that for anyn∈Æ ∗

pxn, xn1≤ ₁₋l

l−kpxn−1, xn. 2.40

Let us takeδl/1−l−k. Hence, we deduce

psxn, xn1≤2pxn, xn1≤2δnpx0, x1. 2.41

Under the condition 0≤k2l <1, we get 0 ≤δ <1. From this fact, we immediately obtain that{xn}is Cauchy in the complete metric spaceX, ps. Of course, similar arguments apply

to the case of the sequence{yn}in order to prove that

psyn, yn1≤2pyn, yn1≤2δnpy0, y1

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and, thus, that the sequence{yn}is Cauchy inX, ps. Therefore, there existu∗, v∗ ∈Xsuch

that

lim

n→∞p

s_x

n, u∗ lim

n→∞p

s_y

n, v∗

0. 2.43

Thanks to Lemma1.6, we have

lim

n→∞pxn, u

∗ _lim

n→∞pxn, xn pu

∗_{, u}∗_,

lim

n→∞p

yn, v∗

lim

n→∞p

yn, yn

pv∗, v∗.

2.44

The conditionp2together with2.41yield that

pxn, xn≤pxn, xn1≤δnpx0, x1, 2.45

hence lettingn → ∞, we get limn→∞pxn, xn 0. It follows that

pu∗, u∗ lim

n→∞pxn, u

∗ _lim

n→∞pxn, xn 0. 2.46

Similarly, we have

pv∗, v∗ lim

n→∞p

yn, v∗

lim

n→∞p

yn, yn

0. 2.47

Therefore, we have, using2.37,

pFu∗, v∗, u∗≤pFu∗, v∗, xn1 pxn1, u∗

pFu∗, v∗, Fxn, yn

pxn1, u∗

≤kpFu∗, v∗, xn lp

Fxn, yn

, u∗pxn1, u∗

kpFu∗, v∗, xn lpxn1, u∗ pxn1, u∗

≤kpFu∗, v∗, u∗ kpu∗, xn lpxn1, u∗ pxn1, u∗, usingp4.

2.48

Lettingn → ∞yields, using2.46,

pFu∗, v∗, u∗≤kpFu∗, v∗, u∗, 2.49

and sincek <1, we havepFu∗, v∗, u∗ 0, that is,Fu∗, v∗ u∗. Similarly, thanks to2.47, we getFv∗, u∗ v∗, and henceu∗, v∗is a coupled fixed point ofF.

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Corollary 2.6. LetX, pbe a complete partial metric space. Suppose that the mappingF :X×X → Xsatisfies the following contractive condition for allx, y, u, v∈X

2

pFx, y, xpFu, v, u, 2.50

where 0≤k <1. Then,Fhas a unique coupled fixed point.

Corollary 2.7. LetX, pbe a complete partial metric space. Suppose that the mappingF :X×X → Xsatisfies the following contractive condition for allx, y, u, v∈X

2

pFx, y, upFu, v, x, 2.51

where 0≤k <2/3. Then,Fhas a unique coupled fixed point.

Proof. The condition 0≤k <2/3 follows from the hypothesis onkandlgiven in Theorem2.5.

Remark 2.8. i Theorem 2.1 extends the Theorem 2.2 of 4 on the class of partial metric spaces.

iiTheorem2.4extends the Theorem 2.5 of4on the class of partial metric spaces.

Remark 2.9. Note that in Theorem2.4, if the mappingF:X×X → Xsatisfies the contractive condition2.23for allx, y, u, v∈X, thenFalso satisfies the following contractive condition:

pFx, y, Fu, vpFu, v, Fx, y

≤kpFu, v, u lpFx, y, x. 2.52

Consequently, by adding2.23and2.52,Falso satisfies the following:

pFx, y, Fu, v≤ kl

2 pFu, v, u

kl

2 p

Fx, y, x, 2.53

which is a contractive condition of the type 2.50 in Corollary 2.6 with equal constants. Therefore, one can also reduce the proof of general case2.23in Theorem2.4to the special case of equal constants. A similar argument is valid for the contractive conditions2.37in Theorem2.5and2.51in Corollary2.7.

Acknowledgment

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References

1 T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces

and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393, 2006.

2 L. C´ır´ıc and V. Lakshmikantham, “Coupled fixed point theorems for nonlinear contractions in

partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4341–4349, 2009.

3 I. Altun, F. Sola, and H. Simsek, “Generalized contractions on partial metric spaces,” Topology and Its

Applications, vol. 157, no. 18, pp. 2778–2785, 2010.

4 F. Sabetghadam, H. P. Masiha, and A. H. Sanatpour, “Some coupled fixed point theorems in cone

metric spaces,” Fixed Point Theory and Applications, Article ID 125426, 8 pages, 2009.

5 H. Aydi, “Some fixed point results in ordered partial metric spaces,” The Journal of Nonlinear Sciences

and Applications. In press.

6 S. G. Matthews, “Partial metric topology, in: Proc. 8th Summer Conference on General Topology and

Applications,” in Annals of the New York Academy of Sciences, vol. 728, pp. 183–197, 1994.

7 S. J. O’Neill, “Two topologies are better than one,” Tech. Rep., University of Warwick, Coventry, UK,

1995,http://www.dcs.warwick.ac.uk/reports/283.html.

8 S. J. O’Neill, “Partial metrics, valuations and domain theory, in: Proc. 11th Summer Conference on

General Topology and Applications,” in Annals of the New York Academy of Sciences, vol. 806, pp. 304– 315, 1996.

9 S. Oltra and O. Valero, “Banach’s fixed point theorem for partial metric spaces,” Rendiconti dell’Istituto

di Matematica dell’Universit`a di Trieste, vol. 36, no. 1-2, pp. 17–26, 2004.

10 S. Romaguera and M. Schellekens, “Partial metric monoids and semivaluation spaces,” Topology and

Its Applications, vol. 153, no. 5-6, pp. 948–962, 2005.

11 S. Romaguera and O. Valero, “A quantitative computational model for complete partial metric spaces

via formal balls,” Mathematical Structures in Computer Science, vol. 19, no. 3, pp. 541–563, 2009.

12 M. P. Schellekens, “The correspondence between partial metrics and semivaluations,” Theoretical

Computer Science, vol. 315, no. 1, pp. 135–149, 2004.

13 O. Valero, “On Banach fixed point theorems for partial metric spaces,” Applied General Topology, vol.

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