and R. Saadati

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Volume 2010, Article ID 125082,13pages doi:10.1155/2010/125082

Research Article

Common Fixed Point Theorem in Partially Ordered L -Fuzzy Metric Spaces

S. Shakeri,

¹

L. J. B. ´ Ciri´c,

²

and R. Saadati

³

1Young Research Club, Islamic Azad University-Ayatollah Amoli Branch, P.O. Box 678, Amol, Iran

2Faculty of Mechanical Engineering, Kraljice Marije 16, 11 000 Belgrade, Serbia

3Faculty of Sciences, Islamic Azad University-Ayatollah Amoli Branch, P.O. Box 678, Amol, Iran

Correspondence should be addressed to R. Saadati,[email protected] Received 29 October 2009; Accepted 27 January 2010

Academic Editor: Juan Jose Nieto

Copyrightq2010 S. Shakeri 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.

We introduce partially orderedL-fuzzy metric spaces and prove a common fixed point theorem in these spaces.

1. Introduction

The Banach fixed point theorem for contraction mappings has been generalized and extended in many directions 1–43. Recently Nieto and Rodr´ıguez-L ´opez 27–29 and Ran and Reurings 33 presented some new results for contractions in partially ordered metric spaces. The main idea in27–33involves combining the ideas of iterative technique in the contraction mapping principle with those in the monotone technique.

Recall that ifX,≤is a partially ordered set andF : X → X is such that forx, y ∈ X, x≤yimpliesFx≤Fy, then a mappingFis said to be nondecreasing. The main result of Nieto and Rodr´ıguez-L ´opez27–33and Ran and Reurings33is the following fixed point theorem.

Theorem 1.1. LetX,≤be a partially ordered set and suppose that there is a metricdonXsuch that X, dis a complete metric space. Suppose thatFis a nondecreasing mapping with

d

Fx, F y

≤kd x, y

1.1 for allx, y∈X, x≤y,where 0< k <1.Also suppose the following.

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aFis continuous.

bIf{xn} ⊂Xis a nondecreasing sequence withxn → xinX, thenxn≤xfor allnhold.

If there exists anx0∈Xwithx0≤Fx0, thenFhas a fixed point.

The works of Nieto and Rodr´ıguez-L ópez27,28and Ran and Reurings33have motivated Agarwal et al.1, Bhaskar and Lakshmikantham3, and Lakshmikantham and Cirić´ 23to undertake further investigation of fixed points in the area of ordered metric spaces. We prove the existence and approximation results for a wide class of contractive mappings in intuitionistic metric space. Our results are an extension and improvement of the results of Nieto and Rodr´ıguez-L ópez27,28and Ran and Reurings33to more general class of contractive type mappings and include several recent developments.

2. Preliminaries

The notion of fuzzy sets was introduced by Zadeh 44. Various concepts of fuzzy metric spaces were considered in15,16,22,45. Many authors have studied fixed point theory in fuzzy metric spaces; see, for example,7,8,25,26,39,46–48. In the sequel, we will adopt the usual terminology, notation, and conventions ofL-fuzzy metric spaces introduced by Saadati et al. 36which are a generalization of fuzzy metric sapces49and intuitionistic fuzzy metric spaces32,37.

Definition 2.1see46. LetL L,≤Lbe a complete lattice, andUa nonempty set called a universe. AnL-fuzzy setAonUis defined as a mappingA:U → L. For eachuinU,Au represents the degreeinLto whichusatisfiesA.

Lemma 2.2see13,14. Consider the setL^∗and the operation≤L^∗defined by

L^∗

x1, x2:x1, x2∈0,1², x1x2≤1

, 2.1

x1, x2≤L^∗y1, y2 ⇔ x1 ≤ y1, andx2 ≥ y2, for everyx1, x2,y1, y2 ∈ L^∗. ThenL^∗,≤L^∗is a complete lattice.

Classically, a triangular normTon0,1,≤is defined as an increasing, commutative, associative mapping T : 0,1² → 0,1 satisfying T1, x x, for all x ∈ 0,1. These definitions can be straightforwardly extended to any latticeL L,≤L. Define first 0LinfL and 1_L supL.

Definition 2.3. A negation on Lis any strictly decreasing mapping N : L → L satisfying N0L 1_L andN1L 0_L. IfNNx x, for allx ∈L, thenNis called an involutive negation.

In this paper the negationN:L → Lis fixed.

Definition 2.4. A triangular normt-norm onLis a mapping T : L² → Lsatisfying the following conditions:

i for allx∈LTx,1_L x boundary condition;

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ii for allx, y∈L²Tx, y Ty, x commutativity;

iii for allx, y, z∈L³Tx,Ty, z TTx, y, z associativity;

iv for allx, x, y, y∈L⁴x≤Lx andy≤Ly ⇒ Tx, y≤LTx, y monotonicity.

At-normTonLis said to be continuous if for anyx, y ∈ Land any sequences{xn} and{yn}which converge toxandywe have

limn T xn, yn

T x, y

. 2.2

For example,Tx, y minx, yandTx, y xyare two continuoust-norms on0,1. A t-norm can also be defined recursively as ann1-ary operationn∈NbyT¹Tand

Tⁿx1, . . . , x_n1 T

Tⁿ⁻¹x1, . . . , xn, x_n1

2.3

forn≥2 andxi∈L.

At-normTis said to be of Hadˇzi´c type if the family{Tⁿ}_n∈Nis equicontinuous atx1_L, that is,

∀ε∈L\ {0_L,1_L}∃δ∈L\ {0_L,1_L}:a>LNδ ⇒ Tⁿa>LNε n≥1. 2.4

TMis a trivial example of at-norm of Hadˇzi´c type, but there existt-norms of Hadˇzi´c type weaker thanTM50where

TM

x, y

⎧⎨

⎩

x, ifx≤Ly,

y, ify≤Lx. 2.5

Definition 2.5. The 3-tupleX,M,Tis said to be anL-fuzzy metric space ifX is an arbitrary nonempty set, T is a continuoust-norm onL and Mis an L-fuzzy set on X²×0,∞ satisfying the following conditions for everyx, y, zinXandt, sin0,∞:

aMx, y, t>L0_L;

bMx, y, t 1_Lfor allt >0 if and only ifxy;

cMx, y, t My, x, t;

dTMx, y, t,My, z, s≤LMx, z, ts;

eMx, y,·:0,∞ → Lis continuous.

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If theL-fuzzy metric spaceX,M,Tsatisfies the condition:

f

t→ ∞limM x, y, t

1_L, 2.6

thenX,M,Tis said to be MengerL-fuzzy metric space or for short aML-fuzzy metric space.

Let X,M,Tbe an L-fuzzy metric space. For t ∈0,∞, we define the open ball Bx, r, twith centerx∈Xand radiusr∈L\ {0L,1_L}, as

Bx, r, t

y∈X:M x, y, t

>LNr

. 2.7

A subsetA⊆ X is called open if for eachx∈ A, there existt >0 andr ∈ L\ {0_L,1_L}such thatBx, r, t ⊆ A. LetτM denote the family of all open subsets ofX. ThenτM is called the topology induced by theL-fuzzy metricM.

Example 2.6see38. LetX, dbe a metric space. DenoteTa, b a1b1,mina2b2,1 for alla a1, a2andb b1, b2inL^∗and letMandNbe fuzzy sets onX²×0,∞defined as follows:

MM,N

x, y, t

M x, y, t

, N

x, y, t

t td

x, y, d x, y td

x, y

. 2.8

ThenX,MM,N,Tis an intuitionistic fuzzy metric space.

Example 2.7. LetXN. DefineTa, b max0, a1b1−1, a2b2−a2b2for alla a1, a2 andb b1, b2inL^∗, and letMx, y, tonX²×0,∞be defined as follows:

M x, y, t

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ x

y,y−x y

if x≤y, y

x,x−y x

if y≤x

2.9

for allx, y∈Xandt >0. ThenX,M,Tis anL-fuzzy metric space.

Lemma 2.8see49. LetX,M,Tbe anL-fuzzy metric space. Then,Mx, y, tis nondecreasing with respect tot, for allx, yinX.

Definition 2.9. A sequence{xn}_n∈Nin anL-fuzzy metric spaceX,M,Tis called a Cauchy sequence, if for eachε∈L\ {0_L}andt >0, there existsn0∈Nsuch that for allm≥n≥n0 n≥ m≥n0,

Mxm, xn, t>LNε. 2.10

The sequence{xn}_n∈Nis said to be convergent tox∈Xin theL-fuzzy metric spaceX,M,T denoted byxn →M xifMxn, x, t Mx, xn, t → 1_Lwhenevern → ∞for everyt >0. A L-fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.

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Definition 2.10. LetX,M,Tbe anL-fuzzy metric space.Mis said to be continuous onX× X×0,∞if

nlim→ ∞M

xn, yn, tn

M x, y, t

2.11

whenever a sequence {xn, yn, tn} in X ×X×0,∞ converges to a point x, y, t ∈ X × X×0,∞, that is, limnMxn, x, t limnMyn, y, t 1_Land limnMx, y, tn Mx, y, t.

Lemma 2.11. LetX,M,Tbe anL-fuzzy metric space. ThenMis continuous function onX × X×0,∞.

Proof. The proof is the same as that for fuzzy spacessee35, Proposition 1.

Lemma 2.12. If an ML-fuzzy metric spaceX,M,Tsatisfies the following condition:

M x, y, t

C, ∀t >0, 2.12

then one hasC1_Landxy.

Proof. LetMx, y, t Cfor allt > 0. Then byfofDefinition 2.5, we haveC 1_L and by bofDefinition 2.5, we conclude thatxy.

Lemma 2.13see50. LetX,M,Tbe anML-fuzzy metric space in whichTis Hadˇzic’ type.

Suppose

Mxn, x_n1, t≥LM

x0, x1, t kⁿ

2.13

for some 0< k <1 andn∈N. Then{xn}is a Cauchy sequence.

3. Main Results

Definition 3.1. Suppose thatX,≤is a partially ordered set andF, h :X → Xare mappings ofXinto itself. We say thatFish-nondecreasing if forx, y∈X,

hx≤h y

impliesFx≤F y

. 3.1

Now we present the main result in this paper.

Theorem 3.2. LetX,≤be a partially ordered set and suppose that there is anL-fuzzy metric M on X such thatX,M,T is a completeML-fuzzy metric space in which T is Hadˇzic’ type. Let F, h : X → X be two self-mappings ofX such that there existk ∈ 0,1andq ∈0,1such that

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FX⊆hX, Fis ah-nondecreasing mapping and

M

Fx, F y

, kt

≥LTM

M

hx, h y

, t

,Mhx, Fx, t,M h

y , F

y , t

, M

hx, F y

, 1q

t ,M

h y

, Fx, 1−q

t 3.2

for allx, y∈Xfor whichhx≤hyand allt >0.

Also suppose that

if{hxn} ⊂X is a nondecreasing sequence withhxn−→hzinhX,

thenhz≤hhz andhxn≤hz ∀nhold. 3.3

Also suppose thathXis closed. If there exists anx0 ∈X withhx0≤ Fx0, thenFandhhave a coincidence. Further, ifFandhcommute at their coincidence points, thenFandhhave a common fixed point.

Proof. Letx0 ∈Xbe such thathx0≤Fx0.SinceFX⊆hX,we can choosex1 ∈Xsuch thathx1 Fx0.Again fromFX⊆hXwe can choosex2 ∈Xsuch thathx2 Fx1. Continuing this process we can choose a sequence{xn}inXsuch that

hxn1 Fxn ∀n≥0. 3.4

Sincehx0≤Fx0andhx1 Fx0,we havehx0≤hx1.Then from3.1,

Fx0≤Fx1, 3.5

that is, by3.4,hx1≤hx2.Again from3.1,

Fx1≤Fx2, 3.6

that is,hx2≤hx3.Continuing we obtain

Fx0≤Fx1≤Fx2≤Fx3≤ · · · ≤Fxn≤Fxn1≤ · · ·. 3.7 Now we will show that a sequence{MFxn, Fx_n1, t} converges to 1_L for each t >0. IfMFxn, Fxn1, t 1_Lfor somenand for eacht >0, then it is easily to show that MFx_nk, Fx_nk1, t 1_Lfor allk≥0. So we suppose that MFxn, Fx_n1, t<L1_Lfor alln.We show that for eacht >0,

MFxn, Fxn1, kt≥LMFxn−1, Fxn, t ∀n≥1. 3.8

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Since from3.4and3.7we havehxn−1≤hxn,from3.1withxxnandyxn1, MFxn, Fx_n1, kt≥LTM{Mhxn, hx_n1, t,Mhxn, Fxn, t,Mhx_n1, Fx_n1, t,

M

hxn, Fxn1, 1q

t ,M

hxn1, Fxn, 1−q

t .

3.9

So by3.4,

MFxn, Fxn1, kt≥LTM{MFxn−1, Fxn, t,MFxn−1, Fxn, t,MFxn, Fxn1, t, M

Fxn−1, Fxn1, 1q

t ,1_L

.

3.10

Since bydofDefinition 2.5 M

Fx_n−1, Fx_n1, 1q

t

≥LTM

MFx_n−1, Fxn, t,M

Fxn, Fx_n1, qt

, 3.11

we have

MFxn, Fxn1, kt≥LTM{MFxn−1, Fxn, t,MFxn, Fxn1, t, M

Fxn, Fxn1, qt

. 3.12

Ast-norm is continuous, lettingq → 1_Lwe get

MFxn, Fx_n1, kt≥LTM{MFx_n−1, Fxn, t,MFxn, Fx_n1, t}. 3.13

Consequently,

MFxn, Fx_n1, t≥LTM

M

Fx_n−1, Fxn,1 kt

,M

Fxn, Fx_n1,1 kt

. 3.14

By repeating the above inequality, we obtain MFxn, Fxn1, t≥LTM

M

Fxn−1, Fxn,1 kt

,M

Fxn, Fxn1, 1 k^pt

. 3.15

SinceMFxn, Fx_n1,1/k^pt → 1_Lasp → ∞,it follows that MFxn, Fxn1, t≥LM

Fxn−1, Fxn,1 kt

. 3.16

Thus we proved3.7. By repeating the above inequality3.7, we get MFxn, Fx_n1, t≥LM

Fx0, Fx1, 1 kⁿt

. 3.17

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SinceMx, y, t → 1_Last → ∞andk <1, lettingn → ∞in3.17we get

nlim→ ∞MFxn, Fx_n1, t 1_L for eacht >0. 3.18

Now we will prove that{Fxn} is a Cauchy sequence which means that for every δ >0 and∈L\ {0_L,1_L}there existsnδ, ∈Nsuch that

M

Fxn, F xnp

, δ

>LN for everyn≥nδ, and everyp∈N. 3.19 Let∈L\ {0_L,1_L}andδ >0 be arbitrary. For anyp≥1 we have

δδ1−k1k· · ·k^p· · ·> δ1−k

1k· · ·k^p−1

. 3.20

SinceMx, y, tis nondecreasing with respect tot, for allx, yinX, M

Fxn, F x_np

, δ

≥LM

Fxn, F x_np

, δ1−k

1kⁿ· · ·k^p−1

3.21

and hence, bydofDefinition 2.5, M

Fxn, F xnp

, δ

≥LT^p−2_M

MFxn, Fxn1,1−kδ,MFxn1, Fxn2,1−kδk , . . . ,M

F xnp−1

, F xnp

,1−kδk^p−1 .

3.22

From3.17it follows that

MFx_ni, Fx_ni1, t≥LM

Fxn, Fx_n1, t kⁱ

for eachi≥L1_L. 3.23

From3.23witht 1−kδkⁱwe get M

Fxni, Fxni1,1−kδkⁱ

≥LMFxn, Fxn1,1−kδ. 3.24

Thus by3.22, M

Fxn, F xnp

, δ

≥LTⁿ_M{MFxn, Fxn1,1−kδ,MFxn, Fxn1,1−kδ

, . . . ,MFxn, Fxn1,1−kδ}. 3.25

Hence we get

M

Fxn, F xnp

, δ

≥LMFxn, Fxn1,1−kδ. 3.26

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From3.26and3.17, M

Fxn, F xnp

, δ

≥LM

Fx0, Fx1,1−kδ kⁿ

. 3.27

Hence we conclude, asMx, y, t → 1_L ast → ∞andk <1, that there existsnδ, ∈N such that

M

Fxn, F x_np

, δ

>LN for everyn≥nδ, and everyp∈N. 3.28 Thus we proved that{Fxn}is a Cauchy sequence.

SincehXis closed and asFxn hx_n1, there is somez∈Xsuch that

nlim→ ∞hxn hz. 3.29

Now we show thatzis a coincidence ofFandh.Since from3.3and3.29we have hxn≤hzfor alln,then from3.2and bydofDefinition 2.5we have

MFxn, Fz, kt≥LTM{Mhxn, hz, t,Mhxn, Fxn, t,Mhz, Fz, t, M

hxn, Fz, 1q

t , M

hz, Fxn, 1−q

t

. 3.30

Lettingn → ∞we get

Mhz, Fz, kt≥LTM{Mhz, hz, t,Mhz, hz, t,Mhz, Fz, t, M

hz, Fz, 1q

t ,M

hz, hz, 1−q

t 3.31

for allt >0.Therefore,

Mhz, Fz, t≥LM

hz, Fz,1 kt

. 3.32

Hence we get

Mhz, Fz, t≥LM

hz, Fz, 1 kⁿt

−→1_L asn−→ ∞ ∀t >0. 3.33

Hence we conclude thatMhz, Fz, t 1_Lfor allt >0.Then bybofDefinition 2.5we haveFz hz.Thus we proved thatFandhhave a coincidence.

Suppose now thatFandhcommute atz. Setwhz Fz.Then

Fw Fhz hFz hw. 3.34

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Since from3.3we havehz≤hhz hwand ashz Fzandhw Fw,from 3.2we get

Mw, Fw, kt MFz, Fw, kt

≥LTM{Mhz, hw, t,Mhz, Fz, t,Mhw, Fw, t, M

hw, Fz,

1q t

,M

hz, Fw,

1−q t M

Fz, Fw,

1−q t

.

3.35

Lettingq → 0 we get

MFz, Fw, kt≥LMFz, Fw, t. 3.36

Hence, similarly as above, we get MFz, Fw, t≥LM

Fz, Fw, 1

kⁿt

−→1_L asn−→ ∞ ∀t >0. 3.37

Hence we conclude thatFw Fz.SinceFz hz w,we have

Fw hw w. 3.38

Thus we proved thatFandhhave a common fixed point.

Remark 3.3. Note that F is h-nondecreasing and can be replaced by F which is h-non- increasing in Theorem 3.2 provided that hx0 ≤ Fx0 is replaced byFx0 ≥ hx0 in Theorem 3.2.

Corollary 3.4. LetX,≤be a partially ordered set and suppose that there is anL-fuzzy metricM on X such thatX,M,T is a completeML-fuzzy metric space in which T is Hadˇzic’ type. Let F:X → Xbe a nondecreasing self-mappings ofXsuch that there existk∈0,1andq∈0,1such that

M

Fx, F y

, kt

≥LTM

M x, y, t

,Mx, Fx, t,M y, F

y , t

, M

x, F y

, 1q

t ,M

y, Fx, 1−q

t 3.39

for allx, y∈Xfor whichx≤yand allt >0.Also suppose the following.

iIf{xn} ⊂Xis a nondecreasing sequence withxn → zinX, thenxn≤zfor allnhold.

iiFis continuous.

If there exists anx0 ∈Xwithx0≤Fx0, thenFhas a fixed point.

Proof. Takingh I I the identity mapping in Theorem 3.2, then 3.3reduces to the hypothesisi.

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Suppose now thatFis continuous. Since from3.4we havexn1Fxnfor alln≥0, and as from3.29,xn → z,then

Fz F

nlim→ ∞xn

lim

n→ ∞Fxn z. 3.40

Corollary 3.5. LetX,≤be a partially ordered set and suppose that there is anL-fuzzy metricM on X such thatX,M,T is a completeML-fuzzy metric space in which T is Hadˇzic’ type. Let F:X → Xbe a nondecreasing self-mappings ofXsuch that there existk∈0,1andq∈0,1such that

M

Fx, F y

, kt

≥LTM

M x, y, t

,Mx, Fx, t,M y, F

y , t

3.41 for allx, y∈Xfor whichx≤yand allt >0.Also suppose the following.

iIf{xn} ⊂Xis a nondecreasing sequence withxn → zinX, thenxn≤zfor allnhold.

iiFis continuous.

If there exists anx0 ∈Xwithx0≤Fx0, thenFhas a fixed point.

Acknowledgments

This research is supported by Young research Club, Islamic Azad University-Ayatollah Amoli Branch, Amol, Iran. The authors would like to thank Professor J. J. Nieto for giving useful suggestions for the improvement of this paper.

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