Common Fixed Point Theorem in Partially Ordered -Fuzzy Metric Spaces

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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

1_{Young Research Club, Islamic Azad University-Ayatollah Amoli Branch, P.O. Box 678, Amol, Iran} 2_{Faculty of Mechanical Engineering, Kraljice Marije 16, 11 000 Belgrade, Serbia}

3_{Faculty 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

dFx, Fy≤kdx, y 1.1

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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 ´opez27,28and Ran and Reurings33have motivated Agarwal et al.1, Bhaskar and Lakshmikantham3, and Lakshmikantham and

´

Ciri´c 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 ´opez27,28and 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 1L supL.

Definition 2.3. A negation on Lis any strictly decreasing mapping N : L → L satisfying N0L 1L andN1L 0L. 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 _→ _L_{satisfying the}

following conditions:

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ii for allx, y∈L2_T_{x, y} _T_{y, x} _{commutativity}_;

iii for allx, y, z∈L3_T_x,_T_{y, z} _T_T_{x, y}_{, z} _{associativity}_;

iv for allx, x, y, y∈L4_x_≤

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

lim

n T

xn, yn

Tx, 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∈NbyT1_T_and

Tn_x

1, . . . , xn1 T

Tn−1_x

1, . . . , xn, xn1

2.3

forn≥2 andxi∈L.

At-normTis said to be ofHadˇzi´c typeif the family{Tn_}

n∈Nis equicontinuous atx1L,

that is,

∀ε∈L\ {0L,1L}∃δ∈L\ {0L,1L}: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 spaceifX 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>L0L;

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

cMx, y, t My, x, t;

dTMx, y, t,My, z, s≤LMx, z, ts;

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

flim

t→ ∞M

x, y, t1L, 2.6

thenX,M,Tis said to beMengerL-fuzzy metric spaceor 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,1L}, as

Bx, r, t y∈X:Mx, y, t>LNr

. 2.7

A subsetA⊆ X is called openif for eachx∈ A, there existt >0 andr ∈ L\ {0L,1L}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, tMx, y, t, Nx, y, t

t tdx, y,

dx, y tdx, 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:

Mx, 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\ {0L}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 convergenttox∈Xin theL-fuzzy metric spaceX,M,T denoted byxn →M xifMxn, x, t Mx, xn, t → 1Lwhenevern → ∞for everyt >0. A

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

lim

n→ ∞M

xn, yn, tn

Mx, 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 1Land limnMx, y, tn Mx, y, t.

Lemma 2.11. Let X,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 anML-fuzzy metric spaceX,M,Tsatisfies the following condition:

Mx, y, tC, ∀t >0, 2.12

then one hasC1_Landxy.

Proof. LetMx, y, t Cfor allt > 0. Then byfofDefinition 2.5, we haveC 1L and by

bofDefinition 2.5, we conclude thatxy.

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

Mxn, xn1, t≥LM

x0, x1, t

kn

2.13

for some0< k <1andn∈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≤hy impliesFx≤Fy. 3.1

Now we present the main result in this paper.

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FX⊆hX, Fis ah-nondecreasing mapping and

MFx, Fy, kt≥LTM

Mhx, hy, t,Mhx, Fx, t,Mhy, Fy, t,

Mhx, Fy,1qt,Mhy, Fx,1−qt 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, Fxn1, t} converges to 1L for each

t >0. IfMFxn, Fxn1, t 1Lfor somenand for eacht >0, then it is easily to show that

MFxnk, Fxnk1, t 1Lfor allk≥0. So we suppose that MFxn, Fxn1, t<L1Lfor

alln.We show that for eacht >0,

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Since from3.4and3.7we havehxn−1≤hxn,from3.1withxxnandyxn1,

MFxn, Fxn1, kt≥LTM{Mhxn, hxn1, t,Mhxn, Fxn, t,Mhxn1, Fxn1, t,

Mhxn, Fxn1,

1qt,Mhxn1, Fxn,

1−qt.

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So by3.4,

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

MFxn−1, Fxn1,

1qt,1L.

3.10

Since bydofDefinition 2.5

MFxn−1, Fxn1,

1qt≥LTM

MFxn−1, Fxn, t,M

Fxn, Fxn1, qt

, 3.11

we have

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

MFxn, Fxn1, qt

.

3.12

Ast-norm is continuous, lettingq → 1Lwe get

MFxn, Fxn1, kt≥LTM{MFxn−1, Fxn, t,MFxn, Fxn1, t}. 3.13

Consequently,

MFxn, Fxn1, t≥LTM

M

Fxn−1, Fxn,

1 kt

,M

Fxn, Fxn1,

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

kpt

. 3.15

SinceMFxn, Fxn1,1/kpt → 1Lasp → ∞,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, Fxn1, t≥LM

Fx0, Fx1,

1 knt

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

lim

n→ ∞MFxn, Fxn1, t 1L for eacht >0. 3.18

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

MFxn, F

xnp, δ>LN for everyn≥nδ, and everyp∈N. 3.19

Let∈L\ {0L,1L}andδ >0 be arbitrary. For anyp≥1 we have

δδ1−k1k· · ·kp· · ·> δ1−k1k· · ·kp−1. 3.20

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

MFxn, F

xnp, δ≥LM

Fxn, F

xnp, δ1−k

1kn· · ·kp−1 3.21

and hence, bydofDefinition 2.5,

MFxn, F

xnp, δ≥LTpM−2

MFxn, Fxn1,1−kδ,MFxn1, Fxn2,1−kδk

, . . . ,MFxnp−1

, Fxnp,1−kδkp−1

.

3.22

From3.17it follows that

MFxni, Fxni1, t≥LM

Fxn, Fxn1, t

ki

for eachi≥L1L. 3.23

From3.23witht 1−kδki_{we get}

MFxni, Fxni1,1−kδki

≥LMFxn, Fxn1,1−kδ. 3.24

Thus by3.22,

MFxn, F

xnp, δ≥LTnM{MFxn, Fxn1,1−kδ,MFxn, Fxn1,1−kδ

, . . . ,MFxn, Fxn1,1−kδ}.

3.25

Hence we get

MFxn, F

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From3.26and3.17,

MFxn, F

xnp, δ≥LM

Fx0, Fx1,1−kδ

kn

. 3.27

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

such that

MFxn, F

xnp, δ>LN for everyn≥nδ, and everyp∈N. 3.28

Thus we proved that{Fxn}is a Cauchy sequence.

SincehXis closed and asFxn hxn1, there is somez∈Xsuch that

lim

n→ ∞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,

Mhxn, Fz,

1qt, Mhz, Fxn,

1−qt. 3.30

Lettingn → ∞we get

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

Mhz, Fz,1qt,Mhz, hz,1−qt 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

−→1L 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

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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,

Mhw, Fz,1qt,Mhz, Fw,1−qt

MFz, Fw,1−qt.

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

MFx, Fy, kt≥LTM

Mx, y, t,Mx, Fx, t,My, Fy, t,

Mx, Fy,1qt,My, Fx,1−qt 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.

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Suppose now thatFis continuous. Since from3.4we havexn1Fxnfor alln≥0,

and as from3.29,xn → z,then

Fz F

lim

n→ ∞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

MFx, Fy, kt≥LTM

Mx, y, t,Mx, Fx, t,My, Fy, 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.

References

1 R. P. Agarwal, M. A. El-Gebeily, and D. O’Regan, “Generalized contractions in partially ordered metric spaces,”Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008.

2 I. ltun and H. Simsek, “Some fixed point theorems on ordered metric spaces and application,”Fixed Point Theory and Applications, vol. 2010, Article ID 621469, 17 pages, 2010.

3 T. G. 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. 4 T. Gnana Bhaskar, V. Lakshmikantham, and J. Vasundhara Devi, “Monotone iterative technique

for functional diﬀerential equations with retardation and anticipation,”Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 10, pp. 2237–2242, 2007.

5 A. Bj ¨orner, “Order-reversing maps and unique fixed points in complete lattices,”Algebra Universalis, vol. 12, no. 3, pp. 402–403, 1981.

6 Dˇz. Burgić, S. Kalabuˇsić, and M. R. S. Kulenović, “Global attractivity results for mixed-monotone mappings in partially ordered complete metric spaces,”Fixed Point Theory and Applications, vol. 2009, Article ID 762478, 17 pages, 2009.

7 S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, and S. M. Kang, “Coincidence point theorems and minimization theorems in fuzzy metric spaces,”Fuzzy Sets and Systems, vol. 88, no. 1, pp. 119–127, 1997.

8 Y. J. Cho, H. K. Pathak, S. M. Kang, and J. S. Jung, “Common fixed points of compatible maps of type βon fuzzy metric spaces,”Fuzzy Sets and Systems, vol. 93, no. 1, pp. 99–111, 1998.

(12)

10 L. B. ´Ciri´c, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974.

11 L. B. ´Ciri´c, “Coincidence and fixed points for maps on topological spaces,” Topology and Its Applications, vol. 154, no. 17, pp. 3100–3106, 2007.

12 L. B. Ćirić, S. N. Jesić, M. M. Milovanović, and J. S. Ume, “On the steepest descent approximation method for the zeros of generalized accretive operators,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 2, pp. 763–769, 2008.

13 G. Deschrijver, C. Cornelis, and E. E. Kerre, “On the representation of intuitionistic fuzzy t-norms and t-conorms,”IEEE Transactions on Fuzzy Systems, vol. 12, no. 1, pp. 45–61, 2004.

14 G. Deschrijver and E. E. Kerre, “On the relationship between some extensions of fuzzy set theory,” Fuzzy Sets and Systems, vol. 133, no. 2, pp. 227–235, 2003.

15 Z. Deng, “Fuzzy pseudometric spaces,”Journal of Mathematical Analysis and Applications, vol. 86, no. 1, pp. 74–95, 1982.

16 M. A. Erceg, “Metric spaces in fuzzy set theory,”Journal of Mathematical Analysis and Applications, vol. 69, no. 1, pp. 205–230, 1979.

17 D. Qiu, L. Shu, and J. Guan, “Common fixed point theorems for fuzzy mappings underΦ-contraction condition,”Chaos, Solitons & Fractals, vol. 41, no. 1, pp. 360–367, 2009.

18 R. Farnoosh, A. Aghajani, and P. Azhdari, “Contraction theorems in fuzzy metric space,”Chaos, Solitons & Fractals, vol. 41, no. 2, pp. 854–858, 2009.

19 M. B. Ghaemi, B. Lafuerza-Guillen, and A. Razani, “A common fixed point for operators in probabilistic normed spaces,”Chaos, Solitons & Fractals, vol. 40, no. 3, pp. 1361–1366, 2009.

20 N. Hussain, “Common fixed points in best approximation for Banach operator pairs with Ćirić type I-contractions,”Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1351–1363, 2008. 21 S. N. Jeˇsić and N. A. Babaˇcev, “Common fixed point theorems in intuitionistic fuzzy metric spaces

andL-fuzzy metric spaces with nonlinear contractive condition,”Chaos, Solitons & Fractals, vol. 37, no. 3, pp. 675–687, 2008.

22 T. Kamran, “Common fixed points theorems for fuzzy mappings,”Chaos, Solitons & Fractals, vol. 38, no. 5, pp. 1378–1382, 2008.

23 V. Lakshmikantham and L. ´Ciri´c, “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.

24 Z. Liu, Z. Guo, S. M. Kang, and S. K. Lee, “On ´Ciri´c type mappings with nonunique fixed and periodic points,”International Journal of Pure and Applied Mathematics, vol. 26, no. 3, pp. 399–408, 2006. 25 D. Mihet¸, “A Banach contraction theorem in fuzzy metric spaces,”Fuzzy Sets and Systems, vol. 144,

no. 3, pp. 431–439, 2004.

26 E. Pap, O. Hadˇzi´c, and R. Mesiar, “A fixed point theorem in probabilistic metric spaces and an application,”Journal of Mathematical Analysis and Applications, vol. 202, no. 2, pp. 433–449, 1996. 27 J. J. Nieto and R. Rodr´ıguez-L ´opez, “Contractive mapping theorems in partially ordered sets and

applications to ordinary diﬀerential equations,”Order, vol. 22, no. 3, pp. 223–239, 2005.

28 J. J. Nieto and R. Rodr´ıguez-L ´opez, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary diﬀerential equations,”Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205–2212, 2007.

29 J. J. Nieto, R. L. Pouso, and R. Rodr´ıguez-L ´opez, “Fixed point theorems in ordered abstract spaces,” Proceedings of the American Mathematical Society, vol. 135, no. 8, pp. 2505–2517, 2007.

30 D. O’Regan and R. Saadati, “Nonlinear contraction theorems in probabilistic spaces,” Applied Mathematics and Computation, vol. 195, no. 1, pp. 86–93, 2008.

31 D. O’Regan and A. Petrus¸el, “Fixed point theorems for generalized contractions in ordered metric spaces,”Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1241–1252, 2008. 32 J. H. Park, “Intuitionistic fuzzy metric spaces,”Chaos, Solitons & Fractals, vol. 22, no. 5, pp. 1039–1046,

2004.

33 A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,”Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004.

(13)

35 J. Rodr´ıguez-L ´opez and S. Romaguera, “The Hausdorﬀfuzzy metric on compact sets,”Fuzzy Sets and Systems, vol. 147, no. 2, pp. 273–283, 2004.

36 R. Saadati, A. Razani, and H. Adibi, “A common fixed point theorem inL-fuzzy metric spaces,” Chaos, Solitons & Fractals, vol. 33, no. 2, pp. 358–363, 2007.

37 R. Saadati and J. H. Park, “On the intuitionistic fuzzy topological spaces,”Chaos, Solitons & Fractals, vol. 27, no. 2, pp. 331–344, 2006.

38 R. Saadati and J. H. Park, “Intuitionistic fuzzy Euclidean normed spaces,” Communications in Mathematical Analysis, vol. 1, no. 2, pp. 85–90, 2006.

39 R. Saadati, S. Sedghi, and H. Zhou, “A common fixed point theorem forψ-weakly commuting maps inL-fuzzy metric spaces,”Iranian Journal of Fuzzy Systems, vol. 5, no. 1, pp. 47–53, 2008.

40 V. M. Sehgal and A. T. Bharucha-Reid, “Fixed points of contraction mappings on probabilistic metric spaces,”Mathematical Systems Theory, vol. 6, pp. 97–102, 1972.

41 S. Sharma and B. Deshpande, “Common fixed point theorems for finite number of mappings without continuity and compatibility on intuitionistic fuzzy metric spaces,”Chaos, Solitons & Fractals, vol. 40, no. 5, pp. 2242–2256, 2009.

42 S. L. Singh and S. N. Mishra, “On a Ljubomir ´Ciri´c fixed point theorem for nonexpansive type maps with applications,”Indian Journal of Pure and Applied Mathematics, vol. 33, no. 4, pp. 531–542, 2002. 43 R. Vasuki, “A common fixed point theorem in a fuzzy metric space,”Fuzzy Sets and Systems, vol. 97,

no. 3, pp. 395–397, 1998.

44 L. A. Zadeh, “Fuzzy sets,”Information and Computation, vol. 8, pp. 338–353, 1965.

45 I. Kramosil and J. Mich´alek, “Fuzzy metrics and statistical metric spaces,”Kybernetika, vol. 11, no. 5, pp. 336–344, 1975.

46 J. A. Goguen, “L-fuzzy sets,”Journal of Mathematical Analysis and Applications, vol. 18, pp. 145–174, 1967.

47 V. Gregori and A. Sapena, “On fixed-point theorems in fuzzy metric spaces,”Fuzzy Sets and Systems, vol. 125, no. 2, pp. 245–252, 2002.

48 S. B. Hosseini, D. O’Regan, and R. Saadati, “Some results on intuitionistic fuzzy spaces,”Iranian Journal of Fuzzy Systems, vol. 4, no. 1, pp. 53–64, 2007.

49 A. George and P. Veeramani, “On some results in fuzzy metric spaces,”Fuzzy Sets and Systems, vol. 64, no. 3, pp. 395–399, 1994.