Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces

(1)

Volume 2011, Article ID 363716,14pages doi:10.1155/2011/363716

Research Article

Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces

Xin-Qi Hu

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Correspondence should be addressed to Xin-Qi Hu,[email protected] Received 23 November 2010; Accepted 27 January 2011

Academic Editor: Ljubomir B. Ciric

Copyrightq2011 Xin-Qi Hu. 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 prove a common fixed point theorem for mappings underφ-contractive conditions in fuzzy metric spaces. We also give an example to illustrate the theorem. The result is a genuine generalization of the corresponding result of S.Sedghi et al.2010

1. Introduction

Since Zadeh 1 introduced the concept of fuzzy sets, many authors have extensively developed the theory of fuzzy sets and applications. George and Veeramani 2, 3 gave the concept of fuzzy metric space and defined a Hausdorﬀ topology on this fuzzy metric space which have very important applications in quantum particle physics particularly in connection with both string andE-infinity theory.

Bhaskar and Lakshmikantham 4, Lakshmikantham and ´Ciri´c 5 discussed the mixed monotone mappings and gave some coupled fixed point theorems which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem.

Sedghi et al.6gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang7gave some common fixed point theorems underφ-contractions for compatible and weakly compatible mappings in Menger probabilistic metric spaces. Many authors8–

23have proved fixed point theorems inintuitionisticfuzzy metric spaces or probabilistic metric spaces.

In this paper, using similar proof as in7, we give a new common fixed point theorem under weaker conditions than in6and give an example which shows that the result is a genuine generalization of the corresponding result in6.

(2)

2. Preliminaries

First we give some definitions.

Definition 1see2. A binary operation∗ :0,1×0,1 → 0,1is continuoust-norm if∗ is satisfying the following conditions:

1∗is commutative and associative;

2∗is continuous;

3a∗1afor alla∈0,1;

4a∗b≤c∗dwhenevera≤candb≤dfor alla, b, c, d∈0,1.

Definition 2see24. Let sup0<t<1Δt, t 1. At-normΔis said to be of H-type if the family of functions{Δ^mt}^∞_m1is equicontinuous att1, where

Δ¹t tΔt, Δ^{m 1}t tΔΔ^mt, m1,2, . . . , t∈0,1. 2.1

Thet-normΔM min is an example oft-norm of H-type, but there are some other t-normsΔof H-type24.

Obviously,Δis a H-typetnorm if and only if for anyλ∈0,1, there existsδλ∈0,1 such thatΔ^mt>1−λfor allm∈, whent >1−δ.

Definition 3see2. A 3-tupleX, M,∗is said to be a fuzzy metric space ifXis an arbitrary nonempty set,∗is a continuous t-norm, andMis a fuzzy set onX²×0, ∞satisfying the following conditions, for eachx, y, z∈Xandt, s >0:

FM-1Mx, y, t>0;

FM-2Mx, y, t 1 if and only ifxy;

FM-3Mx, y, t My, x, t;

FM-4Mx, y, t∗My, z, s≤Mx, z, t s;

FM-5Mx, y,·:0,∞ → 0,1is continuous.

LetX, M,∗be a fuzzy metric space. Fort > 0, the open ballBx, r, twith a center x∈Xand a radius 0< r <1 is defined by

Bx, r, t

y∈X:M x, y, t

>1−r

. 2.2

A subsetA⊂Xis called open if, for eachx∈A, there existt >0 and 0< r <1 such that Bx, r, t⊂A. Letτdenote the family of all open subsets ofX. Thenτis called the topology onXinduced by the fuzzy metricM. This topology is Hausdorﬀand first countable.

Example 1. LetX, dbe a metric space. Definet-norma∗baband for allx, y∈Xandt >0, Mx, y, t t/t dx, y. ThenX, M,∗is a fuzzy metric space. We call this fuzzy metric Minduced by the metricdthe standard fuzzy metric.

(3)

Definition 4see2. LetX, M,∗be a fuzzy metric space, then

1a sequence{xn}inXis said to be convergent toxdenoted by limn→ ∞xnxif

nlim→ ∞Mxn, x, t 1, 2.3

for allt >0;

2a sequence{xn}inX is said to be a Cauchy sequence if for anyε >0, there exists n0∈, such that

Mxn, xm, t>1−ε, 2.4

for allt >0 and n, m≥n₀;

3a fuzzy metric spaceX, M,∗is said to be complete if and only if every Cauchy sequence inXis convergent.

Remark 1see25. 1For allx, y∈X,Mx, y,·is nondecreasing.

2It is easy to prove that ifxn → x,yn → y,tn → t, then

nlim→ ∞M

x_n, y_n, t_n M

x, y, t

. 2.5

3In a fuzzy metric spaceX, M,∗, wheneverMx, y, t> 1−rforx, yinX,t > 0, 0< r <1, we can find at0, 0< t0< tsuch thatMx, y, t0>1−r.

4For anyr1> r2, we can find anr3such thatr1∗r3≥r2and for anyr4we can find a r5such thatr5∗r5≥r4 r1, r2, r3, r4, r5∈0,1.

Definition 5see6. LetX, M,∗be a fuzzy metric space.Mis said to satisfy then-property onX²×0,∞if

nlim→ ∞

M

x, y, kⁿt_n^p

1, 2.6

wheneverx, y∈X,k >1 andp >0.

Lemma 1. LetX, M,∗be a fuzzy metric space andMsatisfies then-property; then

tlim→ ∞M x, y, t

1, ∀x, y∈X. 2.7

Proof. If not, sinceMx, y,·is nondecreasing and 0 ≤Mx, y,·≤ 1, there existsx₀, y₀ ∈ X such that lim_t→_∞Mx0, y₀, t λ < 1, then fork >1,kⁿt → ∞whenn → ∞ast >0 and we get limn→ ∞Mx0, y0, kⁿtⁿ^p 0, which is a contraction.

(4)

Remark 2. Condition2.7cannot guarantee then-property. See the following example.

Example 2. LetX, dbe an ordinary metric space,a∗b ≤ abfor all a, b ∈ 0,1, andψ be defined as following:

ψt

⎧⎪

⎨

⎪⎩ α√

t, 0< t≤4, 1− 1

lnt, t >4, 2.8

whereα 1/21−1/ln 4. Thenψtis continuous and increasing in0,∞,ψt∈ 0,1 and lim_t→_∞ψt 1. Let

M x, y, t

ψt_dx,y

, ∀x, y∈X, t >0, 2.9 thenX, M,∗is a fuzzy metric space and

t→lim ∞M x, y, t

lim

t→ ∞

ψt_dx,y

1, ∀x, y∈X. 2.10

But for anyx /y,p1,k >1,t >0,

nlim→ ∞

M

x, y, kⁿtn^p lim

n→ ∞

ψkⁿt_dx,y·n^p lim

n→ ∞

1− 1 lnkⁿt

_n·dx,y

e^−dx,y/^ln^k/1.

2.11 DefineΦ {φ:R → R }, whereR 0, ∞and eachφ∈Φsatisfies the following conditions:

φ-1φis nondecreasing;

φ-2φis upper semicontinuous from the right;

φ-3_∞

n0φⁿt< ∞for allt >0, whereφ^{n 1}t φφⁿt,n∈. It is easy to prove that, ifφ∈Φ, thenφt< tfor allt >0.

Lemma 2see7. LetX, M,∗be a fuzzy metric space, where∗is a continuoust-norm of H-type.

If there existsφ∈Φsuch that if M

x, y, φt

≥M x, y, t

, 2.12

for allt >0, thenxy.

Definition 6see5. An elementx, y∈X×Xis called a coupled fixed point of the mapping F:X×X → Xif

F x, y

x, F y, x

y. 2.13

(5)

Definition 7see5. An elementx, y∈X×Xis called a coupled coincidence point of the mappingsF:X×X → Xandg:X → Xif

F x, y

gx, F y, x

g y

. 2.14

Definition 8see7. An elementx, y ∈X×X is called a common coupled fixed point of the mappingsF:X×X → Xandg:X → Xif

xF x, y

gx, yF y, x

g y

. 2.15

Definition 9see7. An elementx ∈ X is called a common fixed point of the mappings F:X×X → Xandg:X → Xif

xgx Fx, x. 2.16

Definition 10see7. The mappingsF:X×X → Xandg:X → Xare said to be compatible if

n→ ∞limM gF

xn, yn

, F

gxn, g yn

, t 1,

n→ ∞limM gF

yn, xn

, F g

yn

, gxn , t

1,

2.17

for allt >0 whenever{xn}and{yn}are sequences inX, such that

n→ ∞limF x_n, y_n

lim

n→ ∞gxn x, lim

n→ ∞F y_n, x_n

lim

n→ ∞g y_n

y, 2.18

for allx, y∈Xare satisfied.

Definition 11see7. The mappingsF:X×X → Xandg:X → Xare called commutative if

g F

x, y F

gx, gy

, 2.19

for allx, y∈X.

Remark 3. It is easy to prove that, ifFandgare commutative, then they are compatible.

3. Main Results

For convenience, we denote M

x, y,t_n

Mx, y, t ∗Mx, y, t ∗ · · · ∗Mx, y, t

n

, 3.1

for alln∈.

(6)

Theorem 1. Let X, M,∗ be a complete FM-space, where ∗ is a continuous t-norm of H-type satisfying2.7. LetF : X×X → X andg : X → X be two mappings and there existsφ ∈ Φ such that

M F

x, y

, Fu, v, φt

≥M

gx, gu, t

∗M g

y

, gv, t

, 3.2

for allx, y, u, v∈X,t >0.

Suppose thatFX×X⊆gX, andgis continuous,Fandgare compatible. Then there exist x, y∈Xsuch thatxgx Fx, x, that is,Fandghave a unique common fixed point inX.

Proof. Letx₀, y₀ ∈ X be two arbitrary points in X. SinceFX×X ⊆ gX, we can choose x1, y1 ∈Xsuch thatgx1 Fx0, y0andgy1 Fy0, x0. Continuing in this way we can construct two sequences{xn}and{yn}inXsuch that

gxn 1 F x_n, y_n

, g y_{n 1}

F y_n, x_n

, ∀n≥0. 3.3

The proof is divided into 4 steps.

Step 1. Prove that{gxn}and{gyn}are Cauchy sequences.

Since∗is at-norm of H-type, for anyλ >0, there exists aμ >0 such that 1−μ∗1−μ∗ · · · ∗1−μ

k

≥1−λ,

3.4

for allk∈.

SinceMx, y,·is continuous and lim_{t→ ∞}Mx, y, t 1 for allx, y ∈X, there exists t0>0 such that

M

gx0, gx1, t0

≥1−μ, M

gy0, gy1, t0

≥1−μ. 3.5

On the other hand, sinceφ∈Φ, by conditionφ-3we have_∞

n1φⁿt0<∞. Then for anyt >0, there existsn0∈ such that

t >

∞ kn0

φ^kt0. 3.6

From condition3.2, we have M

gx1, gx2, φt0 M

F x0, y0

, F x1, y1

, φt0

≥M

gx0, gx1, t0

∗M

gy0, gy1, t0

,

M

gy1, gy2, φt0 M

F y0, x0

, F y1, x1

, φt0

≥M

gy₀, gy₁, t₀

∗M

gx₀, gx₁, t₀ .

3.7

(7)

Similarly, we can also get

M

gx₂, gx₃, φ²t0 M

F x₁, y₁

, F x₂, y₂

, φ²t0

≥M

gx1, gx2, φt0

∗M

gy1, gy2, φt0

≥ M

gx₀, gx₁, t₀₂

∗ M

gy₀, gy₁, t₀₂ ,

M

gy2, gy3, φ²t0 M

F y1, x1

, F y2, x2

, φ²t0

≥ M

gy0, gy1, t0

2∗ M

gx0, gx1, t0

2

.

3.8

Continuing in the same way we can get M

gx_n, gx_{n 1}, φⁿt0

≥ M

gx₀, gx₁, t₀₂ⁿ⁻¹

∗ M

gy₀, gy₁, t₀₂ⁿ⁻¹ , M

gy_n, gy_{n 1}, φⁿt0

≥ M

gy₀, gy₁, t₀₂ⁿ⁻¹

∗ M

gx₀, gx₁, t₀₂ⁿ⁻¹ .

3.9

So, from3.5and3.6, form > n≥n0, we have M

gxn, gxm, t

≥M

gxn, gxm, ∞ kn0

φ^kt0

≥M

gx_n, gx_m,

m−1

kn

φ^kt0

≥M

gx_n, gx_{n 1}, φⁿt0

∗M

gx_{n 1}, gx_{n 2}, φ^{n 1}t0

∗ · · · ∗M

gx_m−1, gx_m, φ^m−1t0

≥ M

gy0, gy1, t0

₂ⁿ⁻¹

∗ M

gx0, gx1, t0

₂ⁿ⁻¹

∗ M

gy0, gy1, t0

₂ⁿ

∗ M

gx₀, gx₁, t₀₂ⁿ

∗ · · · ∗ M

gy₀, gy₁, t₀₂^m−2

∗ M

gx₀, gx₁, t₀₂^m−2

M

gy₀, gy₁, t₀₂^{m−nm n−3}

∗ M

gx₀, gx₁, t₀₂^{m−nm n−3}

≥1−μ∗1−μ∗ · · · ∗1−μ

2^{2m−nm n−3}

≥1−λ,

3.10

which implies that

M

gx_n, gx_m, t

>1−λ, 3.11

for allm, n∈withm > n≥n₀andt >0. So{gxn}is a Cauchy sequence.

Similarly, we can get that{gyn}is also a Cauchy sequence.

(8)

Step 2. Prove thatgandFhave a coupled coincidence point.

SinceXcomplete, there existx, y∈Xsuch that

nlim→ ∞F xn, yn

lim

n→ ∞gxn x, lim

n→ ∞F yn, xn

lim

n→ ∞g yn

y. 3.12

SinceFandgare compatible, we have by3.12,

n→ ∞limM gF

xn, yn

, F

gxn, g yn

, t 1,

n→ ∞limM gF

yn, xn

, F g

yn

, gxn , t

1.

3.13

for allt >0. Next we prove thatgx Fx, yandgy Fy, x.

For allt >0, by condition3.2, we have M

gx, F x, y

, φt

≥M

ggx_{n 1}, F x, y

, φk1t

∗M

gx, ggx_{n 1}, φt−φk1t M

gF xn, yn

, F x, y

, φk1t

∗M

gx, ggx_{n 1}, φt−φk1t

≥M gF

xn, yn

, F

gxn, gyn

, φk1t−φk2t

∗M F

gxn, gyn

, F x, y

, φk2t

∗M

gx, ggx_{n 1}, φt−φk1t

≥M gF

x_n, y_n , F

gx_n, gy_n

, φk1t−φk2t

∗M

ggx_n, gx, k₂t

∗M

ggy_n, gy, k₂t

∗M

gx, ggx_{n 1}, φt−φk1t ,

3.14

for all 0< k₂< k₁<1. Letn → ∞, sincegandFare compatible, with the continuity ofg, we get

M gx, F

x, y , φt

≥1, 3.15

which implies thatgxFx, y. Similarly, we can getgyFy, x.

Step 3. Prove thatgxyandgyx.

Since∗is at-norm of H-type, for anyλ >0, there exists anμ >0 such that 1−μ∗1−μ∗ · · · ∗1−μ

k

≥1−λ,

3.16

for allk∈.

SinceMx, y,·is continuous and lim_t→_∞ Mx, y, t 1 for allx, y ∈X, there exists t0>0 such thatMgx, y, t0≥1−μandMgy, x, t0≥1−μ.

(9)

n1φⁿt0<∞. Then for anyt >0, there existsn₀∈ such thatt >_∞

kn0φ^kt0. Since M

gx, gy_{n 1}, φt0 M

F x, y

, F yn, xn

, φt0

≥M

gx, gyn, t0

∗M

gy, gxn, t0

,

3.17

lettingn → ∞, we get M

gx, y, φt0

≥M

gx, y, t₀

∗M

gy, x, t₀

. 3.18

Similarly, we can get M

gy, x, φt0

≥M

gx, y, t₀

∗M

gy, x, t₀

. 3.19

From3.18and3.19we have M

gx, y, φt0

∗M

gy, x, φt0

≥ M

gx, y, t0

₂

∗ M

gy, x, t0

₂

. 3.20

By this way, we can get for alln∈,

M

gx, y, φⁿt0

∗M

gy, x, φⁿt0

≥ M

gx, y, φⁿ⁻¹t0₂

∗

M

gy, x, φⁿ⁻¹t0₂

≥ M

gx, y, t0

₂ⁿ

∗ M

gy, x, t0

₂ⁿ .

3.21

Then, we have

M

gx, y, t

∗M

gy, x, t

≥M

gx, y, ∞ kn0

φ^kt0

∗M

gy, x, ∞ kn0

φ^kt0

≥M

gx, y, φⁿ⁰t0

∗M

gy, x, φⁿ⁰t0

≥ M

gx, y, t0

₂ⁿ0

∗ M

gy, x, t0

₂ⁿ0

≥1−μ∗1−μ∗ · · · ∗1−μ

2²ⁿ⁰

≥1−λ.

3.22

So for anyλ >0 we have

M

gx, y, t

∗M

gy, x, t

≥1−λ, 3.23

for allt >0. We can get thatgxyandgyx.

(10)

Step 4. Prove thatxy.

Since∗is at-norm of H-type, for anyλ >0, there exists anμ >0 such that 1−μ∗1−μ∗ · · · ∗1−μ

k

≥1−λ,

3.24

for allk∈.

SinceMx, y,·is continuous and lim_t→_∞Mx, y, t 1, there existst₀ >0 such that Mx, y, t0≥1−μ.

n1φⁿt0<∞. Then for anyt >0, there existsn₀∈ such thatt >_∞

kn0φ^kt0. Since fort0>0,

M

gx_{n 1}, gy_{n 1}, φt0 M

F xn, yn

, F yn, xn

, φt0

≥M

gx_n, gy_n, t₀

∗M

gy_n, gx_n, t₀ .

3.25

Lettingn → ∞yields M

x, y, φt0

≥M x, y, t0

∗M y, x, t0

. 3.26

Thus we have

M x, y, t

≥M

x, y, ∞ kn0

φ^kt0

≥M

x, y, φⁿ⁰t0

≥ M

x, y, t₀2ⁿ⁰ ∗ M

y, x, t₀2ⁿ⁰

≥1 −μ∗1−μ∗ · · · ∗1−μ

2²ⁿ⁰

≥1−λ,

3.27

which implies thatxy.

Thus we have proved thatFandghave a unique common fixed point inX.

This completes the proof of theTheorem 1.

TakinggIthe identity mappinginTheorem 1, we get the following consequence.

Corollary 1. Let X, M,∗ be a complete FM-space, where ∗ is a continuous t-norm of H-type satisfying2.7. LetF:X×X → Xand there existsφ∈Φsuch that

M F

x, y

, Fu, v, φt

≥Mx, u, t∗M y, v, t

, 3.28

for allx, y, u, v∈X,t >0.

(11)

Then there existx∈Xsuch thatxFx, x, that is,Fadmits a unique fixed point inX.

Letφt kt, where 0< k <1, the following byLemma 1, we get the following.

Corollary 2see6. Leta∗b≥abfor alla, b∈0,1andX, M,∗be a complete fuzzy metric space such thatMhasn-property. LetF:X×X → Xandg:X → Xbe two functions such that

M F

x, y

, Fu, v, kt

≥M

gx, gu, t

∗M

gy, gv, t

, 3.29

for allx, y, u, v∈X, where 0< k <1,FX×X⊂gXandgis continuous and commutes withF.

Then there exists a uniquex∈Xsuch thatxgx Fx, x.

Next we give an example to demonstrateTheorem 1.

Example 3. LetX −2,2,a∗babfor alla, b∈0,1.ψis defined as2.8. Let M

x, y, t

ψt_|x−y|

, 3.30

for allx, y∈0,1. ThenX, M,∗is a complete FM-space.

Letφt t/2,gx xandF:X×X → Xbe defined as

F x, y

x² 8

y²

8 −2, ∀x, y∈X. 3.31

ThenFsatisfies all the condition ofTheorem 1, and there exists a pointx2−2√

3 which is the unique common fixed point ofgandF.

In fact, it is easy to see thatFX×X −2,−1, M

F x, y

, Fu, v, φt

ψφt_|x²_−u²_y²_−v²_|/8

, 3.32

For allt >0 andx, y∈−2,2.3.28is equivalent to

ψ t

2

_|x²_−u²_y²_−v²_|/8

≥

ψt_|x−u|

·

ψt_|y−v|

. 3.33

Sinceψt∈0,1, we can get

ψ t

2

_|x²_−u²_y²_−v²_|/8

≥

ψ t

2

_|x−u|/2

·

ψ t

2

_|y−v|/2

. 3.34

From3.33, we only need to verify the following:

ψ

t 2

_|x−u|/2

≥

ψt_|x−u|

, 3.35

(12)

that is,

ψ t

2

≥ ψt₂

, ∀t >0. 3.36

We consider the following cases.

Case 10< t≤4. Then3.36is equivalent to

α t

2 ≥ α√

t₂

, 3.37

it is easy to verified.

Case 2t≥8. Then3.36is equivalent to

1− 1 lnt/2 ≥

1− 1

lnt ₂

, 3.38

which is

2 lnt·ln t

2 ≥ln²t ln t

2, 3.39

since

ln²t ln²t

2−2 lnt·ln t 2 lnt

2 −ln²t

2 ≤0, 3.40

that is

ln²2 ln t 2−ln²t

2 ≤0, 3.41

holds for allt≥8. So3.36holds fort≥8.

Case 34< t <8. Then3.36is equivalent to

α t

2 ≥

1− 1 lnt

₂

. 3.42

Lette^x, we only need to verify

√α 2e^x/2−

1− 1

x ₂

≥0, 3.43

for allxthat 2 ln 2< x <3 ln 2. We can verify it holds.

(13)

Thus it is verified that the functions F,g,φ satisfy all the conditions of Theorem 1;

x2−2√

3 is the common fixed point ofFandginX.

Acknowledgment

The author is grateful to the referees for their valuable comments and suggestions.

References

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

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

3 A. George and P. Veeramani, “On some results of analysis for fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 90, no. 3, pp. 365–368, 1997.

4 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.

5 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.

6 S. Sedghi, I. Altun, and N. Shobe, “Coupled fixed point theorems for contractions in fuzzy metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1298–1304, 2010.

7 J.-X. Fang, “Common fixed point theorems of compatible and weakly compatible maps in Menger spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1833–1843, 2009.

8 L. B. ´Ciri´c, D. Mihet¸, and R. Saadati, “Monotone generalized contractions in partially ordered probabilistic metric spaces,” Topology and its Applications, vol. 156, no. 17, pp. 2838–2844, 2009.

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

10 S. Jain, S. Jain, and L. Bahadur Jain, “Compatibility of typePin modified intuitionistic fuzzy metric space,” Journal of Nonlinear Science and its Applications, vol. 3, no. 2, pp. 96–109, 2010.

11 L. B. Ćirić, S. N. Jeˇsić, and J. S. Ume, “The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy metric spaces,” Chaos, Solitons and Fractals, vol. 37, no. 3, pp. 781–791, 2008.

12 L. ´Ciri´c and V. Lakshmikantham, “Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Stochastic Analysis and Applications, vol. 27, no. 6, pp. 1246–1259, 2009.

13 L. Ćirić, N. Cakić, M. Rajović, and J. S. Ume, “Monotone generalized nonlinear contractions in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 131294, 11 pages, 2008.

14 A. Aliouche, F. Merghadi, and A. Djoudi, “A related fixed point theorem in two fuzzy metric spaces,”

Journal of Nonlinear Science and its Applications, vol. 2, no. 1, pp. 19–24, 2009.

15 L. ´Ciri´c, “Common fixed point theorems for a family of non-self mappings in convex metric spaces,”

Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 5-6, pp. 1662–1669, 2009.

16 K. P. R. Rao, A. Aliouche, and G. R. Babu, “Related fixed point theorems in fuzzy metric spaces,”

Journal of Nonlinear Science and its Applications, vol. 1, no. 3, pp. 194–202, 2008.

17 L. Ćirić and N. Cakić, “On common fixed point theorems for non-self hybrid mappings in convex metric spaces,” Applied Mathematics and Computation, vol. 208, no. 1, pp. 90–97, 2009.

18 L. ´Ciri´c, “Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces,” Chaos, Solitons and Fractals, vol. 42, no. 1, pp. 146–154, 2009.

19 S. Shakeri, L. J. B. ´Ciri´c, and R. Saadati, “Common fixed point theorem in partially orderedL-fuzzy metric spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 125082, 13 pages, 2010.

20 L. ´Ciri´c, B. Samet, and C. Vetro, “Common fixed point theorems for families of occasionally weakly compatible mappings,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 631–636, 2011.

(14)

21 L. ´Ciri´c, M. Abbas, R. Saadati, and N. Hussain, “Common fixed points of almost generalized contractive mappings in ordered metric spaces,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5784–5789, 2011.

22 L. Ćirić, M. Abbas, B. Damjanović, and R. Saadati, “Common fuzzy fixed point theorems in ordered metric spaces,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1737–1741, 2011.

23 T. Kamran and N. Caki´c, “Hybrid tangential property and coincidence point theorems,” Fixed Point Theory, vol. 9, no. 2, pp. 487–496, 2008.

24 O. Hadˇzi´c and E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, vol. 536 of Mathematics and its Applications, Kluwer Academic, Dordrecht, The Netherlands, 2001.

25 M. Grabiec, “Fixed points in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 27, no. 3, pp. 385–389, 1988.