Tripled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces

(1)

Volume 2012, Article ID 329298,14pages doi:10.1155/2012/329298

Research Article

Tripled Coincidence Point Theorems for Nonlinear Contractions in Partially Ordered Metric Spaces

Binayak S. Choudhury,

¹

Erdal Karapınar,

²

and Amaresh Kundu

³

1Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah 711103, India

2Department of Mathematics, Atilim University, ˙Incek 06836, Ankara, Turkey

3Department of Mathematics, Siliguri Institute of Technology, Darjeeling 734009, India

Correspondence should be addressed to Amaresh Kundu,[email protected] Received 12 April 2012; Accepted 19 May 2012

Academic Editor: Paolo Ricci

Copyrightq2012 Binayak S. Choudhury 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.

Tripled fixed points are extensions of the idea of coupled fixed points introduced in a recent paper by Berinde and Borcut, 2011. Here using a separate methodology we extend this result to a triple coincidence point theorem in partially ordered metric spaces. We have defined several concepts pertaining to our results. The main results have several corollaries and an illustrative example.

The example shows that the extension proved here is actual and also the main theorem properly contains all its corollaries.

1. Introduction and Preliminaries

In recent times coupled fixed point theory has experienced a rapid growth in partially ordered metric spaces. The speciality of this line of research is that the problems herein utilize both order theoretic and analytic methods. References1–19are some instances of these works.

Definition 1.1see14. A functiong : R → R is said to be monotone nondecreasingor increasingifx≥yimpliesgx≥gy.

Definition 1.2see14. LetX be a nonempty set. LetF : X×X → X be a mapping. An elementx, yis called a coupled fixed point ofFif

F x, y

x, F y, x

y. 1.1

(2)

Recently, Berinde and Borcut20extended the idea of coupled fixed points to tripled fixed points. The definition is as follows.

Definition 1.3see20. LetXbe a nonempty set. LetF:X×X×X → Xbe a mapping. An elementx, y, zis called a tripled fixed point ofFif

F x, y, z

x, F y, x, y

y, F z, y, x

z. 1.2 They also extended the mixed monotone property to functions with three arguments.

Definition 1.4see20. LetX,be a partially ordered set andF : X×X×X → X. The mappingFis said to have the mixed monotone property if for anyx, y, z∈X

x₁, x₂∈X, x₁x₂⇒F

x₁, y, z F

x₂, y, z , y₁, y₂∈X, y₁y₂ ⇒F

x, y₁, z F

x, y₂, z , z1, z2 ∈X, z1z2⇒F

x, y, z1

F

x, y, z2

.

1.3

Our purpose here is to establish tripled coincidence point results in metric spaces with partial ordering. For that purpose we define mixedg-monotone property in the following.

Mixed g-monotone property was already defined in the context of coupled fixed points14.

Here in the spirit ofDefinition 1.4we have made an extension of that.

Definition 1.5. LetX,be a partially ordered set. Letg :X → XandF :X×X×X → X.

The mappingFis said to have the mixedg-monotone property if for anyx, y, z∈X.

x1, x2∈X, gx1gx2⇒F

x1, y, z F

x2, y, z , y₁, y₂∈X, gy₁gy₂⇒F

x, y₁, z F

x, y₂, z , z1, z2∈X, gz1gz2⇒F

x, y, z1

F

x, y, z2

.

1.4

Coupled coincidence point was defined by Lakshmikantham and ´Ciri´c14. We also extend the concept of coupled coincidence point to tripled coincidence point in the following.

Definition 1.6. LetXbe any nonempty set. Letg:X → XandF :X×X×X → X. An element x, y, zis called a tripled coincidence point ofgandFif

F x, y, z

gx, F y, x, y

gy, F z, y, x

gz. 1.5 We extend the concept of commuting mappings given by Lakshmikantham and ´Ciri´c 14, in the following definition.

Definition 1.7. LetX be a nonempty set. Then one says that the mappingsg : X → Xand F:X×X×X → Xare commuting if for allx, y, z∈X

g F

x, y, z F

gx, gy, gz

. 1.6

(3)

The following is the definition of compatible mappings which is an extension of the compatibility defined by Choudhury and Kundu in8.

Definition 1.8see8. LetX, dbe a metric space. The mappingsgandF, whereg:X → X andF:X×X×X → Xare said to be compatible if

n→ ∞limd gF

xn, yn, zn

, F

gxn, gyn, gzn

0,

nlim→ ∞d gF

yn, xn, yn

, F

gyn, gxn, gyn

0,

n→ ∞limd gF

z_n, y_n, x_n , F

gz_n, gy_n, gx_n 0,

1.7

whenever{xn},{yn},{zn}are sequences inXsuch that

nlim→ ∞F

x_n, y_n, z_n

gx_nx,

nlim→ ∞F

yn, xn, yn

gyny,

nlim→ ∞F

zn, yn, xn

gznz.

1.8

2. Main Results

Theorem 2.1. LetX,be a partially ordered set and suppose there is a metricdonX such that X, dis a complete metric space. SupposeF:X×X×X → Xandg :X → Xare such that,gis monotone increasing,Fhas the mixedg-monotone property and

d F

x, y, z

, Fu, v, w

≤ψ max

d gx, gu

, d gy, gv

, d

gz, gw

2.1 for allx, y, z∈Xfor whichgxgu,gygvandgzgw, whereψ:0,∞ → 0,∞is such thatψtis monotone,ψt< tand limr→tψr< tfor allt >0. SupposeFX×X×X⊆gX,g is continuous, and{g, F}is a compatible pair. Suppose either

aF is continuous or

bXhas the following properties:

iif a nondecreasing sequence{αn} → α, thenαnαfor alln, iiif a nonincreasing sequence{βn} → β, thenβ_nβfor alln.

If there existx0, y0, z0 ∈ X such thatgx0 Fx0, y0, z0,gy0 Fy0, x0, y0, andgz0 Fz0, y₀, x₀, then there existx, y, z∈Xsuch that

F x, y, z

gx, F y, x, y

gy, F z, y, x

gz, 2.2

that is,gandFhave a tripled coincidence point.

(4)

Proof. By a condition of the theorem, there existx0, y0, z0 ∈X such thatgx0 Fx0, y0, z0, gy₀ Fy0, x₀, y₀, and gz₀ Fz0, y₀, x₀. Since FX ×X ×X ⊆ gX, we can choose x₁, y₁, z₁∈Xsuch that

gx₁F

x₀, y₀, z₀

, gy₁F

y₀, x₀, y₀

, gz₁F

z₀, y₀, x₀

. 2.3

Continuing this process, we can construct sequences{xn},{yn}, and{zn}inXsuch that gx_n1F

xn, yn, zn

, gy_n1F

yn, xn, yn

, gz_n1F

zn, yn, xn

. 2.4

Next we will show that, forn≥0,

gx_ngx_n1, gy_ngy_n1, gz_n gz_n1. 2.5

Since,gx0Fx0, y0, z0, gy0Fy0, x0, y0, andgz0 Fz0, y0, x0, by2.3, we get

gx0gx1, gy0gy1, gz0gz1, 2.6

that is,2.5holds forn0.

We presume that2.5holds for somen m > 0. As F has the mixed g-monotone property andgxmgx_m1, gymgy_m1andgzmgz_m1, we obtain

gx_m1F

x_m, y_m, z_m F

x_m1, ym, zm

F

x_m1, y_m, z_m1 F

x_m1, y_m1, z_m1

gx_m2,

2.7

gy_m1F

y_m, x_m, y_m F

ym, xm, y_m1 F

y_m1, x_m, y_m1 F

y_m1, x_m1, y_m1

gy_m2,

2.8

gz_m1F

z_m, y_m, x_m F

z_m1, y_m, x_m F

z_m1, y_m1, xm

F

z_m1, y_m1, x_m1

gz_m2.

2.9

Thus,2.5holds fornm1. Then, by induction, we conclude that2.5holds forn≥1.

If for somen∈N,

gxngx_n1, gyngy_n1, gzn gz_n1, 2.10

(5)

then, by2.4,xn, yn, znis a tripled coincidence point ofgandF. Therefore we assume, for anyn∈N,

gxn, gyn, gzn

/

gx_n1, gy_n1, gz_n1

. 2.11

Setδ_nmax{dgxn, gx_n1, dgyn, gy_n1, dgzn, gz_n1}.

Then

δ_n>0 ∀n≥0. 2.12

Then, by2.1,2.4and2.5, we have d

gx_n, gx_n1 d

F

x_n−1, y_n−1, z_n−1 , F

x_n, y_n, z_n

≤ψ max

d

gx_n−1, gxn

, d

gy_n−1, gyn

, d

gz_n−1, gzn

,

d

gy_n, gy_n1 d

F

y_n−1, x_n−1, y_n−1 , F

y_n, x_n, y_n

≤ψ max

d

gy_n−1, gy_n , d

gx_n−1, gx_n , d

gy_n−1, gy_n , d

gzn, gz_n1 d

F

z_n−1, y_n−1, x_n−1 , F

zn, yn, xn

≤ψ max

d

gz_n−1, gz_n , d

gy_n−1, gy_n , d

gx_n−1, gx_n .

2.13

Thus, from2.13we obtain that δnmax

d

gxn, gx_n1 , d

gyn, gy_n1 , d

gzn, gz_n1

≤ψ max

d

gx_n−1, gx_n , d

gy_n−1, gy_n , d

gz_n−1, gz_n

. 2.14

It then follows from2.12and a propertyψ, that for alln≥1,

δ_n≤ψδ_n−1< δ_n−1. 2.15

Thus,{δn}is a monotone decreasing sequence of nonnegative real numbers. So, there exist a δ≥0 such that

nlim→ ∞δ_nδ. 2.16

Supposeδ >0. Lettingn → ∞in2.14, using2.15,2.16, and a property ofψ, we get

δ≤ψδ< δ, 2.17

which is a contradiction. Thusδ0, or

n→ ∞limδn0, 2.18

(6)

or

nlim→ ∞d

gx_n1, gx_n 0,

nlim→ ∞d

gy_n1, gyn

0,

nlim→ ∞d

gz_n1, gzn

0.

2.19

Now, we will prove that{gxn}, {gyn}, and{gzn} are Cauchy sequences. Suppose, to the contrary, that at least one of {gxn},{gyn}, and{gzn}is not a Cauchy sequence. So, there exists anε >0 for which we can find subsequences{gx_nk}of{gxn},{gy_nk}of{gyn}, and {gz_nk}of{gzn}withnk> mk≥ksuch that

αkmax d

gx_nk, gx_mk , d

gy_nk, gy_mk , d

gz_nk, gz_mk

≥ε. 2.20

Additionally, corresponding tomk, we may choosenksuch that it is the smallest integer satisfying2.20. Then, for allk≥0,

max d

gx_nk−1, gx_mk , d

gy_nk−1, gy_mk , d

gz_nk−1, gz_mk

< ε. 2.21

By using2.20and2.21we have fork≥0,

ε≤αkmax d

gx_nk, gx_mk , d

gy_nk, gy_mk , d

gz_nk, gz_mk

≤ max d

gx_nk, gx_nk−1 d

gx_nk−1, gx_mk , d

gy_nk, gy_nk−1 d

gy_nk−1, gy_mk , d

gz_nk, gz_nk−1 d

gz_nk−1, gz_mk

≤ max d

gx_nk, gx_nk−1 , d

gy_nk, gy_nk−1 , d

gz_nk, gz_nk−1 ε

≤δ_nk−1ε.

2.22

Lettingk → ∞in2.22, and using2.19, we get

klim→ ∞α_k lim

k→ ∞max d

gx_nk, gx_mk , d

gy_nk, gy_mk , d

gz_nk, gz_mk

ε. 2.23

Let, fork≥0,

βkmax d

gx_nk1, gx_mk1 , d

gy_nk1, gy_mk1 , d

gz_nk1, gz_mk1

. 2.24

(7)

Again, for allk≥0,

α_k max d

gx_nk, gx_mk , d

gy_nk, gy_mk , d

gz_nk, gz_mk

≤ max d

gx_nk, gx_nk1 d

gx_nk1, gx_mk1 d

gx_mk1, gx_mk , gy_nk, gy_nk1

d

gy_nk1, gy_mk1 d

gy_mk1, gy_mk , d

gz_nk, gz_nk1 d

gz_nk1, gz_mk1 d

gz_mk1, gz_mk

≤ max d

gx_nk, gx_nk1 , d

gy_nk, gy_nk1 , d

gz_nk, gz_nk1 max

d

gz_nk1, gz_mk1 max

d

gx_mk, gx_mk1 , d

gy_mk, gy_mk1 , d

gz_mk, gz_mk1

≤δ_nk1βkδ_mk1.

2.25

Analogously we have fork≥0,

β_k max d

gy_nk, gy_mk1 , d

gz_nk1, gz_mk1

≤ max d

gx_nk1, gx_nk d

gx_nk, gx_mk d

gx_mk, gx_mk1 , d

gy_nk1, gy_nk d

gy_nk, gy_mk d

gy_mk, gy_mk1 , d

gz_nk1, gz_nk d

gz_nk, gz_mk d

gz_mk, gz_mk1

≤ max d

gx_nk, gx_nk1 , d

gy_nk, gy_nk1 , d

gz_nk, gz_nk1 max

d

gx_nk, gx_mk , d

gy_nk, gy_mk , d

gz_nk, gz_mk max

d

gx_mk, gx_mk1 , d

gy_mk, gy_mk1 , d

gz_mk, gz_mk1

≤δ_nk1α_kδ_mk1.

2.26

Lettingk → ∞in2.25and2.26, we get that

k→ ∞lim max d

gz_nk1, gz_mk1 lim

k→ ∞βkε lim

k→ ∞αk. 2.27

Sincenk> mk, fork≥0, we have

gx_nkgx_mk, gy_nkgy_mk,

gz_nkgz_mk. 2.28

(8)

Then from2.1,2.4, and2.28, we have fork≥0, d

gx_nk1, gx_mk1 d

F

x_nk, y_nk, z_nk , F

x_mk, y_mk, z_mk

≤ψ max

d

gx_nk, gx_mk , d

gy_nk, gy_mk , d

gz_nk, gz_mk , d

gy_nk1, gy_mk1 d

F

y_nk, x_nk, y_nk , F

y_mk, x_mk, y_mk

≤ψ max

d

gy_nk, gy_mk , d

gx_nk, gx_mk , d

gy_nk, gy_mk , d

gz_nk1, gz_mk1 d

F

z_nk, y_nk, x_nk , F

z_mk, y_mk, x_mk

≤ψ max

d

gz_nk, gz_mk , d

gy_nk, gy_mk , d

gx_nk, gx_mk . 2.29

From2.29fork≥0, we get β_k≤ψ

max d

gx_nk, gx_mk , d

gy_nk, gy_mk , d

gz_nk, gz_mk

ψαk. 2.30 Lettingk → ∞in2.30, using2.20,2.27, and a property ofψ, we get

ε≤ψε< ε, 2.31

which is a contradiction. This shows that{gxn},{gyn}, and{gzn}are Cauchy sequences.

SinceXis complete, there existx, y, z∈Xsuch that

nlim→ ∞gxnx, lim

n→ ∞gyny, lim

n→ ∞gznz. 2.32

From2.4and2.32, using the continuity ofg, we have gx lim

n→ ∞g gx_n1

lim

n→ ∞g F

x_n, y_n, z_n

, 2.33

gy lim

n→ ∞g gy_n1

lim

n→ ∞g F

y_n, x_n, y_n

, 2.34

gz lim

n→ ∞g gz_n1

lim

n→ ∞g F

zn, yn, xn

. 2.35

Now we will show thatgxFx, y, z, gyFy, x, y, andgzFz, y, x.

Sinceg andF are compatible, in addition with2.33,2.34, and2.35, respectively imply

nlim→ ∞d g

F

x_n, y_n, z_n , F

gxn, g y_n

, gzn

0, 2.36

nlim→ ∞d g

F

yn, xn, yn

, F g

yn

, gxn, g yn

0, 2.37

nlim→ ∞d g

F

zn, yn, xn

, F

gzn, g yn

, gxn

0. 2.38

Suppose now the assumptionaholds, that is,Fis continuous.

(9)

For alln≥0, we have d

gx, F

gx_n, gy_n, gz_n

≤d gx, g

F

x_n, y_n, z_n d

g F

x_n, y_n, z_n , F

gx_n, gy_n, gz_n . 2.39 Taking the limit as n → ∞, using 2.32, 2.33, 2.36, and the facts that g and F are continuous, we havedgx, Fx, y, z 0.

Similarly, by using2.32,2.34, and2.37and2.32,2.35, and2.38, respectively, and also the facts that g and F are continuous, we have dgy, Fy, x, y 0 and dgz, Fz, y, x 0.

Thus we have proved thatgandFhave a tripled coincidence point.

Suppose that the assumption b holds. Since{gxn}, {gzn} are nondecreasing and gx_n → xwithgz_n → zand also{gyn}is nonincreasing withgy_n → y, by assumptionb we have for alln

gx_nx, gy_ny, gz_nz. 2.40

By virtue of monotone increasing property ofgwe have

ggx_ngx, ggy_ngy, ggz_ngz. 2.41

Now using2.4we have d

gx, F

x, y, z

≤d gx, g

gx_n1 d

g

gxn1 , F

x, y, z

≤d gx, g

gx_n1 d

g F

xn, yn, zn

, F

gxn, gyn, gzn

d

F

gx_n, gy_n, gz_n , F

x, y, z

≤d gx, g

gx_n1 d

g F

xn, yn, zn

, F

gxn, gyn, gzn

ψ

max d

ggx_n, gx , d

ggy_n, gy , d

ggz_n, gz

,

by2.1,2.41 . 2.42

Taking the limit asn → ∞in the above inequality, using2.33,2.36, and2.41we have d

gx, F

x, y, z

≤ lim

n→ ∞ψ max

d

ggxn, gx , d

ggyn, gy , d

ggzn, gz

. 2.43

By2.33,2.34,2.35, and the property ofψ, we have d

gx, F

x, y, z

≤ψ0 0, 2.44

that is

gxF x, y, z

. 2.45

(10)

In a similar manner using 2.33,2.34,2.35, and 2.36, 2.37, 2.38, respectively, we obtain

gyF y, x, y

, gzF

z, y, x

. 2.46

Thus, we proved thatgandFhave a tripled coincidence point.

This completes the proof of the theorem.

Corollary 2.2. LetX,be a partially ordered set and suppose there is a metricdonXsuch that X, dis a complete metric space. SupposeF:X×X×X → Xandg:X → Xare such thatFhas the mixedg-monotone property and

d F

x, y, z

, Fu, v, w

≤ψ max

d gx, gu

, d gy, gv

, d

gz, gw

2.47 for anyx, y, z∈Xfor whichgxgu,gy gvandgz gw, whereψ : 0,∞ → 0,∞be such thatψtis monotone,ψt< tand lim_r_→_tψr< tfor allt >0. SupposeFX×X×X⊆gX, gis continuous, andFandgare commuting. Suppose either

aFis continuous, or

bXhas the following property:

iif a nondecreasing sequence{xn} → x, thenxnxfor alln, iiif a nonincreasing sequence{yn} → y, thenynyfor alln.

If there existx₀, y₀, z₀ ∈ X such thatgx₀ Fx0, y₀, z₀,gy₀ Fy0, x₀, y₀, andgz₀ Fz0, y0, x0, then there existx, y, z∈Xsuch that

F x, y, z

gx, F y, x, y

gy, F z, y, x

gz, 2.48 that is,Fandghave a tripled coincidence point.

Proof. Since a commuting pair is also a compatible pair, the result of theCorollary 2.2follows fromTheorem 2.1.

Later, by an example, we will show that the Corollary 2.2 is properly contained in Theorem 2.1.

Corollary 2.3. LetX,be a partially ordered set and suppose there is a metricdonXsuch that X, dis a complete metric space. SupposeF:X×X×X → Xbe such thatFhas the mixed monotone property and

d F

x, y, z

, Fu, v, w

≤ψ max

dx, u, d y, v

, dz, w

2.49 for anyx, y, z∈X for whichxu, yvandzw, whereψ :0,∞ → 0,∞be such that ψtis monotone,ψt< tand lim_r_→_tψr< tfor allt >0. Suppose

(11)

aFis continuous, or

iif a nondecreasing sequence{xn} → x, thenxnxfor alln, iiif a nonincreasing sequence{yn} → y, theny_nyfor alln.

If there exist x0, y0, z0 ∈ X such that x0 Fx0, y0, z0, y0 Fy0, x0, y0, and z0Fz0, y0, x0, then there existx, y, z∈Xsuch that

F x, y, z

x, F y, x, y

y, F z, y, x

z, 2.50 that is,Fhas a tripled fixed point.

Proof. Takinggx xinTheorem 2.1we obtainCorollary 2.3.

Corollary 2.4. LetX,be a partially ordered set and suppose there is a metricdonXsuch that X, dis a complete metric space. SupposeF:X×X×X → Xandg:X → Xare such thatFhas the mixed monotone property and

d F

x, y, z

, Fu, v, w

≤kmax

dx, u, d y, v

, dz, w

2.51

for anyx, y, z∈Xfor whichxu, yvandzw, where 0< k <1. Suppose either aFis continuous, or

iif a nondecreasing sequence{xn} → x, then x_nxfor alln, iiif a nonincreasing sequence{yn} → y, theny_nyfor alln.

If there exist x₀, y₀, z₀ ∈ X such that x₀ Fx0, y₀, z₀, y₀ Fy0, x₀, y₀, and z₀ Fz0, y₀, x₀, then there existx, y, z∈Xsuch that

F x, y, z

x, F y, x, y

y, F z, y, x

z, 2.52 that is,Fhas a tripled coincidence point.

Proof. Takingψt kt,t >0 where 0< k <1, inCorollary 2.3we obtainCorollary 2.4.

The following corollary is the result of Berinde and Borcut in20.

Corollary 2.5. LetX,be a partially ordered set and suppose there is a metricdonXsuch that X, dis a complete metric space. SupposeF:X×X×X → Xbe such thatFhas the mixed monotone property and

d F

x, y, z

, Fu, v, w

≤a₁dx, u a₂d y, v

a₃dz, w 2.53

for anyx, y, z∈Xfor whichxu, yvandzw, wherea₁a₂a₃<1. Suppose either

(12)

aFis continuous, or

iif a nondecreasing sequence{xn} → x, thenx_nxfor alln iiif a nonincreasing sequence{yn} → y, theny_nyfor alln.

If there exist x₀, y₀, z₀ ∈ X such that x₀ Fx0, y₀, z₀, y₀ Fy0, x₀, y₀, and z₀Fz0, y₀, x₀, then there existx, y, z∈Xsuch that

F x, y, z

x, F y, x, y

y, F z, y, x

z, 2.54 that is,Fhas a tripled fixed point.

Proof. The proof follows fromCorollary 2.4, since the inequality inCorollary 2.5implies that Corollary 2.4.

Remark 2.6. The method used in the proof of Corollary 2.5 is diﬀerent from that used by Berinde and Borcut20.

Next we discuss an example.

Example 2.7. LetX. ThenX,is a partially ordered set with the partial ordering defined byxyif and only if|x| ≤ |y|andx·y≥0.

Letdx, y |x−y|forx, y∈ . ThenX, dis a complete metric space.

Letg:X → Xbe defined asgx x²/10, for allx∈X. LetF:X×X×X → Xbe defined as

F x, y, z

x²−y²z²

9 , ∀x, y, z∈X. 2.55 Then F obeys the mixedg-monotone property.

Letψ:0,∞ → 0,∞be defined asψt 1/3tfor allt∈0,∞.

Let,{xn}, {yn}, and{zn}be three sequences inXsuch that

nlim→ ∞F

xn, yn, zn

lim

n→ ∞gxn a,

nlim→ ∞F

yn, xn, yn

lim

n→ ∞g yn

b,

nlim→ ∞F

z_n, y_n, x_n lim

n→ ∞gzn c.

2.56

Then explicitly,

nlim→ ∞

x_n²−y²_nz²_n

9 lim

n→ ∞

x²_n

10, ∀x, y, z∈X, or,

10a−10b10c

9 aimply a−10b10c0.

2.57

(13)

Again,

nlim→ ∞

y_n²−x_n²y²_n

9 lim

n→ ∞

y²_n

10, ∀x, y, z∈X, or,

10b−10a10b

9 bimply 11b−10a0.

2.58

And

nlim→ ∞

z²_n−y²_nx²_n

9 lim

n→ ∞

z²_n

10, ∀x, y, z∈X, or,

10c−10b10a

9 cimplyc−10b10a0.

2.59

Then from the above relations we have,a0, b0, andc0.

Therefore,

d g

F

x_n, y_n, z_n , F

gx_n, gy_n, gz_n

−→0 asn−→ ∞, d

g F

yn, xn, yn

, F

gyn, gxn, gyn

−→0 asn−→ ∞, d

g F

z_n, y_n, x_n , F

gz_n, gy_n, gx_n

−→0 asn−→ ∞.

2.60

Hence, the pairg, Fis compatible inX.

Also,x0 0, z0 c> 0, andy0 0 are three points inX such thatgx0 g0 0< c²/9F0,0, c Fx0, y₀, z₀, gy0 g0 0 F0,0,0 Fy0, x₀, y₀, andgz0

gc c²/10< c²/9Fc,0,0 Fz0, y₀, x₀.

We next verify inequality2.1ofTheorem 2.1. We takex, y, z, u, v, w∈ X, such that gxgu, gzgwandgygv, that is,x²≤u², z²≤w², andy²≥v².

LetAmax{dgx, gu, dgy, gv, dgz, gw}max{|x²−u²|,|y²−v²|,|z²−w²|}.

ThendFx, y, z, Fu, v, w dx² −y²z²/9,u² −v²w²/9 |x²−u²− y²−v² z²−w²/3| ≤|x²−u²||y²−v²||z²−w²|/9≤3A/9 A/3ψA ψmax{dgx, gu, dgy, gv, dgz, gw}.

Thus it is verified that the functions g, F, and ψ satisfy all the conditions of Theorem 2.1. Here0,0,0is the tripled coincidence point ofgandFinX.

Remark 2.8. It is observed that in Example 2.7 the function F and g do not commute, but they are compatible. HenceCorollary 2.2cannot be applied to this example. This shows that Theorem 2.1properly containsCorollary 2.2. Alsog /I, so the results of Berinde and Borcut 20cannot be applied to this example. This shows that result in20is eﬀectively generalised.

(14)

References

1 M. Abbas, A. R. Khan, and T. Nazir, “Coupled common fixed point results in two generalized metric spaces,” Applied Mathematics and Computation, vol. 217, no. 13, pp. 6328–6336, 2011.

2 H. Aydi, “Some coupled fixed point results on partial metric spaces,” International Journal of Mathe- matics and Mathematical Sciences, Article ID 647091, 11 pages, 2011.

3 V. Berinde, “Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 18, pp. 7347–

7355, 2011.

4 B. S. Choudhury and P. Maity, “Coupled fixed point results in generalized metric spaces,”

Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 73–79, 2011.

5 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.

6 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.

7 B. S. Choudhury, N. Metiya, and A. Kundu, “Coupled coincidence point theorems in ordered metric spaces,” Annali dell’Universit´a di Ferrara, vol. 57, no. 1, pp. 1–16, 2011.

8 B. S. Choudhury and A. Kundu, “A coupled coincidence point result in partially ordered metric spaces for compatible mappings,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 8, pp. 2524–2531, 2010.

9 D. Dori´c, Z. Kadelburg, and S. Radenovi´c, “Coupled fixed point results for mappings without mixed monotone property,” Applied Mathematics Letters. In press.

10 J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3403–3410, 2009.

11 M. Jleli, V. Ćojbaˇsić-Rajić, B. Samet, and C. Vetro, “Fixed point theorems on ordered metric spaces and applications to Nonlinear beam equations,” Journal of Fixed Point Theory and its Applications. In press.

12 E. Karapınar, “Couple fixed point on cone metric spaces,” Gazi University Journal of Science, vol. 24, no. 1, pp. 51–58, 2011.

13 E. Karapınar, “Couple fixed point theorems for nonlinear contractions in cone metric spaces,”

Computers & Mathematics with Applications. An International Journal, vol. 59, no. 12, pp. 3656–3668, 2010.

14 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.

15 N. V. Luong and N. X. Thuan, “Coupled fixed points in partially ordered metric spaces and application,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 3, pp. 983–992, 2011.

16 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.

17 S. Radenović, Z. Kadelburg, D. Jandrlić, and A. Jandrlić, “Some results on weakly contractive maps,”

Bulletin of Iranian Mathematical Society. In press.

18 B. Samet, “Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 12, pp. 4508–

4517, 2010.

19 B. Samet and C. Vetro, “Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 12, pp. 4260–4268, 2011.

20 V. Berinde and M. Borcut, “Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces,” Nonlinear Analysis. Theory, Methods & Applications, vol. 74, no. 15, pp. 4889–

4897, 2011.

(15)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

^{Journal of}