New Meir-Keeler Type Tripled Fixed-Point Theorems on Ordered Partial Metric Spaces

(1)

Volume 2012, Article ID 409872,17pages doi:10.1155/2012/409872

Research Article

New Meir-Keeler Type Tripled Fixed-Point Theorems on Ordered Partial Metric Spaces

Hassen Aydi

¹

and Erdal Karapınar

²

1Institut Sup´erieur d’Informatique et des Technologies de Communication de Hammam Sousse, Universit´e de Sousse, Route GP1 4011, Tunisia

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

Correspondence should be addressed to Erdal Karapınar,[email protected]

Received 3 February 2012; Accepted 27 March 2012 Academic Editor: Zheng-Guang Wu

Copyrightq2012 H. Aydi and E. Karapınar. 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.

In this paper, we prove some new Meir-Keeler type tripled fixed-point theorems on a partially ordered complete partial metric space. Also, as application, some results of integral type are given.

1. Introduction and Preliminaries

In the last century, the theory of fixed points has appeared as a crucial technique in the study of nonlinear phenomena. Particularly, the tools in fixed-point theory have an application in such diverse fields as biology, chemistry, physics, economics, computer sciences, and engineering.

Recently, fixed-point theorems are considered on partial metric spaces on which self- distance of some points may not be zero. This phenomenon was discovered by Matthews1 when he considered the tools of metric spaces in the field of semantics and domain theory in computer sciencesee, e.g.,2,3. After the initial results of Mathews, other papers have been released on partial metric spacessee e.g.,4–20.

Another important development is reported in fixed-point theory via ordered metric spaces. Fixed-point theorems in ordered sets were discussed by Ran and Reurings 21.

Subsequently, many results in this direction were givensee, e.g.,22–31.

In this paper, we combine two recent trends, partial metric spaces and ordered sets, and discuss the existence and uniqueness of some new Meir-Keeler type tripled fixed-point theorems in the context of partially ordered partial metric spaces.

LetXbe a nonempty set. A partial metric is a functionp:X×X → 0,∞satisfying the following conditions:

(2)

P1ifpx, x px, y py, y, thenxy,

P2px, y py, x,

P3px, x≤px, y,

P4px, z py, y≤px, y py, z,

for allx, y, z∈X. Then,X, pis called a partial metric space.

Ifpis a partial metricponX, then the functiondp:X×X → 0,∞given by dp

x, y 2p

x, y

−px, x−p y, y

1.1 is a metric onX. Each partial metricponXgenerates aT0topologyτponXwith a base of the family of openp-balls{Bpx, ε:x∈X, ε >0}, whereBpx, ε {y∈X:px, y< px, x ε}

for allx ∈ X andε > 0. Similarly, closedp-ball is defined asBpx, ε {y ∈ X : px, y ≤ px, x ε}. For more details, see1,5.

Definition 1.1see, e.g.,1,5,15. LetX, pbe a partial metric space.

iA sequence{xn}inXconverges tox∈Xwhenever lim_n_{→ ∞}px, xn px, x.

iiA sequence{xn} inX is called Cauchy whenever lim_{n,m→ ∞}pxn, xmexistsand finite.

iii X, pis said to be complete if every Cauchy sequence{xn}inX converges, with respect toτp, to a pointx∈X, that is, limn,m_→∞pxn, xm px, x.

Lemma 1.2see, e.g.,1,5,15. LetX, pbe a partial metric space.

aA sequence{xn}is Cauchy if and only if{xn}is a Cauchy sequence in the metric space X, dp.

b X, pis complete if and only if the metric spaceX, dpis complete. Moreover,

nlim→ ∞dpx, xn 0⇐⇒ lim

n→ ∞px, xn lim

n,m→ ∞pxn, xm px, x. 1.2

Lemma 1.3see, e.g.,4,15,16. LetX, pbe a partial metric space. Then, Aifpx, y 0, thenxy,

Bifx /y, thenpx, y>0.

Remark 1.4. Ifxy,px, ymay not be 0.

Lemma 1.5see, e.g.,4,15,16. Letxn → zasn → ∞in a partial metric spaceX, pwhere pz, z 0. Then, lim_n_{→ ∞}pxn, y pz, yfor everyy∈X.

X, p,≤is called a partially ordered partial metric space ifX,≤is a partially ordered set andX, pis a partial metric space. Further, ifX, pis a complete partial metric space, then X, p,≤is called a partially ordered complete partial metric space. Hereafter, we assume that X /∅and we use the notation

X^kX ×X× · · · ×X

k-many

. 1.3

(3)

Also, take the mappingP:X³×X³ → 0,∞such that Px,y:max

p x1, y1

, p x2, y2

, p

x3, y3 , 1.4

wherex x1, x2, x3andy y1, y2, y3∈X³.

Let X,≤ be a partially ordered set. We consider the following partial orderalso denoted by≤on the product spaceX³:

u, v, w≤ x, y, z

iﬀx≥u, y≤v, z≥w, 1.5

whereu, v, w,x, y, z∈X³. Moreover, we say thatx, y, zis equal tou, v, rif and only if xu,v, andzr. In the sequel, we need the following definitions.

Definition 1.6 see 32. Let X,≤ be a partially ordered set and F : X³ → X a given mapping. We say that F has the mixed monotone property if Fx, y, z is monotone nondecreasing inxandz, and it is monotone nonincreasing iny, that is, for anyx, y, z∈X,

x1, x2∈X, x1≤x2⇒F

x1, y, z

≤F

x2, y, z , y1, y2∈X, y1≤y2 ⇒F

x, y1, z

≥F

x, y2, z , z1, z2 ∈X, z1≤z2⇒F

x, y, z1

≤F x, y, z2

.

1.6

Definition 1.7see32. An elementx, y, z∈X³is called a tripled fixed point ofF:X³ → X if

F x, y, z

x, F y, x, y

y, F z, y, x

z. 1.7

Berinde and Borcut32proved the following theorem.

Theorem 1.8. LetX,≤be a partially ordered set andX, da complete metric space. LetF:X³ → Xbe a mapping having the mixed monotone property onX. Assume that there exist constantsa, b, c∈ 0,1such thatabc <1 for which

d F

x, y, z

, Fu, v, w

≤adx, u bd y, v

cdz, w 1.8

for allx≥u,y≤vandz≥w. Assume thatXhas the following properties:

iif a nondecreasing sequencexn → x, thenxn≤xfor alln, iiif a nonincreasing sequenceyn → y, thenyn≥yfor alln. If there existx0, y0, z0∈Xsuch that

x0 ≤F

x0, y0, z0

, y0≥F

y0, x0, y0

, z0≤F

z0, y0, x0

, 1.9

then there existx, y, z∈Xsuch that F

x, y, z

x, F y, x, y

y, F z, y, x

z. 1.10

(4)

Recently,Theorem 1.8is extended to cone metric spaces by Rao and Kishore33. On the other hand, very recently, Aydi et al.34introduced the following concepts.

Definition 1.9see34. LetX,≤be a partially ordered set andF :X³ → X. We say thatF has the mixed strict monotone property if, for anyx, y, z∈X,

x1, x2∈X, x1< x2⇒F

x1, y, z

< F

x2, y, z , y1, y2∈X, y1< y2 ⇒F

x, y1, z

> F

x, y2, z , z1, z2 ∈X, z1< z2⇒F

x, y, z1

< F x, y, z2

.

1.11

Definition 1.10 see34. LetX, d,≤be a partially ordered metric space. A mapping F : X³ → Xis said to be a generalized Meir-Keeler type contraction if, for anyε >0, there exists aδε>0 such that

ε≤max

dx, u, d y, v

, dz, r < εδε ⇒d F

x, y, z

, Fu, v, r

< ε, 1.12

for allx, y, z, u, v, r∈Xwithx≤u,y≥vandz≤r.

In the following, we consider the partial case ofDefinition 1.10and we introduce the following.

Definition 1.11. LetX, p,≤be a partially ordered partial metric space. A mappingF:X³ → X is said to be a generalizedp-Meir-Keeler type contraction if, for anyε > 0, there exists a δε>0 such that

ε≤max

px, u, p y, v

, pz, r < εδε ⇒p F

x, y, z

, Fu, v, r

< ε, 1.13

for allx, y, z, u, v, r∈Xwithx≤u,y≥vandz≤r.

Remark 1.12. It is immediate to show that ifF :X³ → Xis a generalizedp-Meir-Keeler type contraction, then

p F

x, y, z

, Fu, v, r

<max

px, u, p y, v

, pz, r 1.14

for allx, u, y, v, z, r,∈Xwithx < u, y≥v, z < rorx≤u, y > v, z≤r.

Proposition 1.13. LetX, p,≤be a partially ordered partial metric space andF :X³ → Xa given mapping. If1.8is satisfied, thenFis a generalizedp-Meir-Keeler type function.

Proof. Assume that1.8is satisfied. For allε >0, one can check that1.13is satisfied with

δε 1/abc−1ε.

In the sequel, we use the following notations given in34. LetF:X³ → X³be such that, fora, b, c∈X,

Fa, b, c Fa, b, c, Fb, a, b, Fc, b, a. 1.15

(5)

Letx0, y0, z0∈Xbe such that x0< F

x0, y0, z0

, y0≥F

y0, x0, y0

, z0< F

z0, y0, x0

. 1.16

We consider sequences{xn},{yn}, and{zn}such that

⎡

⎣xn

yn

zn

⎤

⎦

An

⎡

⎣F

x_n−1, y_n−1, z_n−1 F

yn−1, xn−1, yn−1 F

zn−1, yn−1, xn−1

⎤

⎦

FA n−1

⎡

⎣Fⁿx0, y0, z0 Fⁿy0, x0, y0 Fⁿz0, y0, x0

⎤

⎦

FⁿA0

, 1.17

forn1,2,3, . . ..

Our first auxiliary result is as follows.

Proposition 1.14. LetX, p,≤be a partially ordered partial metric space, and letF :X³ → Xbe a given mapping such that the following hypotheses hold:

iFhas the mixed strict monotone property, iiFis a generalizedp-Meir-Keeler type function,

iii∃x, y, z,u, v, r∈X³such thatx < u,y≥vandz < r.

Then,

P Fⁿ

x, y, z

,Fⁿu, v, r

−→0, asn−→∞. 1.18

Proof. Letx, y, z x0, y0, z0andu, v, r u0, v0, r0. We show that xnFⁿ

x0, y0, z0,

< Fⁿu0, v0, r0 un, ynFⁿ

y0, x0, y0

> Fⁿv0, u0, v0 vn, znFⁿ

z0, y0, x0

< Fⁿr0, v0, u0 rn,

∀n1,2, . . . , 1.19

withFF¹.

Due to the fact that F has the mixed strict monotone property, together with the assumption thatx < u,y≥vandz < r, we obtain

x1F x, y, z

F

x0, y0, z0

< F

u0, y0, z0

⇒F

x0, y0, z0

< Fu0, v0, z0 ⇒F

x0, y0, z0

< Fu0, v0, r0 u1.

1.20

Analogously, we have y1F

y0, x0, y0

> Fv0, u0, v0 v1, z1F

z0, y0, x0

< Fr0, v0, u0 r1. 1.21

(6)

Thus,1.19holds forn1. By using the same arguments, we show that1.19holds also for n2. In fact,

x2F²

x0, y0, z0

F

x1, y1, z1

F

x0, y0, z0

, F

y0, x0, y0

, F

z0, y0, x0

< F

Fu0, v0, r0, F

y0, x0, y0

, F

z0, y0, x0

< F

Fu0, v0, r0, Fv0, u0, v0, F

z0, y0, x0

< FFu0, v0, r0, Fv0, u0, v0, Fr0, v0, u0 F²u0, v0, r0 Fu1, v1, r1 u2.

1.22

Similarly, we find

y2F²

y0, x0, y0

≥F²v0, u0, v0 v2, z2F²

z0, y0, x0

< F²r0, v0, u0 r2. 1.23

Inductively, we get that1.19holds.

ByRemark 1.12, together with1.19, we have

px_n2, u_n2 p Fⁿ²

x0, y0, z0

, Fⁿ²u0, v0, r0 p

F

xn1, yn1, zn1

, Fu_n1, vn1, rn1

<max

pxn1, un1, p

yn1, vn1

, pzn1, rn1 ,

1.24

pz_n2, r_n2 p Fⁿ²

z0, y0, x0

, Fⁿ²r0, v0, u0 p

F

z_n1, y_n1, x_n1

, Fr_n1, v_n1, u_n1

<max

pzn1, rn1, p

yn1, vn1

, pxn1, un1 ,

1.25

p

y_n2, v_n2 p

Fⁿ²

y0, x0, y0

, Fⁿ²v0, u0, v0 p

F

y_n1, x_n1, y_n1

, Fv_n1, u_n1, v_n1

<max p

yn1, vn1

, px_n1, un1, p

yn1, vn1

≤max

pzn1, rn1, p

yn1, vn1

, pxn1, un1 .

1.26

LetΔn1:max{pxn1, un1, pyn1, vn1, pzn1, rn1}. Combining1.24–1.26, we get

Δ_n2<Δ_n1, ∀n1,2. . . . 1.27

If we denoteBn un, vn, rn, then, by definition of the partial metricPand1.27, we have

PAn2, Bn2< PAn1, Bn1. 1.28

(7)

Consequently, the sequence{tn} {PAn, Bn}is decreasing. Hence,{tn}converges, say to ε ≥0. Clearly, ifε 0, we have finished. Suppose, on the contrary,ε > 0. Thus, there exists k∈ {1,2, . . .}such that

ε≤tnPAn, Bn< εδε for any n≥k. 1.29

In particular, fornk, we have

ε≤tkPAk, Bk< εδε, 1.30

that is equal to

ε≤Δk< εδε. 1.31

It follows from1.19and the hypothesisiithat p

F

xk, yk, zk

, Fuk, vk, rk

< ε 1.32

which is equivalent to

pxk1, uk1< ε. 1.33

Moreover, we have

p

yk1, vk1

< ε, pzk1, rk1< ε. 1.34

Combining1.33and1.34, we have

Δ_k1< ε. 1.35

Thus,t_k1PA_k1, B_k1< εwhich is a contradiction with respect to1.29, and soε0.

We conclude that

PAn, Bn P Fⁿ

x, y, z

,Fⁿu, v, r

−→0, asn−→∞. 1.36

Remark 1.15. The previous proposition remains true if, iniii, we change the assumption

∃ x, y, z

,u, v, r∈X³ such thatx < u, y≥v, z < r 1.37

with the following

∃ x, y, z

,u, v, r∈X³ such thatx≤u, y > v, z≤r. 1.38

(8)

2. Existence of Tripled Fixed Point

The following theorem is our first main result.

Theorem 2.1. LetX, p,≤be a partially ordered complete partial metric space. Suppose thatX has the following properties:

aif{xn}is a sequence such thatxn1 > xnfor eachn1,2, . . .andxn → x, thenxn < x for eachn1,2, . . .,

bif{yn}is a sequence such thaty_n1 < yn for eachn1,2, . . .andyn → y, thenyn > y for eachn1,2, . . ..

Assume thatF:X³ → Xsatisfies the following hypotheses:

iFhas the mixed strict monotone property, iiFis a generalizedp-Meir-Keeler type function, iiithere existx0, y0, z0∈Xsuch that

x0< F

x0, y0, z0

, y0≥F

y0, x0, y0

, z0< F

z0, y0, x0,

. 2.1

Then,Fhas a tripled fixed point, that is, there existx, y, z∈Xsuch that F

x, y, z

x, F y, x, y

y, F z, y, x

z. 2.2

Also,px, x py, y pz, z 0.

Proof. Letx0, y0, z0 ∈Xbe as iniii. We construct sequences{xn},{yn}, and{zn}according to1.17.

We claim that, for alln≥2, we have

· · ·> xn> xn−1>· · ·> x1> x0,

· · ·< yn< yn−1<· · ·< y1≤y0,

· · ·> zn> zn−1>· · ·> z1> z0.

2.3

Indeed, we will use a mathematical induction to prove2.3. Clearly, we have x0< F

x0, y0, z0

x1, y0≥F

y0, x0, y0

y1, z0< F

z0, y0, x0

z1. 2.4

Suppose now that the inequalities in2.3hold for somen≥2. By the mixed strict monotone property ofF, together with1.17, we have

xnF

x_n−1, y_n−1, z_n−1

< F

xn, yn, zn

x_n1, ynF

yn−1, xn−1, yn−1

> F

yn, xn, yn

yn1, znF

zn−1, yn−1, xn−1

< F

zn, yn, xn

zn1.

2.5

Thus,2.3holds for alln≥2.

(9)

Puttingx, y, z A0andu, v, r A1and byProposition 1.14, we get

P

FⁿA0,FⁿA1

−→0, asn−→∞, 2.6

which is equivalent to

PAn, A_n1−→0, asn−→∞. 2.7

Take an arbitraryε >0. It follows from2.7that there existsk∈Nsuch that

PAk, A_k1< δε. 2.8

Without loss of the generality, assume thatδε≤εand define the following set:

Π:

A

x, y, z

∈X³:P

FA k,FA

< εδε, x > xk, y≤yk, z > zk

. 2.9

We claim that

FA ∈Π ∀A∈Π. 2.10

TakeA∈Π. Then, by2.8and the triangle inequalitywhich still holds for partial metrics, we have

P

Ak,FA

max p

xk, F

x, y, z , p

yk, F

y, x, y , p

zk, F z, y, x

≤max

pxk, xk1 p

xk1, F

x, y, z , p

yk, yk1 p

y_k1, F

y, x, y

, pzk, z_k1 p z_k1, F

z, y, x max

pxk, x_k1 p F

xk, yk, zk

, F

x, y, z , p

yk, y_k1 p

F

yk, xk, yk

, F

y, x, y

, pzk, zk1 p F

zk, yk, xk

, F z, y, x

≤PAk, Ak1 P

FA k,FA

< δε P

FA k,FA .

2.11

We consider the following two cases.

Case 1PAk, A≤ε. ByRemark 1.12and the definition ofΠ, the inequality2.11turns into

P

Ak,FA

< δε P

FA k,FA

< δε PAk, A

< δε ε.

2.12

(10)

Case 2ε < PAk, A< δε ε. That is,

ε <max

px, xk, p y, yk

, pz, zk < δε ε. 2.13

Sincex > xk, z > zk, y≤yk, then, byii, we have

p F

x, y, z , F

xk, yk, zk

< ε, p

F y, x, y

, F

yk, xk, yk

< ε, p

F z, y, x

, F

zk, yk, xk

< ε.

2.14

Hence, combining2.14and2.11, we get

P

Ak,FA

< δε ε. 2.15

On the other hand, usingi, one can easily check that

F x, y, z

> xk, F y, x, y

≤yk, F z, y, x

> zk. 2.16

Hence, we conclude that2.10holds. By2.8, we have thatAk1 ∈Π, and so, by2.10we get

Ak1∈Π ⇒FA k1 Ak2∈Π ⇒FA _k2 A_k3∈Π

· · ·

⇒An∈Π ∀n > k.

2.17

Then, for alln, m > k, we have

PAn, Am≤PAn, Ak PAk, Am<2εδε≤4ε. 2.18

By definition ofP, we have

n,mlim→ ∞pxn, xm lim

n,m→ ∞p yn, ym

lim

n,m→ ∞pzn, zm 0. 2.19

Consequently, by definition of the metricdp,dpx, y≤2px, y, so we get

n,m→ ∞lim dpxn, xm lim

n,m→ ∞dp

yn, ym

lim

n,m→ ∞dpzn, zm 0. 2.20

(11)

Therefore,{xn},{yn}, and{zn}are Cauchy sequences in the metric spaceX, dp. SinceX, p is a complete partial metric space, then, byLemma 1.2,X, dpis also a complete metric space.

Hence, there exists a pointx, y, z∈X³such that

dpxn, x, dp

yn, y

, dpzn, z−→0 asn−→∞. 2.21

Again, byLemma 1.2and2.19, we obtain px, x lim

n→ ∞pxn, x lim

n,m→ ∞pxn, xm 0, p

y, y lim

n→ ∞p yn, y

lim

n,m→ ∞p yn, ym

0, pz, z lim

n→ ∞pzn, z lim

n,m→ ∞pzn, zm 0.

2.22

We will prove that

F x, y, z

x, F y, x, y

y, F z, y, x

z. 2.23

To this aim, take an arbitraryε >0. Since xnFⁿ

x0, y0, z0

−→x, ynFⁿ

y0, x0, y0

−→y, znFⁿ

z0, y0, x0

−→z, 2.24

then there existn1, n2, n3 ∈Nsuch that by2.22 pxl, x p

F^l

x0, y0, z0

, x

< px, x εε, p

yq, y p

F^q

y0, x0, y0

, y

< p y, y

εε, pzr, z p

F^r

z0, y0, x0

, z

< pz, z εε,

2.25

for alll≥n1,q≥n2,r ≥n3. Now, takingnmax{n1, n2, n3}and usingRemark 1.12with the assumption

xnFⁿ

x0, y0, z0

< x, ynFⁿ

y0, x0, y0

> y, znFⁿ

z0, y0, x0

< z, 2.26

by2.25, we get p

x, F

x, y, z

≤px, x_n1 p

x_n1, F

x, y, z

px, x_n1 p

Fⁿ¹

x0, y0, z0

, F

x, y, z px, xn1 p

F

xn, yn, zn

, F

x, y, z

< px, xn1 max

pxn, x, p yn, y

, pzn, z

<2ε.

2.27

(12)

Analogously, we get that

p y, F

y, x, y

<2ε, p z, F

z, y, x

<2ε, 2.28

which yield that

F x, y, z

x, F y, x, y

y, F z, y, x

z. 2.29

Remark 2.2. Theorem 2.1remains true if we replaceivwith one of the following statements.

There existx0, y0, z0∈Xsuch that

1

⎧⎪

⎪⎨

⎪⎪

⎩ x0≤F

x0, y0, z0

, y0 > F

y0, x0, y0

, z0< F

z0, y0, x0

,

2

⎧⎪

⎪⎨

⎪⎪

⎩ x0≤F

x0, y0, z0

, y0 > F

y0, x0, y0

, z0≤F

z0, y0, x0

,

3

⎧⎪

⎪⎨

⎪⎪

⎩ x0≤F

x0, y0, z0

, y0 ≥F

y0, x0, y0

, z0< F

z0, y0, x0

,

4

⎧⎪

⎪⎨

⎪⎪

⎩ x0< F

x0, y0, z0

, y0 ≥F

y0, x0, y0

, z0≤F

z0, y0, x0

,

5

⎧⎪

⎪⎨

⎪⎪

⎩ x0< F

x0, y0, z0

, y0 > F

y0, x0, y0

, z0≤F

z0, y0, x0

.

2.30

3. Uniqueness of Tripled Fixed Point

In this section, we will prove the uniqueness of the tripled fixed point.

Theorem 3.1. In addition to hypotheses ofTheorem 2.1, assume that, for allx, y, z,u, v, r∈X³, there existsa, b, c∈ X³that is comparable tox, y, zandu, v, r. Then,F has a unique tripled fixed point.

Proof. The set of tripled fixed points ofFis not empty due toTheorem 2.1. We suppose that A x, y, z, A^∗ x^∗, y^∗, z^∗ ∈ X³ are two tripled fixed points of F. We distinguish the following two cases.

(13)

Case 1. x, y, zis comparable tox^∗, y^∗, z^∗with respect to the ordering inX³, where

nlim→∞p Fⁿ

x0, y0, z0

, x

px, x 0,

nlim→∞P Fⁿ

y0, x0, y0

, y p

y, y 0,

n→lim∞p Fⁿ

z0, y0, x0

, z

pz, z 0.

3.1

Without loss of the generality, we may assume that

xF x, y, z

< F

x^∗, y^∗, z^∗ x^∗, yF

y, x, y

≥F

y^∗, x^∗, y^∗ y^∗, zF

z, y, x

< F

z^∗, y^∗, x^∗ z^∗.

3.2

By this, definition ofP,Lemma 1.3, andRemark 1.12, we have

0< PA, A^∗ P x, y, z

,

x^∗, y^∗, z^∗ max

px, x^∗, p y, y^∗

, pz, z^∗ max

p F

x, y, z , F

x^∗, y^∗, z^∗ , p

F y, x, y

, F

y^∗, x^∗, y^∗ , p

F z, y, x

, F

z^∗, y^∗, x^∗

<max

px, x^∗, p y, y^∗

, pz, z^∗ PA, A^∗,

3.3

which is a contradiction and therefore must beAA^∗.

Case 2. x, y, zis not comparable tox^∗, y^∗, z^∗. By assumption, there existsB a, b, c∈X³ which is comparable to bothAandA^∗. Without loss of the generality, we may assume that

xF x, y, z

< a, F

x^∗, y^∗, z^∗

x^∗< a, yF

y, x, y

≥b, F

y^∗, x^∗, y^∗

y^∗≥b, zF

z, y, x

< c, F

z^∗, y^∗, x^∗

z^∗< c.

3.4

FromProposition 1.14and3.4, we have

nlim→∞P

FⁿA,FⁿB 0,

nlim→∞P

FⁿA^∗,FⁿB

0. 3.5

(14)

By triangle inequality, we derive

PA, A^∗ lim

n→∞P

FⁿA,FⁿA^∗

≤ lim

n→∞P

FⁿA,FⁿB lim

n→∞P

FⁿB,FⁿA^∗

0. 3.6

ByLemma 1.3, we getAA^∗.

4. Results of Integral Type

Motivated by Suzuki 35and on the same lines of 31, Theorem 3.1, one can prove the following result.

Theorem 4.1. LetX, p,≤be a partially ordered complete partial metric space, and letF:X³ → X be a given mapping. Assume that there exists a function θ from 0,∞into itself satisfying the following:

Iθ0 0 andθt>0 for everyt >0, IIθis nondecreasing and right continuous, IIIfor everyε >0, there existsδε>0 such that ε≤θ

max

px, u, p y, v

, pz, r < εδε ⇒θ p

F x, y, z

, Fu, v, r

< ε, 4.1

for allx≥u, y≤vandz≥r.

Then,Fis a generalizedp-Meir-Keeler type function.

The following result is an immediate consequence of Theorems2.1and4.1.

Corollary 4.2. LetX, p,≤be a partially ordered complete partial metric spaceF :X³ → X be a mapping satisfying the following hypotheses:

iFhas the mixed strict monotone property, iifor everyε >0, there existsδε>0 such that

ε≤

max{px,u,py,v,pz,r}

0

φtdt < εδε ⇒

pFx,y,z,Fu,v,r

0

φtdt < ε, 4.2

for allx ≥ u, y ≤ v andz ≥ r, whereφ : 0,∞ → 0,∞is a locally integrable function satisfying_s

0φtdt >0 for alls >0, iiithere existx0, y0, z0∈Xsuch that

x0< F

x0, y0, z0

, y0≥F

y0, x0, y0

, z0< F

z0, y0, x0

. 4.3

Assume that the hypotheses (a) and (b) given inTheorem 2.1 hold. Then,F has a tripled fixed point.

(15)

To end this paper, we give the following corollary.

Corollary 4.3. LetX, d,≤be a partially ordered complete partial metric spaceF : X³ → X be a mapping satisfying the following hypotheses:

iFhas the mixed strict monotone property, iifor all,x≥u, y≤vandz≥r,

pFx,y,z,Fu,v,r

0

φtdt≤k

max{px,u,py,v,pz,r}

0

φtdt, 4.4

wherek ∈0,1andφis a locally integrable function from0,∞into itself satisfying _s

0φtdt >0 for alls >0,

iiithere existx0, y0, z0∈Xsuch that

x0 < F

x0, y0, z0

, y0≥F

y0, x0, y0

, z0< F

z0, y0, x0

. 4.5

Assume that the hypotheses (a) and (b) of Theorem 2.1 hold. Then,F has a tripled fixed point.

Proof. For allε >0, we takeδε 1/k−1εand we applyCorollary 4.2.

Remark 4.4. By taking φt 1, we retrieve the analogous of Theorem 1.8of Berinde and Borcut on ordered partial metric spaceswitha b c k/3. In fact, assume that1.8 holds forabck/3, that is,

p F

x, y, z

, Fu, v, w

≤ k 3

px, u p y, v

pz, w

4.6

for allx≥u, y≤v, z≥w. From this inequality, we get that p

F x, y, z

, Fu, v, w

≤kmax

px, u, p y, v

, pz, w , 4.7

which corresponds to4.4withφt 1. Then, we may applyCorollary 4.3.

References

1 S. G. Matthews, “Partial metric topology,” in Papers on General Topology and Applications (Flushing, NY, 1992), vol. 728, pp. 183–197, Annals of the New York Academy of Sciences, New York, NY, USA, 1994.

2 R. D. Kopperman, S. G. Matthews, and H. Pajoohesh, “What do partial metrics represent?” in Proceedings of the 19th Summer Conference on Topology and Its Applications, University of CapeTown, 2004.

3 H. P. A. K ¨unzi, H. Pajoohesh, and M. P. Schellekens, “Partial quasi-metrics,” Theoretical Computer Science, vol. 365, no. 3, pp. 237–246, 2006.

4 T. Abdeljawad, E. Karapnar, and K. Tas¸, “Existence and uniqueness of a common fixed point on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1900–1904, 2011.

5 I. Altun and A. Erduran, “Fixed point theorems for monotone mappings on partial metric spaces,”

Fixed Point Theory and Applications, vol. 2011, Article ID 508730, 10 pages, 2011.

(16)

6 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.

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

8 H. Aydi, “Some fixed point results in ordered partial metric spaces,” Journal of Nonlinear Science, vol.

4, no. 3, pp. 210–217, 2011.

9 H. Aydi, “Fixed point results for weakly contractive mappings in ordered partial metric spaces,”

Journal of Advanced Mathematical Studies, vol. 4, no. 2, pp. 1–12, 2011.

10 H. Aydi, “Fixed point theorems for generalized weakly contractive condition in ordered partial metric spaces,” Journal of Nonlinear Analysis and Optimization, vol. 2, no. 2, pp. 33–48, 2011.

11 H. Aydi, “Common fixed point results for mappings satisfyingψ,φ-weak contractions in ordered partial metric spaces,” International Journal of Mathematics and Statistics, vol. 12, no. 2, pp. 53–64, 2012.

12 H. Aydi, E. Karapınar, and W. Shatanawi, “Coupled fixed point results forψ,φ-weakly contractive condition in ordered partial metric spaces,” Computers & Mathematics with Applications, vol. 62, pp.

4449–4460, 2011.

13 L. J. ´Ciri´c, B. Samet, H. Aydi, and C. Vetro, “Common fixed points of generalized contractions on partial metric spaces and an application,” Applied Mathematics and Computation, vol. 218, no. 6, pp.

2398–2406, 2011.

14 E. Karapinar, “Weakφ-contraction on partial metric spaces,” Journal of Computational Analysis and Applications, vol. 14, no. 2, pp. 206–210, 2012.

15 E. Karapnar and I. M. Erhan, “Fixed point theorems for operators on partial metric spaces,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1894–1899, 2011.

16 E. Karapinar, “Generalizations of Caristi Kirk’s theorem on partial metric spaces,” Fixed Point Theory and Applications, vol. 2011, article 4, 2011.

17 E. Karapınar and U. Yuksel, “Some common fixed point theorems in partial metric spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 263621, 17 pages, 2011.

18 E. Karapınar, “A note on common fixed point theorems in partial metric spaces,” Miskolc Mathematical Notes, vol. 12, no. 2, pp. 185–191, 2011.

19 S. J. O’Neill, “Two topologies are better than one,” Tech. Rep., Univesity of Warwick, 1995.

20 W. Shatanawi, B. Samet, and M. Abbas, “Coupled fixed point theorems for mixed monotone mappings in ordered partial metric spaces,” Mathematical and Computer Modelling, vol. 55, pp. 680–

687, 2012.

21 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.

22 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.

23 H. Aydi, “Coincidence and common fixed point results for contraction type maps in partially ordered metric spaces,” International Journal of Mathematical Analysis, vol. 5, no. 2, pp. 631–642, 2011.

24 H. Aydi, M. Postolache, and W. Shatanawi, “Coupled fixed point results forψ,φ-weakly contractive mappings in ordered G-metric spaces,” Computers & Mathematics with Applications, vol. 63, pp. 298–

309, 2012.

25 H. Aydi, B. Damjanovi´c, B. Samet, and W. Shatanawi, “Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces,” Mathematical and Computer Modelling, vol. 54, pp.

2443–2450, 2011.

26 T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis, vol. 65, no. 7, pp. 1379–1393, 2006.

27 V. Lakshmikantham and L. ´Ciri´c, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis, vol. 70, no. 12, pp. 4341–4349, 2009.

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

29 H. K. Nashine and B. Samet, “Fixed point results for mappings satisfyingψ,φ-weakly contractive condition in partially ordered metric spaces,” Nonlinear Analysis, vol. 74, no. 6, pp. 2201–2209, 2011.

30 J. J. Nieto and R. R. L ´opez, “Contractive mapping theorems in partially ordered sets and applications to ordinary diﬀerential equations,” Order. A Journal on the Theory of Ordered Sets and its Applications, vol. 22, no. 3, pp. 223–239, 2005.

31 B. Samet, “Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces,” Nonlinear Analysis, vol. 72, no. 12, pp. 4508–4517, 2010.

(17)

32 V. Berinde and M. Borcut, “Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces,” Nonlinear Analysis, vol. 74, no. 15, pp. 4889–4897, 2011.

33 K. P. R. Rao and G. N. V. Kishore, “A unique common triple fixed point theorem in partially ordered cone metric spaces,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 4, pp. 213–222, 2011.

34 H. Aydi, E. Karapinar, and C. Vetro, “Meir-Keeler type contractions for tripled fixed points,” preprint.

35 T. Suzuki, “Meir-Keeler contractions of integral type are still Meir-Keeler contractions,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 39281, 6 pages, 2007.

(18)

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}

New Meir-Keeler Type Tripled Fixed-Point Theorems on Ordered Partial Metric Spaces

Research Article