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Volume 2010, Article ID 134897, 11 pagesdoi:10.1155/2010/134897 Research Article Coupled Coincidence Point and Coupled Common Fixed Point Theorems in Partially Ordered Metric 1 Departmen

Volume 2010, Article ID 134897, 11 pages

doi:10.1155/2010/134897

Research Article

Coupled Coincidence Point and Coupled Common Fixed Point Theorems in Partially Ordered Metric

1 Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, Pakistan

2 Department of Mathematics, Faculty of Sciences and Mathematics, University of Niˆs, Viˆsegradska 33,

18000 Niˆs, Serbia

Correspondence should be addressed to Dejan Ili´c,ilicde@ptt.rs

Received 7 April 2010; Accepted 18 October 2010

Academic Editor: Hichem Ben-El-Mechaiekh

Copyrightq 2010 Mujahid Abbas 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 the concept of a w-compatible mapping to obtain a coupled coincidence point and a

coupled point of coincidence for nonlinear contractive mappings in partially ordered metric spaces

equipped with w-distances Related coupled common fixed point theorems for such mappings are

also proved Our results generalize, extend, and unify several well-known comparable results in the literature

1 Introduction and Preliminaries

In 1996, Kada et al.1 introduced the notion of w-distance They elaborated, with the help

of examples, that the concept of w-distance is general than that of metric on a nonempty set.

They also proved a generalization of Caristi fixed point theorem employing the definition of

w-distance on a complete metric space Recently, Ili´c and Rakoˇcevi´c2 obtained fixed point

and common fixed point theorems in terms of w-distance on complete metric spacessee also

3 9

Definition 1.1 Let X, d be a metric space A mapping p : X × X → 0, ∞ is called a w-distance on X if the following are satisfied:

w1 px, z ≤ px, y  py, z for all x, y, z ∈ X,

w2 for any x ∈ X,px, · : X → 0, ∞ is lower semicontinuous,

w3 for any ε > 0 there exists δε > 0 such that pz, x ≤ δ and pz, y ≤ δ imply

p x, y ≤ ε, for any x, y, z ∈ X.

The metric d is a w-distance on X For more examples of w-distances, we refer to10

Definition 1.2 Let X be a nonempty set with a w-distance on X Ones denotes the w-closure

of a subset B of X by cl ω B which is defined as

clω B x ∈ X : px n , x  −→ 0 for some sequence {x n } in B∪ B. 1.1 The next Lemma is crucial in the proof of our results

Lemma 1.3 see 1 Let X, d be a metric space, and let p be a w-distance on X Let {xn } and {y n } be sequences in X, let α n and β n be sequences in 0, ∞ converging to 0, and let x, y, z ∈ X.

Then the following hold.

1 If px n , y  ≤ α n and p x n , z  ≤ β n for any n ∈ N, then y  z In particular, if px, y 

0, px, z  0 then y  z.

2 If px n , y n  ≤ α n and p x n , z  ≤ β n for any n ∈ N, then y n converges to z.

3 If px n , x m  ≤ α n for any m, n ∈ N with n ≺ m, then x n is a Cauchy sequence.

4 If py, x n  ≤ α n for any n ∈ N, then x n is a Cauchy sequence.

Bhaskar and Lakshmikantham in11 introduced the concept of coupled fixed point

of a mapping F : X ×X → X and investigated some coupled fixed point theorems in partially

ordered sets They also discussed an application of their result by investigating the existence and uniqueness of solution for a periodic boundary value problem Sabetghadam et al in12 introduced this concept in cone metric spaces They investigated some coupled fixed point theorems in cone metric spaces Recently, Lakshmikantham and ´Ciri´c13 proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings

in partially ordered complete metric spaces which extend the coupled fixed point theorem given in11 The following are some other definitions needed in the sequel

Definition 1.4see 12 Let X be any nonempty set Let F : X × X → X and g : X → X be two mappings An ordered pairx, y ∈ X × X is called

1 a coupled fixed point of a mapping F : X × X → X if x  Fx, y and y  Fy, x,

2 a coupled coincidence point of hybrid pair {F, g} if gx  Fx, y and gy 

F y, x and gx, gy is called coupled point of coincidence,

3 a common coupled fixed point of hybrid pair {F, g} if x  gx  Fx, y and

y  gy  Fy, x.

Note that ifx, y is a coupled fixed point of F, then y, x is also a coupled fixed point of the mapping F.

Definition 1.5 Let X be any nonempty set Mappings F : X × X → X and g : X → X are called w-compatible if gFx, y  Fgx, gy whenever gx  Fx, y and gy  Fy, x.

Definition 1.6 Let X, d be a metric space with w-distance p A mapping F : X × X → X

is said to be w-continuous at a point x, y ∈ X × X with respect to mapping g : X → X

if for every ε > 0 there exists a δε > 0 such that pgu, gx  pgv, gy < δ implies that

p Fx, y, Fu, v < ε for all u, v ∈ X.

Definition 1.7 Let X be a partially ordered set Mapping g : X → X is called strictly

monotone increasing mapping if

x  y ⇐⇒ gx  gy or equivalentlyx  y ⇐⇒ gx  gy. 1.2

Definition 1.8 Let X be a partially ordered set A mapping F : X × X → X is said to be a mixed monotone if Fx, y is monotone nondecreasing in x and monotone nonincreasing in

y, that is, for any x, y ∈ X,

x1, x2∈ X, x1 x2⇒ Fx1, y

 Fx2, y

,

y1, y2∈ X, y1 y2 ⇒ Fx, y1



 Fx, y2



Kada et al.1 gave an example to show that p is not symmetric in general We denote by

M X and M1X, respectively, the class of all distances on X and the class of all w-distances on X which are symmetric for comparable elements in X Also in the sequel, we

will consider thatx, y and u, v are comparable with respect to ordering in X × X if x  u and y  v.

2 Coupled Coincidence Point

In this section, we prove coincidence point results in the frame work of partially ordered

metric spaces in terms of a w-distance.

Theorem 2.1 Let X, d be a partially ordered metric space with a w-distance p and g : X → X a

strictly monotone increasing mapping Suppose that a mixed monotone mapping F : X × X → X is

w-continuous with respect to g such that

p

F

x, y

, F u, v≤ a1p

gu, gx

 a2p

gv, gy

for all x, y, u, v ∈ X with x  u, y  v or x  u, y  v and a1 a2< 1 Let F X × X ⊆ gX and

p y, x  0 whenever px, y  0, for some x, y ∈ cl ω FX × X If gX is complete and there exist

x0, y0∈ X such that gx0 Fx0, y0 and Fy0, x0  gy0, then F and g have a coupled coincidence point.

Proof Let gx1  Fx0, y0 and gy1  Fy0, x0 for some x1, y1 ∈ X; this can be done since

F X × X ⊆ gX Following the same arguments, we obtain gx2  Fx1, y1 and gy2 

F y1, x1 Put

F1

x0, y0



 gx1, F2

x0, y0



 Fx1, y1



 gx2,

F2

y0, x0



 Fy1, x1



 gy2.

2.2

Similarly for all n ∈ N,

gx n1 F n1

x0, y0

, gy n1 F n1

y0, x0

Since g is strictly monotone increasing and F has the mixed monotone property, we have

gx2 F2

x0, y0

 Fx1, y1

 Fx0, y0

 gx1, gy2 gy1. 2.4 Similarly

gx0  Fx0, y0

 gx1 F2

x0, y0

 gx2 · · ·

 F n1

x0, y0



 gx n1 · · · ,

gy0  Fy0, x0

 gy1 F2

y0, x0

 gy2 · · ·

 F n1

y0, x0



 · · ·

2.5

Now for all n≥ 2, using 2.1, we get

p

F n

x0, y0

, F n1

x0, y0

 pF

x n−1, y n−1

, F

x n , y n

≤ a1p

gx n , gx n−1

 a2p

gy n , gy n−1

 a1



p

F n

x0, y0

, F n−1

x0, y0

 a2



p

F n

y0, x0

, F n−1

y0, x0

,

p

F n

y0, x0

, F n1

y0, x0

≤ a1



p

F n

y0, x0

, F n−1

y0, x0

 a2



p

F n

x0, y0

, F n−1

x0, y0

.

2.6

From2.6,

p

F n

x0, y0



, F n1

x0, y0



 pF n

y0, x0



, F n1

y0, x0



≤ hp

F n

x0, y0



, F n−1

x0, y0



 pF n

y0, x0



, F n−1

y0, x0



,

2.7

where h  a1 a2 Continuing, we conclude that

p

F n

x0, y0

, F n1

x0, y0

 pF n

y0, x0

, F n1

y0, x0

≤ h n

p

gx1, gx0



 pgy1, gy0



 h n δ1

2.8

if n is odd, where δ1 pgx1, gx0  pgy1, gy0 Also,

p

F n

x0, y0



, F n1

x0, y0



 pF n

y0, x0



, F n1

y0, x0



≤ h n

p

gx0, gx1

 pgy0, gy1

 h n δ2

2.9

if n is even, where

δ2 pgx0, gx1

 pgy0, gy1

Let δ n  pF n x0, y0, F n1x0, y0  pF n y0, x0, F n1y0, x0; then for every n in N we have

δ n ≤ h n δ0, 2.11 where

δ0 max{δ1, δ2}. 2.12 Hence,

p

F n

x0, y0

, F n1

x0, y0

−→ 0, pF n

y0, x0

, F n1

y0, x0

−→ 0 as n −→ ∞. 2.13

For m > n, we get

p

F n

x0, y0

, F m

x0, y0

 pF n

y0, x0

, F m

y0, x0

≤ pF n

x0, y0

, F n1

x0, y0

 pF n1

x0, y0

, F n2

x0, y0

 · · ·

 pF m−1

x0, y0

, F m

x0, y0

 pF n

y0, x0

, F n1

y0, x0

 pF n1

y0, x0

, F n2

y0, x0

 · · ·

 pF m−1

y0, x0

, F m

y0, x0

 δ n  δ n1 · · ·  δ m−1≤ h n δ0 h n1δ0 · · ·  h m−1δ0≤ h n

1− h δ0

2.14

which further implies that

p

F n

x0, y0



, F m

x0, y0



≤ h n

1− h δ0

p

F n

y0, x0



, F m

y0, x0



≤ h n

1− h δ0.

2.15

Lemma 1.33 implies that {Fn x0, y0}  {gx n } and {F n y0, x0}  {gy n} are Cauchy

sequences in gX Since gX is complete, there exist x, y ∈ X such that gx n → gx and

gy n → gy Since pgx n ,· is lower semicontinuous, we have

p

F n

x0, y0

, gx

≤ lim inf

m→ ∞ p

gx n , gx m

≤ h n

1− h δ0 2.16 which implies that

p

F n

x0, y0



, gx

−→ 0 as n −→ ∞. 2.17 Similarly

p

F n

y0, x0



, gy

−→ 0 as n −→ ∞. 2.18

Let ε > 0 be given Since F is w-continuous at x, y with respect to g, there exists δ > 0 such that for each n

p

gx n , gx

 pgy n , gy

< δ implies that p

F

x, y

, F

x n , y n



< ε

2. 2.19

Since pgx n , gx  → 0 and pgy n , gy  → 0, for γ  minε/2, δ/2, there exists n0such that,

for all n ≥ n0,

p

gx n , gx

< γ, p

gy n , gy

Now,

p

F

x, y

, gx

≤ pF

x, y

, F n0 1

x0, y0



 pF n0 1

x0, y0



, gx

 pF

x, y

, F

x n0, y n0



 pgx n0 1, gx

< ε

2 γ  ε

2.21

implies that pFx, y, gx  0 Since

p

F n

x0, y0



, F

x, y

≤ pF n

x0, y0



, gx

 pgx, F

x, y

≤ h n

1− h δ0,

2.22

usingLemma 1.31, we obtain Fx, y  gx Similarly, we can prove that Fy, x  gy Hence

x, y is coupled coincidence point of F and g.

Theorem 2.2 Let X, d be a partially ordered metric space with a w-distance p having the following

properties.

1 If {x n } is in X with x n  x n1for all n and x n → x for some x ∈ X, then x n  x for all

n.

2 If {y n } is in X with y n1 y n for all n and y n → y for some y ∈ X, then y  y n for all n.

Let F : X × X → X be a mixed monotone and g : X → X a strict monotone increasing

mapping such that

p

F

x, y

, F u, v≤ a1p

gu, gx

 a2p

gv, gy

, 2.23

for all x, y, u, v ∈ X with x  u, y  v or x  u, y  v and a1 a2< 1 Let F X × X ⊆ gX and py, x  0 whenever px, y  0, for some x, y ∈ cl ω FX × X If gX is complete and there exist x0, y0 ∈ X such that gx0  Fx0, y0 and Fy0, x0  gy0, then F and g have a

coupled coincidence point

Proof Construct two sequences {gx n }  {F n x0, y0} and {gy n }  {F n y0, x0} such that

gx n  gx n1and gy n  gy n1for all n and gx n → gx and gy n → gy for some x ∈ X, as

given in the proof ofTheorem 2.1 Now, we need to show that Fx, y  gx and Fy, x  gy

Let ε > 0 Since pF n x0, y0, gx → 0 and pF n y0, x0, gy → 0, there exists n1 ∈ N such that, for all n ≥ n1, we have

p

F n

x0, y0

, gx

< ε

3, p



F n

y0, x0

, gy

< ε

3. 2.24 Consider

p

F

x, y

, gx

≤ pF

x, y

, F n1

x0, y0



 pF n1

x0, y0



, gx

 pF

x, y

, F

x n , y n

 pF n1

x0, y0

, gx

≤ a1p

gx n , gx

 a2p

gy n , gy

 pF n1

x0, y0

, gx

 a1p

F n

x0, y0

, gx

 a2p

F n

y0, x0

, gy

 pF n1

x0, y0

, gx

< a1ε

3  a2ε

3 ε 3

< ε,

2.25

which implies that pFx, y, gx  0 Also, fromTheorem 2.1, we have

p

F n

x0, y0



, gx

≤ h n

1− h δ0. 2.26

p

F n

x0, y0



, F

x, y

≤ pF n

x0, y0

, gx

 pgx, F

x, y

≤ h n

1− h δ0

2.27

implies that gx  Fx, y Similarly, we can prove that Fy, x  gy Hence x, y is coupled coincidence point of F and g.

3 Coupled Common Fixed Point

In this section, using the concept of w-compatible maps, we obtain a unique coupled common

fixed point of two mappings

Theorem 3.1 Let all the hypotheses of Theorem 2.1 (resp., Theorem 2.2 ) hold with a1 a2< 1/2 If for every x, y, x∗, y∗ ∈ X × X there exists u, v ∈ X × X that is comparable to x, y and x∗, y∗

with respect to ordering in X × X, then there exists a unique coupled point of coincidence of F and g.

Moreover if F and g are w-compatible, then F and g have a unique coupled common fixed point Proof Let gx∗, gy∗ be another coupled coincidence point of F and g We will discuss the

following two cases

Case 1 If x, y is comparable to x∗, y∗ with respect to ordering in X × X, then

p

gx, gx∗

 pgy, gy∗

 pF

x, y

, F

x∗, y∗

 pF

y, x

, F

y∗, x∗

≤ a1p

gx∗, gx

 a2p

gy∗, gy

 a1p

gy∗, gy

 a2p

gx∗, gx

≤ a1 a2 p

gx, gx∗

 pgy, gy∗

3.1

implies that pgx, gx∗  pgy, gy∗  0 Hence pgx, gx∗  0  pgy, gy∗ Also,

p

gx, gx

 pgy, gy

 pFx, x, Fx, x  pF

y, y

, F

y, y

≤ 2a1p

gx, gx

 2a2p

gives that pgx, gx  0  pgy, gy The result follows usingLemma 1.31

Case 2 If x, y is not comparable to x∗, y∗, then there exists an upper bound or lower bound u, v of x, y, x∗, y∗ Again since g is strictly monotone increasing mapping and

F satisfies mixed monotone property, therefore, for all n  0, 1, ,F n u, v, F n v, u is

comparable toF n x, y, F n y, x  gx, gy and F n y, x, F n x, y  gy, gx Following

similar arguments to those given in the proof ofTheorem 2.1, we obtain

p

gx, gx∗

 pgy, gy∗

 pF n

x, y

, F n

x∗, y∗

 pF n

y, x

, F n

y∗, x∗

≤ p

F n

x, y

, F n u, v pF n u, v, F n

x∗, y∗

 p

F n

y, x

, F n v, u pF n v, u, F n

y∗, x∗

 p

F n

x, y

, F n u, v pF n

y, x

, F n v, u

 p

F n u, v, F n

x∗, y∗

 pF n v, u, F n

y∗, x∗

≤ h n β0 h n γ0,

3.3

where β0  max{pgu, gx  pgv, gy, pgx, gu  pgy, gv} and γ0  max{pgx∗, gu 

p gy∗, gv , pgu, gx∗  pgv, gy∗} On taking limit as n → ∞ on both sides of 3.3, we

have

p

gx, gx∗

 pgy, gy∗

and pgx, gx∗  0  pgy, gy∗ By the same lines as in Case1, we prove that pgx, gx 

0  pgy, gy AgainLemma 1.31 implies that gx  gx∗and gy  gy∗ Hencegx, gy is unique coupled point of coincidence of F and g Note that if gx, gy is a coupled point of coincidence of F and g, then gy, gx are also a coupled points of coincidence of F and g Then gx  gy and therefore gx, gx is unique coupled point of coincidence of F and g Let

u  gx Since F and g are w-compatible, we obtain

gu  ggx

 gFx, x  Fgx, gx

 Fu, u. 3.5

Consequently gu  gx Therefore u  gu  Fu, u Hence u, u is a coupled common fixed point of F and g.

Remark 3.2 If in addition to the hypothesis ofTheorem 2.1resp.,Theorem 2.2 we suppose

that p ∈ M1X, x0and y0are comparable, then gx  gy.

Proof Recall that gx0  Fx0, y0 Now, if x0  y0, then gx0  gy0 We claim that, for all

n ∈ N, gx n  gy n Since g is strictly monotone increasing and F satisfies mixed monotone

property, we have

gx1 Fx0, y0

 Fy0, x0

 gy1. 3.6

Assuming that gx n  gy n , since g is strictly monotone increasing, so x n  y n By the mixed

monotone property of F, we have

gx n1 F n1

x0, y0



 Fx n , y n



 Fy n , x n



 gy n1. 3.7

gx n  gy n ∀n. 3.8

Letting ε > 0, there exists an n0∈ N such that pgx, F n x0, y0 < ε/4 and pF n y0, x0, gy <

ε/4 for all n ≥ n0 Now,

p

gx, gy

≤ pgx, F n0 1

x0, y0



F n0 1

x0, y0



, gy

≤ pgx, F n0 1

x0, y0



 pF n0 1

x0, y0



, F n0 1

y0, x0



F n0 1

y0, x0



, gy

< ε

4 hpF n0

x0, y0

, F n0

y0, x0

ε 4

≤ ε

2 h p

F n0

x0, y0

, gx

 pgx, gy

gy, F n0

y0, x0

< ε

2 h ε

4  hpgx, gy

 h ε

4

< ε  hpgx, gy

3.9

implies that1 − hpgx, gy < ε Since h < 1, therefore pgx, gy  0 Similarly we can prove that pgx, gx  0 Hence byLemma 1.31, we have gx  gy Similarly, if gx0  gy0, we can

show that gx n  gy n for each n and gx  gy.

Acknowledgment

The present version of the paper owes much to the precise and kind remarks of the learned referees

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