Coupled Coincidence Point Results for

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Volume 2012, Article ID 496103,19pages doi:10.1155/2012/496103

Research Article

Coupled Coincidence Point Results for

ψ, α, β -Weak Contractions in Partially Ordered Metric Spaces

A. Razani and V. Parvaneh

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Correspondence should be addressed to A. Razani,razani@ipm.ir Received 24 June 2012; Accepted 6 August 2012

Academic Editor: Tai-Ping Chang

Copyrightq2012 A. Razani and V. Parvaneh. 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 coupled coincidence points of mappings satisfying a nonlinear contractive condition in the framework of partially ordered metric spaces are obtained. Our results extend the results of Harjani et al.2011. Moreover, an example of the main result is given. Finally, some coupled coincidence point results for mappings satisfying some contraction conditions of integral type in partially ordered complete metric spaces are deduced.

1. Introduction and Mathematical Preliminaries

The existence of fixed points for certain mappings in ordered metric spaces has been studied and applied by Ran and Reurings 1and then by Nieto and Rodr´ıguez-L ´opez2. So far, many researchers have obtained fixed point and common fixed point results for mappings under various contractive conditions in diﬀerent metric spacessee, e.g.,3–8.

Existence of coupled fixed points in partially ordered metric spaces was first investigated in 2006 by Bhaskar and Lakshmikantham9and then by Lakshmikantham and Ciri´c´ 10. Further results in this direction under weak contraction conditions in diﬀerent metric spaces were proved in, for example,4,5,10–15.

Bhaskar and Lakshmikantham9introduced the following definitions.

Definition 1.1see9. LetX,be a partially ordered set andF:X×X → Xbe a self-map.

One can say thatFhas the mixed monotone property ifFx, yis monotone nondecreasing inxand is monotone nonincreasing iny, that is, for allx1, x2∈X,x1x2impliesFx1, y Fx2, yfor anyy ∈ X, and for ally₁, y₂ ∈ X,y₁ y₂ impliesFx, y1 Fx, y2for any x∈X.

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Definition 1.2see9. An elementx, y∈X×Xis called a coupled fixed point of mapping F:X×X → XifxFx, yandyFy, x.

The main results of Bhaskar and Lakshmikantham in9 are the following coupled fixed point theorems.

Theorem 1.3see9. LetX,be a partially ordered set and suppose there exists a metricdonX such thatX, dis a complete metric space. LetF:X×X → Xbe a continuous mapping having the mixed monotone property onX. Assume that there exists ak∈0,1with

d F

x, y

, Fu, v

≤ k 2

dx, u d y, v

, 1.1

for allxuandyv. If there exist two elementsx0, y0 ∈Xwithx0Fx0, y0andy0Fy0, x0, thenFhas a coupled fixed point.

Theorem 1.4see9. LetX,be a partially ordered set and suppose that there is a metricdin Xsuch thatX, dis a complete metric space. Assume thatXhas the following properties:

iif a nondecreasing sequencexn → x, thenxnx, for alln;

iiif a nonincreasing sequenceyn → y, thenyyn, for alln.

LetF:X×X → Xbe a mapping having the mixed monotone property onX.

Assume that there exists ak∈0,1with d

F x, y

, Fu, v

≤ k 2

dx, u d y, v

, 1.2

for allxuandyv.

If there existx₀, y₀ ∈ Xsuch thatx₀ Fx0, y₀andy₀ Fy0, x₀, thenFhas a coupled fixed point.

Recently, Abbas et al.11have introduced the concept ofw-compatible mappings to obtain coupled coincidence point for nonlinear contractive mappings in a cone metric space.

Definition 1.5 see 11. The mappings F : X×X → X and g : X → X are called w- compatible ifgFx, y Fgx, gy, whenevergx Fx, yandgy Fy, x.

Ciri´c et al.´ 3have presented the concepts of a mixedg-monotone mapping, coupled coincidence point, and commutative mapping. They proved some coupled coincidence and coupled common fixed point theorems for mixed g-monotone nonlinear contractive mappings in partially ordered complete metric spaces. The results of Lakshmikantham and Ciri´c are generalizations of Theorems´ 1.3and1.4.

Definition 1.6see10. An elementx, y∈X×Xis called

1a coupled coincidence point of mappings F : X ×X → X and g : X → X if gx Fx, yandgy Fy, x,

2a common coupled fixed point of mappings F : X ×X → X andg : X → X if xgx Fx, yandygy Fy, x.

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Definition 1.7see3. LetX, be a partially ordered set andF : X ×X → X and g : X → X be two self-mappings.F has the mixedg-monotone property ifF is monotoneg- nondecreasing in its first argument and is monotoneg-nonincreasing in its second argument, that is, for allx1, x2 ∈ X,gx1 gx2impliesFx1, y Fx2, yfor anyy ∈ X, and for all y1, y2∈X,gy1 gy2impliesFx, y1Fx, y2for anyx∈X.

Definition 1.8see3. LetXbe a nonempty set. One can say that the mappingsF :X×X → Xandg:X → Xare commutative ifgFx, y Fgx, gy, for allx, y∈X.

Theorem 1.9Corollary 2.13. LetX,be a partially ordered set and suppose there is a metric donXsuch thatX, dis a complete metric space. SupposeF :X×X → Xandg : X → X are such thatFhas the mixedg-monotone property and assume that there exists ak∈0,1with

d F

x, y

, Fu, v

≤ k 2

d gx, gu

d

gy, gv

, 1.3

for allx, y, u, v ∈ X for whichgx guandgy gv. SupposeFX×X ⊆ gX,g is continuous and commutes withFand also suppose either

aFis continuous, or,

bXhas the following properties,

iif a nondecreasing sequencex_n → x, thenx_nxfor alln∈N, iiif a nonincreasing sequencey_n → y, theny_n yfor alln∈N.

If there existx0, y0∈Xsuch thatgx0Fx0, y0andgy0Fy0, x0, then there existx, y∈X such thatgx Fx, yandgy Fy, x, that is,Fandghave a coupled coincidence point.

Harjani et al. 7 obtained the following theorem for mappings with the mixed monotone property.

Theorem 1.10see7. LetX,be a partially ordered set and suppose that there exists a metricd inXsuch thatX, dis a complete metric space. LetF:X×X → Xbe a mapping having the mixed monotone property onXand continuous such that

ψ d

F x, y

, Fu, v

≤ψ max

dx, u, d y, v

−ϕ max

dx, u, d y, v

, 1.4

for allx, y, u, v ∈Xwithx uandy v, whereψ andϕare altering distance functions. If there existx₀, y₀∈Xwithx₀ Fx0, y₀andy₀Fy0, x₀, thenFhas a coupled fixed point.

Also, they proved that the above theorem is still valid forFnot necessarily continuous, assuming the following hypothesis.

If{xn}is a nondecreasing sequence withx_n → x, thenx_nx, for alln∈N.

If{yn}is a nonincreasing sequence withyn → y, thenyny, for alln∈N.

Theorem 1.11 see7. If inTheorem 1.10 one substitutes the continuity of F by the condition mentioned above one also obtains the existence of a coupled fixed point forF.

The aim of this paper is to study necessary conditions for the existence of coupled coincidence and common coupled fixed points of ψ, α, β-weak contractions in ordered

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metric spaces. For more details onψ, α, β-weakly contractive mappings we refer the reader to16.

2. Main Results

The notion of an altering distance function was introduced by Khan et al.17as follows.

Definition 2.1. The functionψ :0,∞ → 0,∞is called an altering distance function, if the following properties are satisfied:

1ψis continuous and nondecreasing, 2ψt 0 if and only ift0.

Now, we establish an existence theorem for coupled coincidence point of mappings satisfyingψ, α, β-weak contraction condition in the setup of partially ordered metric spaces.

Note thatψ, α, β-weak contraction condition was first appeared in16.

Theorem 2.2. LetX,, dbe a partially ordered complete metric space and letF :X² → X and g:X → Xbe such thatFX²⊆gXandFis continuous. Assume that

ψ d

F x, y

, Fu, v

≤α max

d gx, gu

, d

gy, gv

−β max

d gx, gu

, d

gy, gv

, 2.1

for everyx, y, u, v∈Xwithgxguandgygv, whereψ, α, β:0,∞ → 0,∞are such that, ψis an altering distance function,αis continuous,βis lower semicontinuous,α0 β0 0 and ψt−αt βt>0 for allt >0.

Assume that

1Fhas the mixedg-monotone property, 2gis continuous and commutes withF.

If there existx₀, y₀∈Xsuch thatgx₀Fx0, y₀andgy₀ Fy0, x₀, thenFandghave a coupled coincidence point inX.

Proof. Letx0, y0∈Xbe such thatgx0Fx0, y0andgy0Fy0, x0. Definex1, y1 ∈Xsuch thatgx₁Fx0, y₀andgy₁Fy0, x₀and in this way, we construct the sequences{an}and {bn}as follows:

a_ngx_nF

x_n−1, y_n−1 , bngynF

y_n−1, x_n−1

, 2.2

for alln≥0.

We will do the proof in two steps.

Step I. We will show that{an}and{bn}are Cauchy. Let

δnmax{dan−1, an, dbn−1, bn}. 2.3

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Asgx_n−1gxnandgy_n−1gyn, using2.1we obtain that ψdan, a_n1 ψ

d F

x_n−1, y_n−1 , F

xn, yn

≤α max

d

gx_n−1, gx_n , d

gy_n−1, gy_n

−β max

d

gx_n−1, gx_n , d

gy_n−1, gy_n αmax{dan−1, an, dbn−1, bn}

−βmax{da_n−1, a_n, db_n−1, b_n}.

2.4

In a similar way, sincegyngy_n−1andgxn gx_n−1, we have ψdbn1, bn ψ

d F

yn, xn

, F

y_n−1, x_n−1

≤α max

d

gy_n, gy_n−1 , d

gx_n, gx_n−1

−β max

d

gy_n, gy_n−1 , d

gx_n, gx_n−1 αmax{dan−1, an, dbn−1, bn}

−βmax{da_n−1, a_n, db_n−1, b_n}.

2.5

If for ann≥1,δn0, then the conclusion of the theorem follows. So, we assume that

δn/0, 2.6

for alln≥1.

Let, for somen,δ_n−1< δn. So, from2.4and2.5asψis nondecreasing, we have ψmax{dan−1, an, dbn−1, bn}< ψmax{dan, a_n1, dbn, b_n1}

max

ψdan, a_n1, ψdbn, b_n1

≤αmax{dan−1, an, dbn−1, bn}

−βmax{da_n−1, a_n, db_n−1, b_n},

2.7

that is,ψδn−αδnβδn≤0. By our assumptions, we haveδ_n0, which contradicts2.6.

Therefore, for alln≥1 we deduce that

δ_n1≤δn, 2.8

that is,{δn}is a nonincreasing sequence of nonnegative real numbers. Thus, there exists an r≥0 such that lim_n_{→ ∞}δ_nr.

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Takingn → ∞in2.7and using the lower semicontinuity ofβand the continuity of ψandα, we obtainψr≤αr−βr, which further implies thatr0, from our assumptions aboutψ,α, andβ. Therefore,

nlim→ ∞max{da_n−1, a_n, db_n−1, b_n}0. 2.9 Next, we claim that{an}and{bn}are Cauchy.

We will show that for everyε >0, there existsk∈Nsuch that ifm, n≥k,

max{dan, a_m, dbn, b_m}< ε. 2.10 Suppose the above statement is false.

Then, there exists anε >0 for which we can find subsequences{a_mk}and{a_nk}of {an}and{b_mk}and{b_nk}of{bn}such thatnk> mk> kand

max d

a_mk, a_nk , d

b_mk, b_nk

≥ε, 2.11

wherenkis the smallest index with this property, that is, max

d

a_mk, a_nk−1 , d

b_mk, b_nk−1

< ε. 2.12

From triangle inequality, d

a_mk, a_nk

≤d

a_mk, a_nk−1 d

a_nk−1, a_nk

. 2.13

Similarly,

d

b_mk, b_nk

≤d

b_mk, b_nk−1 d

b_nk−1, b_nk

. 2.14

So, max

d

a_mk, a_nk , d

b_mk, b_nk

≤max d

a_mk, a_nk−1 , d

b_mk, b_nk−1 max

d

a_nk−1, a_nk , d

b_nk−1, b_nk

. 2.15

Lettingk → ∞, as limn→ ∞δn0, from2.11and2.12, we conclude that

klim→ ∞max d

a_mk, a_nk , d

b_mk, b_nk

ε. 2.16

Since d

a_nk1, a_mk1

≤d

a_nk1, a_nk d

a_nk, a_mk d

a_mk, a_mk1 , d

b_nk1, b_mk1

≤d

b_nk1, b_nk d

b_nk, b_mk d

a_mk, a_mk1

, 2.17

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we obtain that max

d

a_nk1, a_mk1 , d

b_nk1, b_mk1

≤ max d

a_nk1, a_nk , d

b_nk1, b_nk max

d

a_nk, a_mk , d

b_nk, b_mk max

d

a_mk, a_mk1 , d

b_mk, b_mk1 . 2.18

If in the above inequality,k → ∞, as limn→ ∞δn0, from2.16we have

k→ ∞lim max d

a_nk1, a_mk1 , d

b_nk1, b_mk1

≤ε. 2.19

Again, since d

a_nk, a_mk

≤d

a_nk, a_nk1 d

a_nk1, a_mk1 d

a_mk1, a_mk , d

b_nk, b_mk

≤d

b_nk, b_nk1 d

b_nk1, a_mk1 d

b_mk1, b_mk

, 2.20

we have max

d

a_nk, a_mk , d

b_nk, b_mk

≤max d

a_nk, a_nk1 , d

b_nk, b_nk1 max

d

a_nk1, a_mk1 , d

b_nk1, b_mk1 max

d

a_mk1, a_mk , d

b_mk1, b_mk .

2.21

Lettingk → ∞, we have ε≤ lim

k→ ∞max d

a_nk1, a_mk1 , d

b_nk1, b_mk1

. 2.22

Now, from2.19and2.22, we have

k→ ∞lim max d

a_nk1, a_mk1 , d

b_nk1, b_mk1

ε. 2.23

Asnk > mk, we have gx_mk gx_nk and gy_mk gy_nk. Puttingx x_mk, yy_mk,ux_nk, andvy_nkin2.1, we have

ψ d

a_mk1, a_nk1 ψ

d F

x_mk, y_mk , F

x_nk, y_nk

≤α max

d

a_mk, a_nk , d

b_mk, b_nk

−β max

d

a_mk, a_nk , d

b_mk, b_nk .

2.24

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Also, we have ψ

d

b_nk1, b_mk1 ψ

d F

y_nk, x_nk , F

y_mk, x_mk

≤α max

d

b_nk, b_mk , d

a_nk, a_mk

−β max

d

b_nk, b_mk , d

a_nk, a_mk .

2.25

Therefore,

ψ max

d

a_mk1, a_nk1 , d

b_mk1, b_nk1

≤α max

d

a_mk, a_nk , d

b_mk, b_nk

−β max

d

a_mk, a_nk , d

b_mk, b_nk .

2.26

Then, in2.26, ifk → ∞, from2.16and2.23, we haveψε ≤αε−βε. Thus, ψε−αε βε≤0, and henceε0, which is a contradiction. Consequently,{an}and{bn} are Cauchy.

Completeness of X, d implies that {an} and {bn} converge to some x, y ∈ X, respectively.

Step II. We will show thatFandghave a coupled coincidence point.

From the above step, we have

n→ ∞limF x_n, y_n

lim

n→ ∞gx_n lim

n→ ∞a_nx,

nlim→ ∞F yn, xn

lim

n→ ∞gyn lim

n→ ∞bny.

2.27

Sincegis continuous, by2.27, we have

nlim→ ∞g gx_n

gx, lim

n→ ∞g gy_n

gy. 2.28

Commutativity ofFandgyields that g

gx_n1 g

F xn, yn

F

gxn, gyn

, g

gy_n1 g

F

y_n, x_n F

gy_n, gx_n

. 2.29

From the continuity of F, {ggx_n1} is convergent to Fx, y and {ggy_n1} convergent to Fy, x. From2.28 and by uniqueness of the limit, we haveFx, y gx andFy, x gy, that is,gandFhave a coupled coincidence point.

This completes the proof of the theorem.

In the following theorem we omit the continuity assumption ofFandg.

Theorem 2.3. LetX,, dbe a partially ordered complete metric space and letF :X² → X and g :X → Xbe such thatFX²⊆gX. Assume thatFandgsatisfy2.1for everyx, y, u, v∈X

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withgxguandgygv, whereψ, α, β:0,∞ → 0,∞are such that,ψis an altering distance function,αis continuous,βis lower semicontinuous,α0 β0 0, andψt−αt βt>0 for allt >0.

Assume that

1Fhas the mixedg-monotone property, 2gXis a closed subset ofX.

Also, suppose that

iif a nondecreasing sequencexn → x, thenxnx, for alln∈N;

iiif a nonincreasing sequencey_n → y, theny_nyfor alln∈N.

If there existx₀, y₀∈Xsuch thatgx₀Fx0, y₀andgy₀ Fy0, x₀, thenFandghave a coupled coincidence point inX.

Proof. Following the proof of the previous theorem, sincegXis closed and{an}{gxn} ⊆ gX, there existsu∈Xsuch that

nlim→ ∞gxngux. 2.30

Similarly, there existsv∈Xsuch that

nlim→ ∞gy_ngvy. 2.31

Fromiandii, we havegxnguandgyngv.

Now, we prove thatFu, v guandFv, u gv. Using2.1, we have ψ

d

gx_n1, Fu, v ψ

d F

x_n, y_n

, Fu, v

≤α max

d

gx_n, gu , d

gy_n, gv

−β max

d

gxn, gu , d

gyn, gv .

2.32

In the above inequality, ifn → ∞, from properties ofψ,α, andβ, ψ

d

gu, Fu, v

≤α max

d gu, gu

, d

gv, gv

−β max

d gu, gu

, d

gv, gv α0−β0 0.

2.33

Hence,dgu, Fu, v 0, that is,gu Fu, v. Analogously, we can show thatgv Fv, u.

Theorem 2.4. Under the hypotheses ofTheorem 2.3, suppose thatgy₀ gx₀. Then, it follows that gu Fu, v Fv, u gv. Moreover, ifFandg bew-compatible, thenF andg have a coupled coincidence point of the formt, t.

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Proof. Ifgy0gx0, thengvgyngy0gx0 gxn gufor alln∈N. Thus, ifgu /gv, by inequality2.1, we have

ψ d

gv, gu

ψdFv, u, Fu, v

≤α max

d gv, gu

, d

gu, gv

−β max

d gv, gu

, d

gu, gv ψ

d

gu, gv

−β d

gu, gv .

2.34

Thus, from properties of functionsψ, α, βwe obtain dgu, gv 0, a contradiction.

Hence,gu gv, that is,gu Fu, v Fv, u gv. Now, lett gu gv. SinceF andg arew-compatible, thengt ggu gFu, v Fgu, gv Ft, t. Thus,Fandghave a coupled coincidence point of the formt, t.

Remark 2.5. In Theorems2.2and2.3, we extend the results of Harjani et al.Theorems 1.10 and1.11, if we takeαt ψt, for allt∈0,∞andgx IXthe identity mapping onX.

The following theorem can be deduced from our previous obtained results.

Theorem 2.6. LetX,, dbe a partially ordered complete metric space and letF :X² → X be a mapping having the mixed monotone property. Assume that

ψ d

F x, y

, Fu, v

≤ dx, u d y, v

2 −β

max

dx, u, d y, v

, 2.35

for everyx, y, u, v∈X withx uandy v, whereψ, β:0,∞ → 0,∞are such thatψ is an altering distance function,βis lower semicontinuous,β0 0, andψt−tβt>0 for allt >0.

Also, suppose that aFis continuous, or,

bXhas the following properties:

iif a nondecreasing sequencex_n → x, thenx_nx, for alln;

iiif a nonincreasing sequencey_n → y, theny_n yfor alln.

If there existx₀, y₀ ∈ Xsuch thatx₀ Fx0, y₀andy₀ Fy0, x₀, thenFhas a coupled fixed point inX.

Proof. IfF satisfies2.35, thenF satisfies2.1withgx IXthe identity mapping onX andαt t, for allt∈0,∞. Then, the result follows from Theorems2.2and2.3.

Note that if X, is a partially ordered set, then we can endow X × X with the following partial order relation:

x, y

u, v⇐⇒xu, yv, 2.36 for allx, y,u, v∈X×Xsee3.

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In the following theorem, we give a suﬃcient condition for the uniqueness of the common coupled fixed point. A similar proof can be found in Theorem 2.2 of3, Theorem 2.4 of12, and Theorem 2.3 of13.

Theorem 2.7. In addition to the hypotheses of Theorem 2.2 suppose that for every x, y and x^∗, y^∗ ∈ X × X, there exists u, v ∈ X², such that Fu, v, Fv, u is comparable with Fx, y, Fy, xand Fx^∗, y^∗, Fy^∗, x^∗. Then,F and g have a unique common coupled fixed point.

Proof. FromTheorem 2.2the set of coupled coincidence points ofF andg is nonempty. We will show that ifx, yandx^∗, y^∗are coupled coincidence points, that is,

gx F x, y

, g

y F

y, x , gx^∗ F

x^∗, y^∗

, g

y^∗ F

y^∗, x^∗

, 2.37

then,gxgx^∗andgygy^∗.

Choose an element u, v ∈ X² such that Fu, v, Fv, u is comparable with Fx, y, Fy, xandFx^∗, y^∗, Fy^∗, x^∗.

Letu0 u,v0 vand chooseu1, v1 ∈Xso thatgu1 Fu0, v0andgv1 Fv0, u0. Then, similarly as in the proof ofTheorem 2.2, we can inductively define sequences{gun}and {gvn}such thatgu_n1 Fun, v_nandgv_n1 Fvn, u_n. Sincegx, gy Fx, y, Fy, x andFu, v, Fv, u gu1, gv1are comparable, we may assume thatgx, gygu1, gv1. Then, gx gu1 and gy gv1. Using the mathematical induction, it is easy to prove that gxgu_nandgygv_n, for alln∈N.

Letγnmax{dgx, gun, dgy, gvn}. We will show that limn→ ∞γn 0. First, assume thatγn0, for ann≥1.

Applying2.1, asgxgu_nandgygv_none obtains that

ψ d

gx, gu_n1 ψ

d F

x, y

, Fun, v_n

≤α max

d

gx, gu_n , d

gy, gv_n

−β max

d gx, gun

, d gy, gvn

.

2.38

Similarly, we have

ψ d

gy, gv_n1 ψ

d F

y, x

, Fvn, un

≤α max

d

gy, gv_n , d

gx, gu_n

−β max

d gy, gvn

, d gx, gun

.

2.39

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From2.38and2.39, we have ψ

γ_n1 ψ

max d

gu_n1, gx , d

gv_n1, gy max

ψ d

gu_n1, gx , ψ

d

gv_n1, gy

≤α max

d

gx, gu_n , d

gy, gv_n

−β max

d

gx, gu_n , d

gy, gv_n α

γn

−β γn

α0−β0 0.

2.40

So, from properties ofψ,α, andβ, we deduceγ_n10. Repeating this process, we can show thatγ_m0, for allm≥n. So, lim_n_{→ ∞}γ_n0.

Now, letγn/0, for allnand letγn< γ_n1, for somen.

Asψis an altering distance function, from2.40 ψ

γn

ψ max

d

gun, gx , d

gvn, gy

< ψ γ_n1 ψ

max d

gu_n1, gx , d

gv_n1, gy max

ψ d

gu_n1, gx , ψ

d

gv_n1, gy

≤α max

d

gx, gu_n , d

gy, gv_n

−β max

d

gx, gu_n , d

gy, gv_n α

γn

−β γn

.

2.41

This implies thatγn0, which is a contradiction.

Hence,γ_n1≤γ_n, for alln≥1. Now, if we proceed as inTheorem 2.2, we can show that

nlim→ ∞max d

gun, gx , d

gvn, gy

0. 2.42

So,{gun} → gxand{gvn} → gy.

Similarly, we can show that

nlim→ ∞max d

gu_n, gx^∗ , d

gv_n, gy^∗

0, 2.43

that is,{gun} → gx^∗ and {gvn} → gy^∗. Finally, since the limit is unique, gx gx^∗ and gygy^∗.

Sincegx Fx, yandgy Fy, x, by commutativity ofFandg, we haveggx gFx, y Fgx, gyandggy gFy, x Fgy, gx. Letgx aandgy b. Then, gaFa, bandgb Fb, a. Thus,a, bis another coupled coincidence point ofFandg.

Then,a gxgaandb gygb. Therefore,a, bis a coupled common fixed point ofF andg.

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To prove the uniqueness of coupled common fixed point, assume thatp, qis another coupled common fixed point of F and g. Then,p gp Fp, q and q gq Fq, p.

Sincep, qis a coupled coincidence point ofF andg, we havegp gaandgq gb. Thus, pgpgaaandqgqgbb. Hence, the coupled common fixed point is unique.

Theorem 2.8. Under the hypotheses of Theorem 2.3, suppose in addition that for every x, y and x^∗, y^∗ in X², there exists u, v ∈ X² such that Fu, v, Fv, u is comparable to Fx, y, Fy, x and Fx^∗, y^∗, Fy^∗, x^∗. If F and g are w-compatible, then F and g have a unique common coupled fixed point of the formt, t.

Proof. ByTheorem 2.3, the set of coupled coincidence points ofFandgis nonempty. Letx, y andx^∗, y^∗be coupled coincidence points ofFandg. Following the proof ofTheorem 2.7, we can prove thatgxgx^∗andgygy^∗. Note that ifx, yis a coupled coicidence point of Fandg, theny, xis also a coupled coincidence point ofF andg. Thus, we havegx gy.

Puttgxgy. SincegxFx, yandgyFy, xandFandgarew-compatible, we have gtggx gFx, y Fgx, gy Ft, t. Thus,t, tis a coupled coincidence point ofF andg. So,gtgxgy tand hence we havetgtFt, t. Therefore,t, tis a common coupled fixed point ofFandg.

To prove the uniqueness of the coupled common fixed point ofFandg, letv, wbe another coupled fixed point ofF andg, that is,v gv Fv, wandw gw Fw, v.

Clearly, we havegt gvandgt gw. Therefore,t v w. Thus,Fandg have a unique common coupled fixed point of the formt, t.

Remark 2.9. Note that Theorems2.4and2.8have been established and proved according to Theorems 2.3 and 2.5 of12.

The following simple example guarantees that our results are proper generalizations of the results of Harjani et al.Theorems1.10and1.11.

Example 2.10. LetX 0,∞. We define a partial order “” on X asx y if and only if x≤yfor allx, y ∈X. Let a metricdonXbe defined bydx, y 0, if and only ifxy, and dx, y xy, ifx /y. ThenX, dis a complete metric space.

DefineF:X×X → Xas follows:

F x, y

x 4 −y

4

, 2.44

for allx, y∈Xandg:X → Xwithgx xfor allx∈X.

Letψ, α, β : 0,∞ → 0,∞ be defined byψt 4t,αt 7t, and βt 7/2t.

Clearly,ψis an altering distance function,αis continuous,βis lower semicontinuous,α0 β0 0, andψt−αt βt t/2>0 for allt >0.

Now, letxuandyv. So, we have

ψ d

F x, y

, Fu, v 4 x

4 −y 4

u 4 − v

4

≤xu yv

≤2 max

xu, yv

.

2.45

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Hence,

ψ d

F x, y

, Fu, v

≤2 max d

gx, gu , d

gy, gv

≤7 max d

gx, gu , d

gy, gv

−7 2max

d gx, gu

, d

gy, gv α

max d

gx, gu , d

gy, gv

−β max

d gx, gu

, d

gy, gv . 2.46

Therefore, all of the conditions of Theorem 2.2are satisfied. Moreover,0,0 is the unique coupled coincidence point ofFandg.

However, inequality1.4inTheorem 1.10 is not satisfied. Indeed, letx, y 0,1 andu, v 0,0. Then,

ψ d

F x, y

, Fu, v 1 ψ

max d

gx, gu , d

gy, gv

−β max

d gx, gu

, d

gy, gv 4−7

2 1 2.

2.47

Example 2.11. LetX 0,∞be endowed with the euclidian metric and the usual ordering.

DefineF:X×X → Xas follows:

F x, y

⎧⎪

⎨

⎪⎩ x−y

4 , ifx≥y 0, ify > x,

2.48

for allx, y∈Xandg:X → Xwithgx xfor allx∈X.

Letψ:0,∞ → 0,∞be the identity mapping andα, β:0,∞ → 0,∞be defined byαt 2tandβt 3/2t.

Letx, y, u, v∈Xare such thatx≤uandy≥v. Now, we have Case 1y > xandv > u. Then,

ψ d

F x, y

, Fu, v

0≤ α max

d gx, gu

, d

gy, gv

−β max

d gx, gu

, d

gy, gv

. 2.49

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Case 2y > xandu≥v.

ψ d

F x, y

, Fu, v 1

4u−v

≤ 1 2

1 2

u−xy−v

1 2

|x−u|y−v 2

≤ 1 2max

|x−u|,y−v 2 max

|x−u|,y−v−3 2max

|x−u|,y−v α

max d

gx, gu , d

gy, gv

−β max

d gx, gu

, d

gy, gv . 2.50

Case 3x≥yandu≥v. Asx≤uandy≥v, we haveu≥x≥y≥v. Hence, ψ

d F

x, y

, Fu, v x

4 −y 4 − u

4 −v 4

≤ 1 2

|x−u|y−v 2

≤ 1 2max

|x−u|,y−v 2 max

|x−u|,y−v−3 2max

|x−u|,y−v α

max d

gx, gu , d

gy, gv

−β max

d gx, gu

, d

gy, gv . 2.51

Case 4x≥yandv > u. Asx≤u, andy≥v, we havexyuv. Hence, ψ

d F

x, y

, Fu, v

0α0−β0 α

max d

gx, gu , d

gy, gv

−β max

d gx, gu

, d

gy, gv .

2.52

Hence, all of the conditions ofTheorem 2.2are satisfied. Moreover,0,0is the coupled coincidence point ofFandg.

In what follows, we obtain some coupled coincidence point theorems for mappings satisfying some contraction conditions of integral type in an ordered complete metric space.

In 18, Branciari obtained a fixed point result for a single mapping satisfying an integral type inequality. Then, Altun et al.19established a fixed point theorem for weakly compatible maps satisfying a general contractive inequality of integral type.

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Denote byΛ the set of all functionsμ : 0,∞ → 0,∞verifying the following conditions:

Iμis a positive Lebesgue integrable mapping on each compact subset of0,∞, IIfor allε >0,_ε

0μtdt >0.

Corollary 2.12. Replace the contractive condition2.1ofTheorem 2.2by the following condition.

There exists aμ∈Λsuch that ψdFx,y,Fu,v

0

μtdt≤

αmax{dgx,gu,dgy,gv}

0

μtdt

−

βmax{dgx,gu,dgy,gv}

0

μtdt.

2.53

If other conditions ofTheorem 2.2hold, thenFandghave a coupled coincidence point.

Proof . Consider the functionΓx _x

0 μtdt. Then2.53becomes Γ

ψ d

F x, y

, Fu, v

≤Γ α

max d

gx, gu , d

gy, gv

−Γ β

max d

gx, gu , d

gy, gv

. 2.54

Takingψ1 Γoψ,α1 Γoαand β1 Γoβ and applyingTheorem 2.2, we obtain the proof.

Corollary 2.13. Substitute the contractive condition2.1ofTheorem 2.2by the following condition.

There exists aμ∈Λsuch that

ψ

_dFx,y,Fu,v

0

μtdt

≤α

max{dgx,gu,dgy,gv}

0

μtdt

−β

max{dgx,gu,dgy,gv}

0

μtdt

.

2.55

ThenFandghave a coupled coincidence point, if other conditions ofTheorem 2.2hold.

Proof. Again, as inCorollary 2.12, define the functionΓx _x

0 φtdt. Then2.55changes to ψ

Γ d

F x, y

, Fu, v

≤α Γ

max d

gx, gu , d

gy, gv

−β Γ

max d

gx, gu , d

gy, gv

. 2.56

Now, if we defineψ1 ψoΓ,α1 αoΓandβ1 ϕoΓ, and applyingTheorem 2.2, then the proof is obtained.

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As in20, letn∈N^∗be fixed. Let{μi}_1≤i≤N be a family ofNfunctions which belong toΛ. For allt≥0, we define

I1t _t

0

μ1sds, I2t

_I₁_t

0

μ2sds ^t

0μ1sds 0

μ2sds, I3t

_I₂_t

0

μ3sds

⁰^t^μ¹^sds

0 μ2sds

0

μ3sds, ...

I_Nt _I_N−1_t

0

μ_Nsds.

2.57

We have the following result.

Corollary 2.14. Replace the inequality2.1ofTheorem 2.2by the following condition:

ψ

_I_N−1_dFx,y,Fu,v

0

μ_Nsds

≤α

_I_N−1max{dgx,gu,dgy,gv}

0

μ_Nsds

−β

_I_N−1max{dgx,gu,dgy,gv}

0

μ_Nsds

.

2.58

Assume further that all other conditions ofTheorem 2.2are also satisfied, thenFandghave a coupled coincidence point.

Proof. ConsiderψψoIN,ααoIN, andββoIN. Then the above inequality becomes

ψ

d F

x, y

, Fu, v

≤α max

d gx, gu

, d

gy, gv

−β max

d gx, gu

, d

gy, gv .

2.59

ApplyingTheorem 2.2, we obtain the desired result.

Other consequence of our theorems is the following result.

Corollary 2.15. Replace the contractive condition2.1ofTheorem 2.2by the following condition.

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There existμ1, μ2, μ3 ∈Λsuch that _dFx,y,Fu,v

0

μ₁tdt≤

max{dgx,gu,dgy,gv}

0

μ₂tdt

−

max{dgx,gu,dgy,gv}

0

μ₃tdt.

2.60

Let other conditions ofTheorem 2.2are satisfied, thenFandghave a coupled coincidence point.

Acknowledgments

The authors thank the referees for the careful reading and useful comments, suggestions, and remarks that contributed to the improvement of the paper. The authors thank the referees for the careful reading and useful comments, suggestions, and remarks that contributed to the improvement of the paper. Also, they give special thanks to Professor S. Jesic for his help in improving the paper by providing Example2.10.

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