R E S E A R C H
Open Access
Discussion on some coupled fixed point
theorems
Bessem Samet
1*, Erdal Karapınar
2, Hassen Aydi
3and Vesna ´Cojba˘si´c Raji´c
4*Correspondence:
[email protected]; [email protected]
1Department of Mathematics, King
Saud University, Riyadh, Saudi Arabia
Full list of author information is available at the end of the article
Abstract
In this paper, we show that, unexpectedly, most of the coupled fixed point theorems (on ordered metric spaces) are in fact immediate consequences of well-known fixed point theorems in the literature.
MSC: 47H10; 54H25
Keywords: coupled fixed point; fixed point; ordered set; metric space
1 Introduction
In recent years, there has been recent interest in establishing fixed point theorems on ordered metric spaces with a contractivity condition which holds for all points that are related by partial ordering.
In [], Ran and Reurings established the following fixed point theorem that extends the Banach contraction principle to the setting of ordered metric spaces.
Theorem .(Ran and Reurings []) Let(X,)be an ordered set endowed with a metric d and T:X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) Tis continuous nondecreasing(with respect to);
(iii) there existsx∈Xsuch thatxTx;
(iv) there exists a constantk∈(, )such that for allx,y∈Xwithxy,
d(Tx,Ty)≤kd(x,y).
Then T has a fixed point.Moreover,if for all(x,y)∈X×X there exists a z∈X such that xz and yz,we obtain uniqueness of the fixed point.
Nieto and López [] extended the above result for a mappingTnot necessarily contin-uous by assuming an additional hypothesis on (X,,d).
Theorem .(Nieto and López []) Let(X,)be an ordered set endowed with a metric d and T:X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) if a nondecreasing sequence{xn}inXconverges to some pointx∈X,thenxnxfor alln;
(iii) Tis nondecreasing;
(iv) there existsx∈Xsuch thatxTx;
(v) there exists a constantk∈(, )such that for allx,y∈Xwithxy,
d(Tx,Ty)≤kd(x,y).
Then T has a fixed point.Moreover,if for all(x,y)∈X×X there exists a z∈X such that xz and yz,we obtain uniqueness of the fixed point.
Theorems . and . are extended and generalized by many authors. Before presenting some of theses results, we need to introduce some functional sets.
Denote bythe set of functionsϕ: [,∞)→[,∞) satisfying the following conditions:
() ϕis continuous nondecreasing;
() ϕ–({}) ={}.
Denote bySthe set of functionsβ: [,∞)→[, ) satisfying the following condition:
β(tn) –→ implies tn–→.
Denote bythe set of functionsθ: [,∞)→[, ) which satisfy the condition:
θ(sn,tn) –→ implies tn,sn–→.
Denote bythe set of functionsψ: [,∞)→[,∞) satisfying the following conditions:
() ψ(t) <tfor allt> ;
() limr→t+ψ(r) <t.
In [], Harjani and Sadarangani established the following results.
Theorem .(Harjani and Sadarangani []) Let(X,)be an ordered set endowed with a metric d and T:X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) Tis continuous nondecreasing; (iii) there existsx∈Xsuch thatxTx;
(iv) there existϕ,ψ∈such that for allx,y∈Xwithxy,
ψd(Tx,Ty)≤ψd(x,y)–ϕd(x,y).
Then T has a fixed point.Moreover,if for all(x,y)∈X×X there exists a z∈X such that xz and yz,we obtain uniqueness of the fixed point.
Theorem .(Harjani and Sadarangani []) Let(X,)be an ordered set endowed with a metric d and T:X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) if a nondecreasing sequence{xn}inXconverges to some pointx∈X,thenxnxfor alln;
(iii) Tis nondecreasing;
(v) there existϕ,ψ∈such that for allx,y∈Xwithxy,
ψd(Tx,Ty)≤ψd(x,y)–ϕd(x,y).
Then T has a fixed point.Moreover,if for all(x,y)∈X×X there exists a z∈X such that xz and yz,we obtain uniqueness of the fixed point.
In [], Amini-Harandi and Emami established the following results.
Theorem .(Amini-Harandi and Emami []) Let(X,)be an ordered set endowed with a metric d and T:X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) Tis continuous nondecreasing; (iii) there existsx∈Xsuch thatxTx;
(iv) there existsβ∈Ssuch that for allx,y∈Xwithxy,
d(Tx,Ty)≤βd(x,y)d(x,y).
Then T has a fixed point.Moreover,if for all(x,y)∈X×X there exists a z∈X such that xz and yz,we obtain uniqueness of the fixed point.
Theorem .(Amini-Harandi and Emami []) Let(X,)be an ordered set endowed with a metric d and T:X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) if a nondecreasing sequence{xn}inXconverges to some pointx∈X,thenxnxfor alln;
(iii) Tis nondecreasing;
(iv) there existsx∈Xsuch thatxTx;
(v) there existsβ∈Ssuch that for allx,y∈Xwithxy,
d(Tx,Ty)≤βd(x,y)d(x,y).
Then T has a fixed point.Moreover,if for all(x,y)∈X×X there exists a z∈X such that xz and yz,we obtain uniqueness of the fixed point.
Remark . Jachymski [] established that Theorem . (resp. Theorem .) follows from Theorem . (resp. Theorem .).
Remark . Theorems . and . hold ifϕ: [,∞)→[,∞) satisfies only the following conditions:ϕis lower semi-continuous andϕ–({}) ={}(see, for example, []).
The following results are special cases of Theorem . in [].
Theorem .(Ćirićet al.[]) Let(X,)be an ordered set endowed with a metric d and T:X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(iv) there exists a continuous functionϕ: [,∞)→[,∞)withϕ(t) <tfor allt> such that for allx,y∈Xwithxy,
d(Tx,Ty)≤ϕd(x,y).
Then T has a fixed point.
Theorem .(Ćirićet al.[]) Let(X,)be an ordered set endowed with a metric d and T:X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) if a nondecreasing sequence{xn}inXconverges to some pointx∈X,thenxnxfor alln;
(iii) Tis nondecreasing;
(iv) there existsx∈Xsuch thatxTx;
(v) there exists a continuous functionϕ: [,∞)→[,∞)withϕ(t) <tfor allt> such that for allx,y∈Xwithxy,
d(Tx,Ty)≤ϕd(x,y).
Then T has a fixed point.
Remark . Theorems . and . hold if we suppose thatϕ∈(see, for example, []).
LetXbe a nonempty set andF:X×X→Xbe a given mapping. We say that (x,y)∈ X×Xis a coupled fixed point ofFif
x=F(x,y) and y=F(y,x).
In [], Bhaskar and Lakshmikantham established some coupled fixed point theorems on ordered metric spaces and applied the obtained results to the study of existence and uniqueness of solutions to a class of periodic boundary value problems. The obtained re-sults in [] have been extended and generalized by many authors (see, for example, [, –]).
In this paper, we will prove that most of the coupled fixed point theorems are in fact immediate consequences of well-known fixed point theorems in the literature.
2 Main results
Let (X,) be a partially ordered set endowed with a metricdandF:X×X→Xbe a given mapping. We endow the product setX×Xwith the partial order:
(x,y), (u,v)∈X×X, (x,y)(u,v) ⇐⇒ xu, yv.
Definition . Fis said to have the mixed monotone property ifF(x,y) is monotone non-decreasing inxand is monotone non-increasing iny, that is, for anyx,y∈X,
x,x∈X, xx =⇒ F(x,y)F(x,y);
LetY=X×X. It is easy to show that the mappingsη,δ:Y×Y→[,∞) defined by
η(x,y), (u,v)=d(x,u) +d(y,v);
δ(x,y), (u,v)=maxd(x,u),d(y,v)
for all (x,y), (u,v)∈Y, are metrics onY. Now, define the mappingT:Y→Yby
T(x,y) =F(x,y),F(y,x) for all (x,y)∈Y.
It is easy to show the following.
Lemma . The following properties hold:
(a) (X,d)is complete if and only if(Y,η)and(Y,δ)are complete;
(b) Fhas the mixed monotone property if and only ifTis monotone nondecreasing with respect to;
(c) (x,y)∈X×Xis a coupled fixed point ofFif and only if(x,y)is a fixed point ofT.
2.1 Bhaskar and Lakshmikantham’s coupled fixed point results
We present the obtained results in [] in the following theorem.
Theorem .(see Bhaskar and Lakshmikantham []) Let(X,)be a partially ordered set endowed with a metric d.Let F:X×X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(X) if a nondecreasing sequence{xn}inXconverges to some pointx∈X,thenxnx for alln,
(X) if a decreasing sequence{yn}inXconverges to some pointy∈X,thenynyfor alln;
(iv) there existx,y∈Xsuch thatxF(x,y)andyF(y,x);
(v) there exists a constantk∈(, )such that for all(x,y), (u,v)∈X×Xwithxuand
yv,
dF(x,y),F(u,v)≤k
d(x,u) +d(y,v).
Then F has a coupled fixed point(x*,y*)∈X×X.Moreover,if for all(x,y), (u,v)∈X×X
there exists(z,z)∈X×X such that(x,y)(z,z)and(u,v)(z,z),we have
unique-ness of the coupled fixed point and x*=y*.
We will prove the following result.
Proof From (v), for all (x,y), (u,v)∈X×Xwithxuandyv, we have
dF(x,y),F(u,v)≤k
d(x,u) +d(y,v)
and
dF(v,u),F(y,x)≤k
d(x,u) +d(y,v).
This implies that for all (x,y), (u,v)∈X×Xwithxuandyv,
dF(x,y),F(u,v)+dF(v,u),F(y,x)≤kd(x,u) +d(y,v),
that is,
ηT(x,y),T(u,v)≤kη(x,y), (u,v)
for all (x,y), (u,v)∈Ywith (x,y)(u,v). From Lemma ., since (X,d) is complete, (Y,η)
is also complete. SinceFhas the mixed monotone property,Tis a nondecreasing mapping with respect to. From (iv), we have (x,y)T(x,y). Now, ifFis continuous, then
Tis continuous. In this case, applying Theorem ., we get thatThas a fixed point, which implies from Lemma . thatFhas a coupled fixed point. If conditions (X) and (X) are
satisfied, thenY satisfies the following property: if a nondecreasing (with respect to)
sequence{un}inYconverges to some pointu∈Y, thenunufor alln. Applying
Theo-rem ., we get thatThas a fixed point, which implies thatFhas a coupled fixed point. If, in addition, we suppose that for all (x,y), (u,v)∈X×Xthere exists (z,z)∈X×Xsuch
that (x,y)(z,z) and (u,v)(z,z), from the last part of Theorems . and ., we
ob-tain the uniqueness of the fixed point ofT, which implies the uniqueness of the coupled fixed point ofF. Now, let (x*,y*)∈X×Xbe a unique coupled fixed point ofF. Since (y*,x*)
is also a coupled fixed point ofF, we getx*=y*.
2.2 Harjani, López and Sadarangani’s coupled fixed point results
We present the results obtained in [] in the following theorem.
Theorem .(see Harjaniet al.[]) Let(X,)be a partially ordered set endowed with a metric d.Let F:X×X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(X) if a nondecreasing sequence{xn}inXconverges to some pointx∈X,thenxnx for alln,
(X) if a decreasing sequence{yn}inXconverges to some pointy∈X,thenynyfor alln;
(v) there existψ,ϕ∈such that for all(x,y), (u,v)∈X×Xwithxuandyv,
ψdF(x,y),F(u,v)≤ψmaxd(x,u),d(y,v)–ϕmaxd(x,u),d(y,v).
Then F has a coupled fixed point(x*,y*)∈X×X.Moreover,if for all(x,y), (u,v)∈X×X
there exists(z,z)∈X×X such that(x,y)(z,z)and(u,v)(z,z),we have
unique-ness of the coupled fixed point and x*=y*.
We will prove the following result.
Theorem . Theorem.follows from Theorems.and..
Proof From (v), for all (x,y), (u,v)∈X×Xwithxuandyv, we have
ψdF(x,y),F(u,v)≤ψmaxd(x,u),d(y,v)–ϕmaxd(x,u),d(y,v)
and
ψdF(v,u),F(y,x)≤ψmaxd(x,u),d(y,v)–ϕmaxd(x,u),d(y,v).
This implies (since ψ is nondecreasing) that for all (x,y), (u,v)∈X×Xwithxuand yv,
ψmaxdF(x,y),F(u,v),dF(v,u),F(y,x)
≤ψmaxd(x,u),d(y,v)–ϕmaxd(x,u),d(y,v),
that is,
ψδT(x,y),T(u,v)≤ψδ(x,y), (u,v)–ϕδ(x,y), (u,v)
for all (x,y), (u,v)∈Y with (x,y)(u,v). Thus we proved that the mappingT satisfies
the condition (iv) (resp. (v)) of Theorem . (resp. Theorem .). The rest of the proof is
similar to the above proof.
2.3 Lakshmikantham and ´Ciri´c’s coupled fixed point results
In [], puttingg=iX(the identity mapping), we obtain the following result.
Theorem .(see Lakshmikantham and Ćirić’s []) Let(X,)be a partially ordered set endowed with a metric d.Let F:X×X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(X) if a decreasing sequence{yn}inXconverges to some pointy∈X,thenynyfor alln;
(iv) there existx,y∈Xsuch thatxF(x,y)andyF(y,x);
(v) there existsϕ∈such that for all(x,y), (u,v)∈X×Xwithxuandyv,
dF(x,y),F(u,v)≤ϕ
d(x,u) +d(y,v)
.
Then F has a coupled fixed point.
We will prove the following result.
Theorem . Theorem.follows from Theorems.and..
Proof From (v), for all (x,y), (u,v)∈X×Xwithxuandyv, we have
dF(x,y),F(u,v)≤ϕ
d(x,u) +d(y,v)
and
dF(v,u),F(y,x)≤ϕ
d(x,u) +d(y,v)
.
This implies that for all (x,y), (u,v)∈X×Xwithxuandyv,
d(F(x,y),F(u,v)) +d(F(v,u),F(y,x))
≤ϕ
d(x,u) +d(y,v)
,
that is,
ηT(x,y),T(u,v)≤ϕη(x,u), (y,v)
for all (x,y), (u,v)∈Y with (x,y)(u,v). Here,η:Y×Y →[,∞) is the metric onY
given by
η(x,y), (u,v)=η((x,y), (u,v))
for all (x,y), (u,v)∈Y.
Thus we proved that the mappingT satisfies the condition (iv) (resp. (v)) of Theorem . (resp. Theorem .). ThenT has a fixed point, which implies thatF has a coupled fixed
point.
2.4 Luong and Thuan’s coupled fixed point results
Luong and Thuan [] presented a coupled fixed point result involving an ICS mapping.
Definition . Let (X,d) be a metric space. A mappingS:X→X is said to be ICS if S is injective, continuous and has the property: for every sequence{xn}inX, if{Sxn}is
convergent, then{xn}is also convergent.
Lemma . Let(X,d)be a metric space and S:X→X be an ICS mapping. Then the mapping dS:X×X→[,∞)defined by
dS(x,y) =d(Sx,Sy) for all x,y∈X,
is a metric on X.Moreover,if(X,d)is complete,then(X,dS)is also complete.
Proof Let us prove thatdSis a metric onX. Letx,y∈Xsuch thatdS(x,y) = . This implies
thatSx=Sy. SinceSis injective, we obtain thatx=y. Other properties of the metric can be easily checked. Now, suppose that (X,d) is complete and let{xn}be a Cauchy sequence in
the metric space (X,dS). This implies that{Sxn}is Cauchy in (X,d). Since (X,d) is complete,
{Sxn}is convergent in (X,d) to some pointy∈X. SinceSis an ICS mapping,{xn}is also
convergent in (X,d) to some pointx∈X. On the other hand, the continuity ofSimplies the convergence of{Sxn}in (X,d) toSx. By the uniqueness of the limit in (X,d), we get that
y=Sx, which implies thatdS(xn,x) –→ asn–→ ∞. Thus{xn}is a convergent sequence
in (X,dS). This proves that (X,dS) is complete.
The obtained result in [] is the following.
Theorem .(see Luong and Thuan []) Let(X,)be a partially ordered set endowed with a metric d.Let S:X→X be an ICS mapping.Let F:X×X→X be a given mapping. Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(X) if a nondecreasing sequence{xn}inXconverges to some pointx∈X,thenxnx for alln,
(X) if a decreasing sequence{yn}inXconverges to some pointy∈X,thenynyfor alln;
(iv) there existx,y∈Xsuch thatxF(x,y)andyF(y,x);
(v) there existsψ∈such that for all(x,y), (u,v)∈X×Xwithxuandyv,
dSF(x,y),SF(u,v)≤ ψ
d(Sx,Su) +d(Sy,Sv).
Then F has a coupled fixed point.
We will prove the following result.
Theorem . Theorem.follows from Theorems.and..
Proof The condition (v) implies that for all (x,y), (u,v)∈X×Xwithxuandyv,
dS
F(x,y),F(u,v)+dS
F(v,u),F(y,x)≤ψdS(x,u) +dS(y,v)
,
that is,
ηS
T(x,y),T(u,v)≤ψηS
for all (x,y), (u,v)∈Y with (x,y)(u,v), whereηSis the metric (see Lemma .) onY
defined by
ηS
(x,y), (u,v)=dS(x,u) +dS(y,v) for all (x,y), (u,v)∈Y.
Thus we proved that the mappingT satisfies the condition (iv) (resp. (v)) of Theorem . (resp. Theorem .). ThenT has a fixed point, which implies thatF has a coupled fixed
point.
2.5 Berind’s coupled fixed point results
The following result was established in [].
Theorem .(see Berinde []) Let(X,)be a partially ordered set endowed with a
met-ric d.Let F:X×X→X be a given mapping.Suppose that the following conditions hold: (i) (X,d)is complete;
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(X) if a nondecreasing sequence{xn}inXconverges to some pointx∈X,thenxnx for alln,
(X) if a decreasing sequence{yn}inXconverges to some pointy∈X,thenynyfor alln;
(iv) there existx,y∈Xsuch thatxF(x,y)andyF(y,x);
(v) there exists a constantk∈(, )such that for all(x,y), (u,v)∈X×Xwithxuand
yv,
dF(x,y),F(u,v)+dF(y,x),F(v,u)≤kd(x,u) +d(y,v).
Then F has a coupled fixed point(x*,y*)∈X×X.Moreover,if for all(x,y), (u,v)∈X×X
there exists(z,z)∈X×X such that(x,y)(z,z)and(u,v)(z,z),we have
unique-ness of the coupled fixed point and x*=y*.
We have the following result.
Theorem . Theorem.follows from Theorems.and..
Proof From the condition (v), the mappingTsatisfies
ηT(x,y),T(u,v)≤kη(x,y), (u,v)
for all (x,y), (u,v)∈Y with (x,y)(u,v). Thus we proved that the mappingT satisfies
the condition (iv) (resp. (v)) of Theorem . (resp. Theorem .). ThenThas a fixed point, which implies thatFhas a coupled fixed point. The rest of the proof is similar to the above
2.6 Rasouli and Bahrampour’s coupled fixed point results
Theorem .(see Rasouli and Bahrampour []) Let(X,)be a partially ordered set
endowed with a metric d.Let F:X×X→X be a given mapping.Suppose that the following conditions hold:
(i) (X,d)is complete;
(ii) Fhas the mixed monotone property;
(iii) Fis continuous orXhas the following properties:
(X) if a nondecreasing sequence{xn}inXconverges to some pointx∈X,thenxnx for alln,
(X) if a decreasing sequence{yn}inXconverges to some pointy∈X,thenynyfor alln;
(iv) there existx,y∈Xsuch thatxF(x,y)andyF(y,x);
(v) there existsβ∈Ssuch that for all(x,y), (u,v)∈X×Xwithxuandyv,
dF(x,y),F(u,v)≤βmaxd(x,u),d(y,v)maxd(x,u),d(y,v).
Then F has a coupled fixed point(x*,y*)∈X×X.Moreover,if for all(x,y), (u,v)∈X×X
there exists(z,z)∈X×X such that(x,y)(z,z)and(u,v)(z,z),we have
unique-ness of the coupled fixed point and x*=y*. We have the following result.
Theorem . Theorem.follows from Theorems.and..
Proof From the condition (v), the mappingTsatisfies
δT(x,y),T(u,v)≤βδ(x,y), (u,v)δ(x,y), (u,v)
for all (x,y), (u,v)∈Y with (x,y)(u,v). Thus we proved that the mappingT satisfies
the condition (iv) (resp. (v)) of Theorem . (resp. Theorem .). ThenThas a fixed point, which implies thatFhas a coupled fixed point. The rest of the proof is similar to the above
proofs.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, King Saud University, Riyadh, Saudi Arabia.2Department of Mathematics, Atılım University,
˙Incek, Ankara 06836, Turkey.3Institut Supérieur d’Informatique et des Technologies de Communication de Hammam
Sousse, Université de Sousse, Route GP1-4011, H. Sousse, Tunisie.4Faculty of Economics, University of Belgrade,
Kameniˇcka 6, Beograd, 11000, Serbia.
Acknowledgements
This work is supported by the Research Center, College of Science, King Saud University.
References
1. Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc.132, 1435-1443 (2004)
2. Nieto, JJ, Rodríguez-López, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order22, 223-239 (2005)
3. Harjani, J, Sadarangani, K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal.72, 1188-1197 (2010)
4. Amini-Harandi, A, Emami, H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal.72, 2238-2242 (2010)
5. Jachymski, J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal.74(3), 768-774 (2011)
6. Cherichi, M, Samet, B: Fixed point theorems on ordered gauge spaces with applications to nonlinear integral equations. Fixed Point Theory Appl.2012, 13 (2012)
7. ´Ciri´c, L, Caki´c, N, Rajovi´c, M, Ume, JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl.2008, Article ID 131294, 11 pages (2008)
8. Lakshmikantham, V, ´Ciri´c, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal.70(12), 4341-4349 (2009)
9. Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal.65, 1379-1393 (2006)
10. Amini-Harandi, A: Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem. Math. Comput. Model.57(9-10), 2343-2348 (2013)
11. Berinde, V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal.74, 7347-7355 (2011)
12. Berinde, V: Coupled fixed point theorems for-contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal.75, 3218-3228 (2012)
13. ´Ciri´c, L, Agarwal, RP, Samet, B: Mixed monotone-generalized contractions in partially ordered probabilistic metric spaces. Fixed Point Theory Appl.2011, 56 (2011)
14. Ding, H-S, Li, L, Radenovi´c, S: Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces. Fixed Point Theory Appl.2012, 96 (2012)
15. Golubovi´c, Z, Kadelburg, Z, Radenovi´c, S: Coupled coincidence points of mappings in ordered partial metric spaces. Abstr. Appl. Anal.2012, Article ID 192581, 18 pages (2012)
16. Harjani, J, López, B, Sadarangani, K: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal.74, 1749-1760 (2011)
17. Karapınar, E: Coupled fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl.59, 3656-3668 (2010)
18. Luong, NV, Thuan, NX: Coupled fixed point theorems in partially ordered metric spaces depended on another function. Bull. Math. Anal. Appl.3(3), 129-140 (2011)
19. Radenovi´c, S, Panteli´c, S, Salimi, P, Vujakovi´c, J: A note on some tripled coincidence point results inG-metric spaces. Int. J. Math. Sci. Eng. Appl.6(VI), 23-38 (2012)
20. Rasouli, SH, Bahrampour, M: A remark on the coupled fixed point theorems for mixed monotone operators in partially ordered metric spaces. J. Math. Comput. Sci.3(2), 246-261 (2011)
21. Samet, B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal.74, 4508-4517 (2010)
22. Samet, B, Yazidi, H: Coupled fixed point theorems in partially orderedε-chainable metric spaces. J. Math. Comput. Sci. 1(3), 142-151 (2010)
23. Shatanawi, W: Coupled fixed point theorems in generalized metric spaces. Hacet. J. Math. Stat.40(3), 441-447 (2011)
doi:10.1186/1687-1812-2013-50
Cite this article as:Samet et al.:Discussion on some coupled fixed point theorems.Fixed Point Theory and Applications
