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Remarks on ‘Coupled coincidence point results for a generalized compatible pair with applications’
Fixed Point Theory and Applications volume 2014, Article number: 207 (2014)
Abstract
Very recently, Hussain et al. (Fixed Point Theory Appl. 2014:62, 2014) announced the existence and uniqueness of some coupled coincidence point. In this short note we remark that the announced results can be derived from the coincidence point results in the literature.
MSC: 47H10, 54H25.
1 Introduction
Recently, a number of studies related to fixed points, coupled fixed points and coupled coincidence points of maps defined via auxiliary functions have appeared in the literature. In particular, the socalled weak φcontractions, contractions defined by means of altering distance functions, \alpha \psitype contractions have been a subject of considerable interest. Studies of this type aim to generalize and improve contractive condition on the maps (see, e.g., [1–15]).
A great deal of these studies investigate contractions on partially ordered metric spaces because of their applicability to initial value problems defined by differential or integral equations. This is the case of the following result.
Theorem 1.1 (Hussain et al. [16], Theorem 15)
Let (X,\u2aaf) be a partially ordered set such that there exists a complete metric d on X. Assume that F,G:X\times X\to X are two generalized compatible mappings such that F is Gincreasing with respect to ⪯, G is continuous and has the mixed monotone property, and there exist two elements {x}_{0},{y}_{0}\in X such that
Suppose that there exist \varphi \in \mathrm{\Phi} and \psi \in \mathrm{\Psi} such that
for all x,y,u,v\in X with G(x,y)\u2aafG(u,v) and G(y,x)\u2ab0G(v,u). Suppose that for any x,y\in X, there exist u,v\in X such that
Also suppose that either

(a)
F is continuous, or

(b)
X has the following property:

(i)
if a ⪯nondecreasing sequence \{{x}_{n}\}\to x, then {x}_{n}\u2aafx for all n\in \mathbb{N},

(ii)
if a ⪯nonincreasing sequence \{{y}_{n}\}\to y, then y\u2aaf{y}_{n} for all n\in \mathbb{N}.
Then F and G have a coupled coincidence point in X.
In this paper we show that the previous result can be easily improved because of the following facts.

(1)
The mixed monotone property is not necessary since F is Gincreasing with respect to ⪯.

(2)
It is possible to consider a pair of mappings satisfying a weaker condition than the generalized compatible property (using monotone sequences).

(3)
In fact, Theorem 1.1 is not a true advance because it can be reduced to its corresponding unidimensional coincidence point theorem.
To prove our main claims, we will show a unidimensional proof of the mentioned theorem.
2 Preliminaries
Firstly, we recall some basic definitions and elementary results needed throughout the paper. Some of them can be found in [17]. In the sequel, we denote by X a nonempty set. Given a natural number n\in \mathbb{N}, let {X}^{n} be the nth Cartesian product X\times X\times \cdots \times X (n times). We employ mappings T,g:X\to X and F:{X}^{n}\to X. For simplicity, if x\in X, we denote T(x) by Tx.
Definition 2.1 (Khan et al. [18])
An altering distance function is a continuous, nondecreasing function \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) such that \varphi (t)=0 if and only if t=0. Let {\mathcal{F}}_{alt} denote the family of all altering distance functions.
A function \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) is said to be subadditive if \varphi (t+s)\le \varphi (t)+\varphi (s) for all t,s\ge 0. Following [16], we introduce the following families of control functions. Let Φ denote the family of all subadditive altering distance functions, that is, functions \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) which satisfy the following:
({\varphi}_{1}) ϕ is continuous and nondecreasing;
({\varphi}_{2}) \varphi (t)=0 if and only if t=0;
({\varphi}_{3}) \varphi (t+s)\le \varphi (t)+\varphi (s) for all t,s\in [0,\mathrm{\infty}).
We denote by Ψ the family of all functions \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) which satisfy the following:

(1)
{lim}_{t\to r}\psi (t)>0 for all r>0;

(2)
{lim}_{t\to {0}^{+}}\psi (t)=0.
Remark 2.1 Let \psi \in \mathrm{\Psi}, c>0 and define {\psi}_{c}:[0,\mathrm{\infty})\to [0,\mathrm{\infty}) by {\psi}_{c}(t)=c\psi (t/c) for all t\ge 0. Then {\psi}_{c}\in \mathrm{\Psi}.
A coincidence point of two mappings T,g:X\to X is a point x\in X such that Tx=gx.
Definition 2.3 (Hussain et al. [16], Definition 10)
A coupled coincidence point of two mappings F,G:{X}^{2}\to X is a point (x,y)\in X such that
Definition 2.4 An ordered metric space (X,d,\u2aaf) is a metric space (X,d) provided with a partial order ⪯.
An ordered metric space (X,d,\u2aaf) is said to be nondecreasingregular (respectively, nonincreasingregular) if for every sequence \{{x}_{m}\}\subseteq X such that \{{x}_{m}\}\to x and {x}_{m}\u2aaf{x}_{m+1} (respectively, {x}_{m}\u2ab0{x}_{m+1}) for all m, we have that {x}_{m}\u2aafx (respectively, {x}_{m}\u2ab0x) for all m. (X,d,\u2aaf) is said to be regular if it is both nondecreasingregular and nonincreasingregular.
Remark 2.2 Notice that condition (b) in Theorem 1.1 means that (X,d,\u2aaf) is regular.
Definition 2.6 Let (X,\u2aaf) be a partially ordered set, and let T,g:X\to X be two mappings. We say that T is (g,\u2aaf) nondecreasing if Tx\u2aafTy for all x,y\in X such that gx\u2aafgy. If g is the identity mapping on X, we say that T is ⪯nondecreasing.
Remark 2.3 If T is (g,\u2aaf) nondecreasing and gx=gy, then Tx=Ty. It follows that
Definition 2.7 (Hussain et al. [16], Definition 7)
Suppose that F,G:X\times X\to X are two mappings, and let ⪯ be a partial order on X. The mapping F is said to be Gincreasing with respect to ⪯ if for all x,y,u,v\in X with G(x,y)\u2aafG(u,v) we have F(x,y)\u2aafF(u,v).
Lemma 2.1 (see [22])
Let (X,d) be a metric space and define {\mathrm{\Delta}}_{n}:{X}^{n}\times {X}^{n}\to [0,\mathrm{\infty}), for all A=({a}_{1},{a}_{2},\dots ,{a}_{n}),B=({b}_{1},{b}_{2},\dots ,{b}_{n})\in {X}^{n}, by
Then {\mathrm{\Delta}}_{n} is metric on {X}^{n} and (X,d) is complete if and only if (X,{\mathrm{\Delta}}_{n}) is complete.
Consider on the product space {X}^{2} the following partial order: for (x,y),(u,v)\in {X}^{2},
Let (X,d,\u2aaf) be an ordered metric space. Two mappings T,g:X\to X are said to be Ocompatible if
provided that \{{x}_{m}\} is a sequence in X such that \{g{x}_{m}\} is ⪯monotone, that is, it is either nonincreasing or nondecreasing with respect to ⪯, and
Definition 2.9 (Hussain et al. [16], Definition 12)
Let F,G:X\times X\to X. We say that the pair \{F,G\} is generalized compatible if for all sequences \{{x}_{n}\},\{{y}_{n}\}\subseteq X such that
we have that
3 Main results
To start with, we highlight the weakness of Theorem 1.1 using the following example.
Example 3.1 Let X=[0,\mathrm{\infty}) endowed with the standard metric d(x,y)=xy for all x,y\in X. Consider the maps F,G:X\times X\to X defined by
Then, for all x,y,u,v\in X with y=v, we have
Thus,
Regarding the properties of the functions in Φ, we derive that
Since the function in the class Ψ takes values on [0,\mathrm{\infty}), it is impossible to verify inequality (2). Hence, Theorem 1.1 cannot be applied to get a coupled coincidence point. However, it is easy to see that (0,0) is a coupled coincidence point of F and G.
Next, we show a unidimensional version of Theorem 1.1. Notice that, indeed, the following result is better than Theorem 1.1 because we reorder the hypotheses obtaining that, in some cases, neither the continuity of, at least, one mapping (T or g) nor the Ocompatibility of the pair (T,g) is necessary. In fact, both hypotheses are omitted in case (c).
Theorem 3.1 Let (X,d,\u2aaf) be an ordered metric space, and let T,g:X\to X be two mappings such that the following properties are fulfilled:

(i)
T(X)\subseteq g(X);

(ii)
T is (g,\u2aaf)nondecreasing;

(iii)
there exists {x}_{0}\in X such that g{x}_{0}\u2aafT{x}_{0};

(iv)
there exist \varphi \in \mathrm{\Phi} and \psi \in \mathrm{\Psi} verifying
\varphi (d(Tx,Ty))\le \varphi (d(gx,gy))\psi (d(gx,gy))\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}x,y\in X\mathit{\text{such that}}gx\u2aafgy.
Also assume that, at least, one of the following conditions holds.

(a)
(X,d) is complete, T and g are continuous and the pair (T,g) is Ocompatible;

(b)
(X,d) is complete and T and g are continuous and commuting;

(c)
(g(X),d) is complete and (X,d,\u2aaf) is nondecreasingregular;

(d)
(X,d) is complete, g(X) is closed and (X,d,\u2aaf) is nondecreasingregular;

(e)
(X,d) is complete, g is continuous and monotone ⪯nondecreasing, the pair (T,g) is Ocompatible and (X,d,\u2aaf) is nondecreasingregular.
Then T and g have, at least, a coincidence point.
We omit the proof of the previous result since its proof is similar to the main theorem in [17] and it can be concluded by following, point by point, all of its arguments.
Next, we show how to deduce an appropriate version of Theorem 1.1 from Theorem 3.1. Given the ordered metric space (X,d,\u2aaf), let us consider the ordered metric space ({X}^{2},{\mathrm{\Delta}}_{2},\u2291), where {\mathrm{\Delta}}_{2} was defined in Lemma 2.1 and ⊑ was introduced in (4). We define the mappings {T}_{F},{T}_{G}:{X}^{2}\to {X}^{2}, for all (x,y)\in {X}^{2}, by
Under these conditions, the following properties hold.
Lemma 3.1 Let (X,d,\u2aaf) be an ordered metric space, and let F,G:{X}^{2}\to X be two mappings. Then the following properties hold.

(1)
(X,d) is complete if and only if ({X}^{2},{\mathrm{\Delta}}_{2}) is complete.

(2)
If (X,d,\u2aaf) is regular, then ({X}^{2},{\mathrm{\Delta}}_{2},\u2291) is also regular.

(3)
If F is dcontinuous, then {T}_{F} is {\mathrm{\Delta}}_{2}continuous.

(4)
If F is Gincreasing with respect to ⪯, then {T}_{F} is ({T}_{G},\u2291)nondecreasing.

(5)
Condition (1) is equivalent to the existence of a point ({x}_{0},{y}_{0})\in {X}^{2} such that {T}_{G}({x}_{0},{y}_{0})\u2291{T}_{F}({x}_{0},{y}_{0}).

(6)
Condition (3) is equivalent to {T}_{F}({X}^{2})\subseteq {T}_{G}({X}^{2}).

(7)
If there exist \varphi \in \mathrm{\Phi} and \psi \in \mathrm{\Psi} such that (2) holds, then
\varphi \left({\mathrm{\Delta}}_{2}({T}_{F}(x,y),{T}_{F}(u,v))\right)\le \varphi \left({\mathrm{\Delta}}_{2}({T}_{G}(x,y),{T}_{G}(u,v))\right){\psi}_{2}\left({\mathrm{\Delta}}_{2}({T}_{G}(x,y),{T}_{G}(u,v))\right)
for all (x,y),(u,v)\in {X}^{2} such that {T}_{G}(x,y)\u2291{T}_{G}(u,v), where {\psi}_{2}\in \mathrm{\Psi} was defined in Remark 2.1.

(8)
If the pair \{F,G\} is generalized compatible, then the mappings {T}_{F} and {T}_{G} are Ocompatible in ({X}^{2},{\mathrm{\Delta}}_{2},\u2291).

(9)
A point (x,y)\in {X}^{2} is a coupled coincidence point of F and G if and only if it is a coincidence point of {T}_{F} and {T}_{G}.
Proof Item (1) follows from Lemma 2.1 and items (2), (3), (5), (6) and (9) are obvious.

(4)
Assume that F is Gincreasing with respect to ⪯, and let (x,y),(u,v)\in {X}^{2} be such that {T}_{G}(x,y)\u2291{T}_{G}(u,v). Then G(x,y)\u2aafG(u,v) and G(y,x)\u2ab0G(v,u). Since F is Gincreasing with respect to ⪯, we deduce that F(x,y)\u2aafF(u,v) and F(y,x)\u2ab0F(v,u). Therefore, {T}_{F}(x,y)\u2291{T}_{F}(u,v) and this means that {T}_{F} is ({T}_{G},\u2291)nondecreasing.

(7)
Suppose that there exist \varphi \in \mathrm{\Phi} and \psi \in \mathrm{\Psi} such that (2) holds, and let (x,y),(u,v)\in {X}^{2} be such that {T}_{G}(x,y)\u2291{T}_{G}(u,v). Therefore G(x,y)\u2aafG(u,v) and G(y,x)\u2ab0G(v,u). Using (2), we have that
\begin{array}{rl}\varphi \left(d(F(x,y),F(u,v))\right)\le & \frac{1}{2}\varphi (d(G(x,y),G(u,v))+d(G(y,x),G(v,u)))\\ \psi \left(\frac{d(G(x,y),G(u,v))+d(G(y,x),G(v,u))}{2}\right).\end{array}(5)
Furthermore, taking into account that G(v,u)\u2aafG(y,x) and G(u,v)\u2ab0G(x,y), the contractivity condition (2) also guarantees that
Since ϕ is subadditive, it follows from (5) and (6) that

(8)
Let \{({x}_{m},{y}_{m})\}\subseteq {X}^{2} be any sequence such that \{{T}_{F}({x}_{m},{y}_{m})\}\stackrel{{\mathrm{\Delta}}_{2}}{\u27f6}(x,y) and \{{T}_{G}({x}_{m},{y}_{m})\}\stackrel{{\mathrm{\Delta}}_{2}}{\u27f6}(x,y) (notice that we do not need to suppose that \{{T}_{G}({x}_{m},{y}_{m})\} is ⊑monotone). Therefore,
\begin{array}{r}\left\{(F({x}_{m},{y}_{m}),F({y}_{m},{x}_{m}))\right\}\stackrel{{\mathrm{\Delta}}_{2}}{\u27f6}(x,y)\phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\left[\{F({x}_{m},{y}_{m})\}\stackrel{d}{\u27f6}x\text{and}\{F({y}_{m},{x}_{m})\}\stackrel{d}{\u27f6}y\right];\\ \left\{(G({x}_{m},{y}_{m}),G({y}_{m},{x}_{m}))\right\}\stackrel{{\mathrm{\Delta}}_{2}}{\u27f6}(x,y)\\ \phantom{\rule{1em}{0ex}}\Rightarrow \phantom{\rule{1em}{0ex}}\left[\{G({x}_{m},{y}_{m})\}\stackrel{d}{\u27f6}x\text{and}\{G({y}_{m},{x}_{m})\}\stackrel{d}{\u27f6}y\right].\end{array}
Therefore
Since the pair \{F,G\} is generalized compatible, we deduce that
In particular,
Hence, the mappings {T}_{F} and {T}_{G} are Ocompatible in ({X}^{2},{\mathrm{\Delta}}_{2},\u2291). □
As a consequence, we conclude that Hussain et al.’s result can be deduced from the corresponding unidimensional result. Furthermore, as we have pointed out, it is not necessary for G to have the mixed monotone property because F is Gincreasing with respect to ⪯.
Corollary 3.1 Theorem 1.1, even avoiding the assumption that G has the mixed monotone property, is a consequence of Theorem 3.1.
Proof It is only necessary to apply Theorem 3.1 to the mappings T={T}_{F} and g={T}_{G} in the ordered metric space ({X}^{2},{\mathrm{\Delta}}_{2},\u2291), taking into account all items of Lemma 3.1. □
The following result is an improved version of Theorem 1.1 in which the contractivity condition is replaced by a more convenient one, which is symmetric on the variables (x,y) and (u,v).
Corollary 3.2 Let (X,\u2aaf) be a partially ordered set such that there exists a complete metric d on X. Assume that F,G:X\times X\to X are two generalized compatible mappings such that F is Gincreasing with respect to ⪯, G is continuous and there exist two elements {x}_{0},{y}_{0}\in X such that
Suppose that there exist \varphi \in \mathrm{\Phi} and \psi \in \mathrm{\Psi} such that
for all x,y,u,v\in X with G(x,y)\u2aafG(u,v) and G(y,x)\u2ab0G(v,u). Suppose that for any x,y\in X, there exist u,v\in X such that
Also assume that either

(a)
F is continuous, or

(b)
(X,d,\u2aaf) is regular.
Then F and G have, at least, a coupled coincidence point, that is, there exist x,y\in X such that G(x,y)=F(x,y) and G(y,x)=F(y,x).
Proof It is only necessary to apply Theorem 3.1 to the mappings T={T}_{F} and g={T}_{G} in the ordered metric space ({X}^{2},{\mathrm{\Delta}}_{2}^{\prime},\u2291), where {\mathrm{\Delta}}_{2}^{\prime}={\mathrm{\Delta}}_{2}/2, taking into account all items of Lemma 3.1. □
In the following example we show that Corollary 3.2 is applicable to the mappings of Example 3.1, when Theorem 1.1 is not useful.
Example 3.2 Let X=[0,\mathrm{\infty}) endowed with the Euclidean metric d(x,y)=xy for all x,y\in X. Consider the maps F,G:X\times X\to X defined by
Then, for all x,y,u,v\in X with y=v, we have
Thus,
Regarding the properties of the functions in Φ, we derive that
To provide inequality (7), it is sufficient to choose \psi (t)=\frac{t}{10}. Hence, Theorem 3.2 can be applied in order to guarantee that F and G have a coupled coincidence point. Indeed, it is easy to check that (0,0) is a coupled coincidence point of F and G.
To finish the paper, we want to point out a pair of details.

(1)
The function in ϕ in Theorem 3.1 is not a true generalization because if \varphi \in \mathrm{\Phi}, then the mapping {d}_{\varphi}:X\times X\to [0,\mathrm{\infty}), defined by {d}_{\varphi}(x,y)=\varphi (d(x,y)) for all x,y\in X, is also a metric on X. For more details, see [26]. Notice also that the assumption of subadditivity ({\varphi}_{2}) is superfluous in most of the published results (see, e.g., [27]).

(2)
Using the same techniques that can be found in [17, 22, 28–31], it is possible to deduce, from Theorem 3.1, tripled, quadrupled and, in general, multidimensional coincidence point theorems.
References
Luong NV, Thuan NX: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 2011, 74: 983–992. 10.1016/j.na.2010.09.055
Abbas M, Rhoades BE: Common fixed point results for noncommuting mappings without continuity in generalized metric spaces. Appl. Math. Comput. 2009, 215: 262–269. 10.1016/j.amc.2009.04.085
Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74: 1799–1803. 10.1016/j.na.2010.10.052
Berinde V: Coupled fixed point theorems for ϕ contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2012, 75: 3218–3228. 10.1016/j.na.2011.12.021
Bilgili N, Karapınar E, Turkoglu D:A note on common fixed points for (\psi ,\alpha ,\beta )weakly contractive mappings in generalized metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 287
Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003
Turkoglu D, Sangurlu M: Coupled fixed point theorems for mixed g monotone mappings in partially ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 348
Roldán A, MartínezMoreno J, Roldán C: Multidimensional fixed point theorems in partially ordered complete metric spaces. J. Math. Anal. Appl. 2012, 396(2):536–545. 10.1016/j.jmaa.2012.06.049
Roldán A, MartínezMoreno J, Roldán C, Karapınar E:Multidimensional fixed point theorems in partially ordered complete partial metric spaces under (\psi ,\phi )contractivity conditions. Abstr. Appl. Anal. 2013., 2013: Article ID 634371
Karapınar E, Roldán A, MartínezMoreno J, Roldán C: MeirKeeler type multidimensional fixed point theorems in partially ordered metric spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 406026
Pathak HK, Shahzad N: Fixed point results for setvalued contractions by altering distances in complete metric spaces. Nonlinear Anal. 2009, 70(7):2634–2641. 10.1016/j.na.2008.03.050
Alshehri S, Aranđelović I, Shahzad N: Symmetric spaces and fixed points of generalized contractions. Abstr. Appl. Anal. 2014., 2014: Article ID 763547
Asl JH, Rezapour S, Shahzad N: On fixed points of α  ψ contractive multifunctions. Fixed Point Theory Appl. 2012., 2012: Article ID 212 10.1186/168718122012212
Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of α  ψ Ćirić generalized multifunctions. Fixed Point Theory Appl. 2013., 2013: Article ID 24 10.1186/16871812201324
Alikhani H, Rezapour S, Shahzad N: Fixed points of a new type contractive mappings and multifunctions. Filomat 2013, 27(7):1315–1319. 10.2298/FIL1307315A
Hussain N, Abbas M, Azam A, Ahmad J: Coupled coincidence point results for a generalized compatible pair with applications. Fixed Point Theory Appl. 2014., 2014: Article ID 62
AlMezel SA, Alsulami H, Karapınar E, Roldán A: Discussion on ‘Multidimensional coincidence points’ via recent publications. Abstr. Appl. Anal. 2014., 2014: Article ID 287492. http://www.hindawi.com/journals/aaa/aip/287492/
Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30: 1–9. 10.1017/S0004972700001659
Goebel K: A coincidence theorem. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 1968, 16: 733–735.
Agarwal RP, Bisht RK, Shahzad N: A comparison of various noncommuting conditions in metric fixed point theory and their applications. Fixed Point Theory Appl. 2014., 2014: Article ID 38
Bhaskar TG, Lakshmikantham V: Fixed point theory in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Roldán A, MartínezMoreno J, Roldán C, Karapınar E: Some remarks on multidimensional fixed point theorems. Fixed Point Theory Appl. 2013., 2013: Article ID 158
Choudhury BS, Kundu A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 2010, 73: 2524–2531. 10.1016/j.na.2010.06.025
Luong NV, Thuan NX: Coupled points in ordered generalized metric spaces and application to integrodifferential equations. Comput. Math. Appl. 2011, 62(11):4238–4248. 10.1016/j.camwa.2011.10.011
Hung NM, Karapınar E, Luong NV: Coupled coincidence point theorem for O compatible mappings via implicit relation. Abstr. Appl. Anal. 2012., 2012: Article ID 796964
Karapınar E, Samet B: A note on ψ Geraghty contractions. Fixed Point Theory Appl. 2014., 2014: Article ID 26
Karapınar E: A discussion on α  ψ Geraghty contracton type mappings and some related fixed point results. Filomat 2014, 28(1):37–48. 10.2298/FIL1401037K
Karapınar E, Roldán A: A note on ‘ n Tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces’. J. Inequal. Appl. 2013., 2013: Article ID 567
Karapınar E, Roldán A, Roldán C, MartínezMoreno J: A note on ‘ N Fixed point theorems for nonlinear contractions in partially ordered metric spaces’. Fixed Point Theory Appl. 2013., 2013: Article ID 310
Karapınar E, Roldán A, Shahzad N, Sintunavarat W: Discussion on coupled and tripled coincidence point theorems for ϕ contractive mappings without the mixed g monotone property. Fixed Point Theory Appl. 2014., 2014: Article ID 92
Samet B, Karapınar E, Aydi H, Rajic VC: Discussion on some coupled fixed point theorems. Fixed Point Theory Appl. 2013., 2013: Article ID 50
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. N Shahzad acknowledges with thanks DSR for financial support. Antonio Roldán has been partially supported by Junta de Andalucía by project FQM268 of the Andalusian CICYE.
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Erhan, İ.M., Karapınar, E., RoldánLópezdeHierro, AF. et al. Remarks on ‘Coupled coincidence point results for a generalized compatible pair with applications’. Fixed Point Theory Appl 2014, 207 (2014). https://doi.org/10.1186/168718122014207
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DOI: https://doi.org/10.1186/168718122014207
Keywords
 fixed point
 coupled coincidence point
 ordered metric space