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An original coupled coincidence point result for a pair of mappings without MMP
Journal of Inequalities and Applications volume 2014, Article number: 61 (2014)
Abstract
The purpose of this paper is to establish a coupled coincidence point theorem for a pair of mappings without MMP (mixed monotone property) in metric spaces endowed with partial order, which is not an immediate consequence of a wellknown theorem in the literature. Also, we present a result on the existence and uniqueness of coupled common fixed points. The results presented in the paper generalize and extend some of the results of Bhaskar and Lakshmikantham (Nonlinear Anal. 65:13791393, 2006), Choudhury, Metiya and Kundu (Ann. Univ. Ferrara 57:116, 2011), Harjani, Lopez and Sadarangani (Nonlinear Anal. 74:17491760, 2011) and of Luong and Thuan (Bull. Math. Anal. Appl. 2:1624, 2010) for the mappings having no MMP. We introduce an example that there exists a common coupled fixed point of the mappings g and F such that F does not satisfy the gmixed monotone property, and also g and F do not commute.
MSC:41A50, 47H10, 54H25.
1 Introduction and preliminaries
Fixed point theory is one of the famous and traditional theories in mathematics and has a large number of applications. The Banach contraction mapping is one of the pivotal results of analysis. It is a very popular tool for solving existence problems in many different fields of mathematics. There are a lot of generalizations of the Banach contraction principle in the literature. Ran and Reurings [1] extended the Banach contraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. While Nieto and RodŕiguezLópez [2] extended the result of Ran and Reurings and applied their main theorems to obtain a unique solution for a firstorder ordinary differential equation with periodic boundary conditions. Bhaskar and Lakshmikantham [3] introduced the concept of mixed monotone mappings and obtained some coupled fixed point results. Also, they applied their results on a firstorder differential equation with periodic boundary conditions. Recently, many researchers have obtained fixed point, common fixed point, coupled fixed point and coupled common fixed point results in cone metric spaces, fuzzy metric spaces, intuitionistic fuzzy normed spaces, partially ordered metric spaces and others (see [1–25]).
Definition 1.1 Let (X,d) be a metric space and F:X\times X\to X and g:X\to X, F and g are said to commute if F(gx,gy)=g(F(x,y)) for all x,y\in X.
Definition 1.2 Let (X,d) be a metric space and let g:X\to X, F:X\times X\to X. The mappings g and F are said to be compatible if {lim}_{n\to \mathrm{\infty}}d(gF({x}_{n},{y}_{n}),F(g{x}_{n},g{y}_{n}))=0 and {lim}_{n\to \mathrm{\infty}}d(gF({y}_{n},{x}_{n}),F(g{y}_{n},g{x}_{n}))=0 hold whenever \{{x}_{n}\} and \{{y}_{n}\} are sequences in X such that {lim}_{n\to \mathrm{\infty}}F({x}_{n},{y}_{n})={lim}_{n\to \mathrm{\infty}}g{x}_{n} and {lim}_{n\to \mathrm{\infty}}F({y}_{n},{x}_{n})={lim}_{n\to \mathrm{\infty}}g{y}_{n}.
Definition 1.3 Let (X,\u2aaf) be a partially ordered set and F:X\times X\to X. The mapping F is said to be nondecreasing if for x,y\in X, x\u2aafy implies F(x)\u2aafF(y) and nonincreasing if for x,y\in X, x\u2aafy implies F(x)\u2ab0F(y).
Definition 1.4 Let (X,\u2aaf) be a partially ordered set and F:X\times X\to X and g:X\to X. The mapping F is said to have the mixed gmonotone property if F(x,y) is monotone gnondecreasing in x and monotone gnonincreasing in y, that is, for any x,y\in X,
and
If g= identity mapping in Definition 1.4, then the mapping F is said to have the mixed monotone property.
Recently, Ðoric et al. [12] showed that the mixed monotone property in coupled fixed point results for mappings in ordered metric spaces can be replaced by another property which is often easy to check. In particular, it is automatically satisfied in the case of a totally ordered space, the case which is important in applications. Hence, these results can be applied in a much wider class of problems.
If elements x, y of a partially ordered set (X,\u2aaf) are comparable (i.e., x\u2aafy or y\u2aafx holds) we will write x\asymp y. Let g:X\to X and F:X\times X\to X. We will consider the following condition:
If g is an identity mapping, for all x, y, v, if x\asymp F(x,y) then F(x,y)\asymp F(F(x,y),v).
Ðoric et al. [12] gave some examples that these conditions may be satisfied when F does not have the gmixed monotone property.
Definition 1.5 An element (x,y)\in X\times X is called a coupled coincidence point of the mappings F:X\times X\to X and g:X\to X if F(x,y)=gx and F(y,x)=gy.
If g= identity mapping in Definition 1.5, then (x,y)\in X\times X is called a coupled fixed point.
The purpose of this paper is to establish some coupled coincidence point results in partially ordered metric spaces for a pair of mappings without mixed monotone property satisfying a contractive condition. Also, we present a result on the existence and uniqueness of coupled common fixed points. Also, we give an example to illustrate the main result in this paper. The results proved generalize some of the results of Bhaskar and Lakshmikantham [3], Choudhury et al. [10], Luong and Thuan [17] and Harjani et al. [13] for the mappings having no mixed monotone property.
2 Main results
2.1 Coupled common fixed point theorems
In this section, we prove some coupled common fixed point theorems in the context of ordered metric spaces.
We denote by Φ the set of functions \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) satisfying:

(i)
ϕ is continuous;

(ii)
\varphi (t)<t for all t>0 and \varphi (t)=0 if and only if t=0.
Theorem 2.1 Let (X,\u2aaf) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Suppose that F:X\times X\to X and g:X\to X are selfmappings on X such that the following conditions hold:

(i)
g is continuous and g(X) is closed;

(ii)
F(X\times X)\subseteq g(X) and g and F are compatible;

(iii)
for all x,y,u,v\in X, if g(x)\asymp F(x,y)=gu, then F(x,y)\asymp F(u,v);

(iv)
there exist {x}_{0},{y}_{0}\in X such that g{x}_{0}\asymp F({x}_{0},{y}_{0}) and g{y}_{0}\asymp F({y}_{0},{x}_{0});

(v)
there exists a nonnegative real number L such that
\begin{array}{rcl}d(F(x,y),F(u,v))& \le & \varphi (max\{d(gx,gu),d(gy,gv)\})\\ +Lmin\{d(F(x,y),gu),d(F(u,v),gx),\\ d(F(x,y),gx),d(F(u,v),gu)\}\end{array}(2.1)
for all x,y,u,v\in X with gx\u2ab0gu and gy\u2aafgv, where \varphi \in \mathrm{\Phi};

(vi)
(a) F is continuous or (b) {x}_{n}\to x, when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for sufficiently large n.
Then there exist x,y\in X such that F(x,y)=g(x) and F(y,x)=g(y), that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Proof Using conditions (ii) and (iv), construct sequences \{{x}_{n}\} and \{{y}_{n}\} in X satisfying g{x}_{n}=F({x}_{n1},{y}_{n1}) and g{y}_{n}=F({y}_{n1},{x}_{n1}) for n=1,2,\dots .
By (iv), g{x}_{0}\asymp F({x}_{0},{y}_{0})=g{x}_{1} and condition (iii) implies that g{x}_{1}=F({x}_{0},{y}_{0})\asymp F({x}_{1},{y}_{1})=g{x}_{2}. Proceeding by induction, we get that g{x}_{n1}\asymp g{x}_{n}, and similarly, g{y}_{n1}\asymp g{y}_{n} for each n\in \mathbb{N}.
Now from the contractive condition (2.1), we have
which implies that d(g{x}_{n+1},g{x}_{n})\le \varphi (max\{d(g{x}_{n},g{x}_{n1}),d(g{y}_{n},g{y}_{n1})\}).
Similarly, we have d(g{y}_{n+1},g{y}_{n})\le \varphi (max\{d(g{y}_{n},g{y}_{n1}),d(g{x}_{n},g{x}_{n1})\}).
Therefore, from the above two inequalities we have
Since \varphi (t)<t for all t>0 and \varphi (t)=0 if and only if t=0, from (2.3) we have
Set {\varrho}_{n}:=max\{d(g{x}_{n+1},g{x}_{n}),d(g{y}_{n+1},g{y}_{n})\}, then \{{\varrho}_{n}\} is a nonincreasing sequence of positive real numbers. Thus, there is d\ge 0 such {lim}_{n\to \mathrm{\infty}}{\varrho}_{n}=d.
Suppose that d>0, letting n\to \mathrm{\infty} two sides of (2.3) and using the properties of ϕ, we have
which is a contradiction. Hence d=0, i.e.,
Now, we shall prove that \{g{x}_{n}\} and \{g{y}_{n}\} are Cauchy sequences. Suppose, to the contrary, that at least one of \{g{x}_{n}\} or \{g{y}_{n}\} is not a Cauchy sequence. This means that there exists an \u03f5>0 for which we can find subsequences \{g{x}_{n(k)}\}, \{g{x}_{m(k)}\} of \{g{x}_{n}\} and \{g{y}_{n(k)},g{y}_{m(k)}\} of \{g{y}_{n}\} with n(k)>m(k)\ge k such that
Further, corresponding to m(k), we can choose n(k) in such a way that it is the smallest integer with n(k)>m(k)\ge k and satisfies (2.6). Then
Using the triangle inequality and (2.7), we have
and
From (2.6), (2.8) and (2.9), we have
Letting k\to \mathrm{\infty} in the inequalities above and using (2.5), we get
By the triangle inequality,
and
From the last two inequalities and (2.6), we have
Again, by the triangle inequality,
and
Therefore,
Taking k\to \mathrm{\infty} in (2.11) and (2.12) and using (2.5), (2.10), we have
Since n(k)>m(k), g{x}_{n(k)1}\u2ab0g{x}_{m(k)1} and g{y}_{n(k)1}\u2aafg{y}_{m(k)1}. Then from (2.1) we have
Similarly,
From (2.14) and (2.15), we have
Letting n\to \mathrm{\infty} in the above inequality and using (2.5), (2.10), (2.13) and the properties of ϕ, we have
which is a contradiction. Therefore, \{g{x}_{n}\} and \{g{y}_{n}\} are Cauchy sequences and since g(X) is closed in a complete metric space (condition (i)), there exist x,y\in g(X) such that {lim}_{n\to \mathrm{\infty}}g{x}_{n}={lim}_{n\to \mathrm{\infty}}F({x}_{n1},{y}_{n1})=x and {lim}_{n\to \mathrm{\infty}}g{y}_{n}={lim}_{n\to \mathrm{\infty}}F({y}_{n1},{x}_{n1})=y.
Compatibility of F and g (condition (ii)) implies that
and
Consider the two possibilities given in condition (vi).

(a)
Suppose that F is continuous. Using the triangle inequality, we get that
d(gx,F(g{x}_{n},g{y}_{n}))\le d(gx,g(F({x}_{n},{y}_{n})))+d(g(F({x}_{n},{y}_{n})),F(g{x}_{n},g{y}_{n})).
By taking limit n\to \mathrm{\infty} and using the continuity of F and g, we have d(gx,F(x,y))=0, i.e., gx=F(x,y) and, in a similar way, we have gy=F(y,x). Thus F and g have a coupled coincidence point.

(b)
In this case g{x}_{n}\asymp u=gx and g{y}_{n}\asymp v=gy for some x,y\in X and n sufficiently large. For such n, using (2.1) we get
\begin{array}{rcl}d(F(x,y),gx)& \le & d(F(x,y),g{x}_{n+1})+d(g{x}_{n+1},gx)\\ =& d(F(x,y),F({x}_{n},{y}_{n}))+d(g{x}_{n+1},gx)\\ \le & \varphi (max\{d(gx,g{x}_{n}),d(gy,g{y}_{n})\})\\ +Lmin\{d(F(x,y),g{x}_{n}),d(F({x}_{n},{y}_{n}),g{x}_{n}),\\ d(F(x,y),gx),d(F({x}_{n},{y}_{n}),g{x}_{n})\}+d(g{x}_{n+1},gx).\end{array}
Taking n\to \mathrm{\infty} in the above inequality and using the compatibility of F and g and the properties of ϕ, we have d(F(x,y),gx)\le \varphi (max\{0,0\})+0+0=0. Hence F(x,y)=gx. Similarly, one can show that F(y,x)=gy. Hence the result. □
Remark 2.1 Very recently, using the equivalence of the three basic metrics, Samet et al. [19] show that many of the coupled fixed point theorems are immediate consequences of wellknown fixed point theorems in the literature.
In our Theorem 2.1, it is easy to see that if L\ne 0 there is no equivalence and this theorem is not a consequence of a known fixed point theorem.
Remark 2.2 In the above theorem, condition (iii) is a substitution for the mixed monotone property that has been used in most of the coupled fixed point results so far. Note that this condition is trivially satisfied if the order ⪯ on X is total, which is the case in most of the examples in articles mentioned in the references.
If g is an identity mapping in the above theorem, we have the following result.
Corollary 2.2 Let (X,d,\le ) be a complete partially ordered metric space and let F:X\times X\to X. Suppose that the following hold:

(i)
for all x,y,v\in X, if x\asymp F(x,y), then F(x,y)\asymp F(F(x,y),v);

(ii)
there exist {x}_{0},{y}_{0}\in X such that {x}_{0}\asymp F({x}_{0},{y}_{0}) and {y}_{0}\asymp F({y}_{0},{x}_{0});

(iii)
there exists a nonnegative real number L such that
\begin{array}{rcl}d(F(x,y),F(u,v))& \le & \varphi (max\{d(x,u),d(y,v)\})\\ +Lmin\{d(F(x,y),u),d(F(u,v),x),\\ d(F(x,y),x),d(F(u,v),u)\}\end{array}
for all x,y,u,v\in X with x\u2ab0u and y\u2aafv, where \varphi \in \mathrm{\Phi};

(iv)
(a) F is continuous or (b) {x}_{n}\to x, when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for sufficiently large n.
Then there exist x,y\in X such that F(x,y)=x and y=F(y,x), that is, F has a coupled fixed point (x,y)\in X\times X.
Remark 2.3 Letting L=0, in inequality (2.1), for all x,y,u,v\in X, \alpha ,\beta \ge 0, \alpha +\beta <1, we have
where \varphi (t)=(\alpha +\beta )(t) for all t\ge 0 is in Φ. Hence Theorem 2.1 generalizes the corresponding coupled fixed point results of Bhaskar and Lakshmikantham [3], Choudhury et al. [10], Luong and Thuan [17] and Harjani et al. [13] for the mappings having no mixed monotone property.
Taking L=0, we have the following result.
Corollary 2.3 Let (X,\u2aaf) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Suppose that F:X\times X\to X and g:X\to X are selfmappings on X such that the following conditions hold:

(i)
g is continuous and g(X) is closed;

(ii)
F(X\times X)\subseteq g(X) and g and F are compatible;

(iii)
for all x,y,u,v\in X, if g(x)\asymp F(x,y)=gu, then F(x,y)\asymp F(u,v);

(iv)
there exist {x}_{0},{y}_{0}\in X such that g{x}_{0}\asymp F({x}_{0},{y}_{0}) and g{y}_{0}\asymp F({y}_{0},{x}_{0});

(v)
F and g satisfy
d(F(x,y),F(u,v))\le \varphi (max\{d(gx,gu),d(gy,gv)\})(2.16)
for all x,y,u,v\in X with gx\u2ab0gu and gy\u2aafgv, where \varphi \in \mathrm{\Phi};

(vi)
(a) F is continuous or (b) {x}_{n}\to x, when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for sufficiently large n.
Then there exist x,y\in X such that F(x,y)=g(x) and gy=F(y,x), that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Corollary 2.4 Let (X,\u2aaf) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Suppose that F:X\times X\to X and g:X\to X are selfmappings on X such that following conditions hold:

(i)
g is continuous and g(X) is closed;

(ii)
F(X\times X)\subseteq g(X) and g and F are compatible;

(iii)
for all x,y,u,v\in X, if g(x)\asymp F(x,y)=gu, then F(x,y)\asymp F(u,v);

(iv)
there exist {x}_{0},{y}_{0}\in X such that g{x}_{0}\asymp F({x}_{0},{y}_{0}) and g{y}_{0}\asymp F({y}_{0},{x}_{0});

(v)
there exist nonnegative real numbers α, β with \alpha +\beta <1 such that
d(F(x,y),F(u,v))\le \alpha d(gx,gu)+\beta d(gy,gv)(2.17)
for all x,y,u,v\in X with gx\u2ab0gu and gy\u2aafgv;

(vi)
(a) F is continuous or (b) {x}_{n}\to x, when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for sufficiently large n.
Then there exist x,y\in X such that F(x,y)=g(x) and gy=F(y,x), that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Taking \alpha =\beta =k\in [0,1), we have the following result.
Corollary 2.5 Let (X,\u2aaf) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Suppose that F:X\times X\to X and g:X\to X are selfmappings on X such that following conditions hold:

(i)
g is continuous and g(X) is closed;

(ii)
F(X\times X)\subseteq g(X) and g and F are compatible;

(iii)
for all x,y,u,v\in X, if g(x)\asymp F(x,y)=gu, then F(x,y)\asymp F(u,v);

(iv)
there exist {x}_{0},{y}_{0}\in X such that g{x}_{0}\asymp F({x}_{0},{y}_{0}) and g{y}_{0}\asymp F({y}_{0},{x}_{0});

(v)
there exists k\in [0,1) such that
d(F(x,y),F(u,v))\le k\phantom{\rule{0.25em}{0ex}}[d(gx,gu)+d(gy,gv)](2.18)
for all x,y,u,v\in X with gx\u2ab0gu and gy\u2aafgv;

(vi)
(a) F is continuous or (b) {x}_{n}\to x, when n\to \mathrm{\infty} in X, then {x}_{n}\asymp x for sufficiently large n.
Then there exist x,y\in X such that F(x,y)=g(x) and gy=F(y,x), that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Now, we shall prove the existence and uniqueness of a coupled common fixed point. Note that if (X,\u2aaf) is a partially ordered set, then we endow the product space X\times X with the following partial order relation:
Theorem 2.6 In addition to hypotheses of Theorem 2.1, suppose that

(vii)
for every (x,y),(u,v)\in X\times X, there exists (w,z)\in X\times X such that (F(w,z),F(z,w)) is comparable to (F(x,y),F(y,x)) and (F(u,v),F(v,u)).
Then F and g have a unique coupled common fixed point, that is, there exists a unique (p,q)\in X\times X such that p=gp=F(p,q) and q=gq=F(q,p).
Proof From Theorem 2.1, there exists (x,y)\in X\times X such that gx=F(x,y) and gy=F(y,x). Suppose that there is also (u,v)\in X\times X such that gu=F(u,v) and gv=F(v,u). We will prove that gx=gu and gy=gv. Condition (vii) implies that there exists (w,z)\in X\times X such that (F(w,z),F(z,w)) is comparable to both (F(x,y),F(y,x)) and (F(u,v),F(v,u)). Put {w}_{0}=w, {z}_{0}=z and, analogously to the proof of Theorem 2.1, choose sequences \{{w}_{n}\}, \{{z}_{n}\} satisfying
for n\in \mathbb{N}. Starting from {x}_{0}=x, {y}_{0}=y and {u}_{0}=u, {v}_{0}=v, choose sequences \{{x}_{n}\}, \{{y}_{n}\} and \{{u}_{n}\}, \{{v}_{n}\}, satisfying g{x}_{n}=F({x}_{n1},{y}_{n1}), g{y}_{n}=F({y}_{n1},{x}_{n1}) and g{u}_{n}=F({u}_{n1},{v}_{n1}), g{v}_{n}=F({v}_{n1},{u}_{n1}) for n\in \mathbb{N}, taking into account properties of coincidence points, it is easy to see that this can be done so that {x}_{n}=x, {y}_{n}=y and {u}_{n}=u, {v}_{n}=v, i.e.,
Since (F(x,y),F(y,x))=(gx,gy) and (F(w,z),F(z,w))=(g{w}_{1},g{z}_{1}) are comparable, then gx\asymp g{w}_{1} and gy\asymp g{z}_{1}, and, in a similar way, we have gx\asymp g{w}_{n} and gy\asymp g{z}_{n}. Thus from (2.1) we have
which implies that d(gx,g{w}_{n+1})\le \varphi (max\{d(gx,g{w}_{n}),d(gy,g{z}_{n})\}).
Similarly, we can prove that d(gy,g{z}_{n+1})\le \varphi (max\{d(gx,g{w}_{n}),d(gy,g{z}_{n})\}).
Therefore, from the above two inequalities we have
Since \varphi (t)\le t for all t\ge 0, from (2.20) we have
Hence the sequence \{{\delta}_{n}\} defined by {\delta}_{n}:=max\{d(gx,g{w}_{n+1}),d(gy,g{z}_{n+1})\} is nonnegative and decreasing and so {lim}_{n\to \mathrm{\infty}}{\delta}_{n}=\delta for some \delta \ge 0.
Now, we show that \delta =0. Assume that \delta >0, letting n\to \mathrm{\infty} two sides of (2.20) and using the properties of ϕ, we have
which is a contradiction. Hence d=0, i.e.,
Similarly, we can prove that
Using relations (2.22) and (2.23), together with the triangle inequality, we have d(gx,gu)=0 and d(gy,gv)=0 and so gx=gu and gy=gv.
Denote gx=p and gy=q. So, we have that
By definition of the sequences \{{x}_{n}\} and \{{y}_{n}\} we have
and so
as well as
Compatibility of g and F implies that
i.e., gF(x,y)=F(gx,gy). This together with (2.24) implies that gp=F(p,q) and, in a similar way, gq=F(q,p). Thus, we have another coincidence, and by the property we have just proved, it follows that gp=gx=p and gq=gy=q. In other words, p=gp=F(p,q) and q=gq=F(q,p), and (p,q) is a common coupled fixed point of g and F.
To prove the uniqueness, assume that (r,s) is another coupled common fixed point. Then by (2.24) we have r=gr=gp=p and s=gs=gq=q. Hence we get the result. □
Example 2.7 Let X=[0,1]. Then (X,\le ) is a partially ordered set with the natural ordering of real numbers. Let d(x,y)=xy for all x,y\in X. Define a mapping g:X\to X by g(x)={x}^{2} and a mapping F:X\times X\to X by
Then it is easy to check all the conditions of Theorems 2.1 and 2.6. In particular, we will check that g and F are compatible.
Let \{{x}_{n}\} and \{{y}_{n}\} be two sequences in X such that
Then \frac{a2b}{4}=a and \frac{b2a}{4}=b, where from it follows that a=b=0. Then
and similarly d(gF({y}_{n},{x}_{n}),F(g{y}_{n},g{x}_{n}))\to 0.
Now, we verify inequality (2.1) of Theorem 2.1 for \varphi (t)=\frac{3}{4}t, t>0 and L=[0,\mathrm{\infty}) for all x,y,u,v\in X with gx\u2ab0gu and gy\u2aafgv.
Thus there exists a common coupled fixed point (0,0) of the mappings g and F. Note that F does not satisfy the gmixed monotone property. Also, g and F do not commute.
References
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004,132(5):1435–1443. 10.1090/S0002993903072204
Nieto JJ, López RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s1108300590185
Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Agarwal RP, ElGebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 1–8. 10.1080/00036810701714164
Chandok S: Some common fixed point theorems for generalized f weakly contractive mappings. J. Appl. Math. Inform. 2011, 29: 257–265.
Chandok, S, Cho, YJ: Coupled common fixed point theorems for mixed gmonotone mappings in partially ordered metric spaces. An. Univ. Oradea, Fasc. Mat. 21(1) (2014, in press)
Chandok S, Dinu S: Common fixed points for weak ψ contractive mappings in ordered metric spaces with applications. Abstr. Appl. Anal. 2013., 2013: Article ID 879084
Chandok, S, Khan, MS, Abbas, M: Common fixed point theorems for nonlinear weakly contractive mappings. Ukr. Math. J. (2013, in press)
Chandok S, Khan MS, Rao KPR: Some coupled common fixed point theorems for a pair of mappings satisfying a contractive condition of rational type without monotonicity. Int. J. Math. Anal. 2013,7(9):433–440.
Choudhury BS, Metiya N, Kundu A: Coupled coincidence point theorems in ordered metric spaces. Ann. Univ. Ferrara 2011, 57: 1–16. 10.1007/s1156501101175
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
Ðoric D, Kadelburg Z, Radenović S: Coupled fixed point results for mappings without mixed monotone property. Appl. Math. Lett. 2012. 10.1016/j.aml.2012.02.022
Harjani J, Lopez B, Sadarangani K: Fixed point theorems for mixed monotone operators and applications to integral equations. Nonlinear Anal. 2011, 74: 1749–1760. 10.1016/j.na.2010.10.047
Jachymski J: Equivalent conditions for generalized contractions on (ordered) metric spaces. Nonlinear Anal. 2011, 74: 768–774. 10.1016/j.na.2010.09.025
Kim JK, Chandok S: Coupled common fixed point theorems for generalized nonlinear contraction mappings with the mixed monotone property in partially ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 307
Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
Luong NV, Thuan NX: Coupled fixed point theorems in partially ordered metric spaces. Bull. Math. Anal. Appl. 2010, 2: 16–24.
Samet B: Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026
Samet B, Karapınar E, Aydi H, Rajic VC: Discussion on some coupled fixed point theorems. Fixed Point Theory Appl. 2013., 2013: Article ID 50
Karapınar E, Luong NV, Thuan NX: Coupled coincidence points for mixed monotone operators in partially ordered metric spaces. Arab. J. Math. 2012, 1: 329–339. 10.1007/s4006501200270
Kutbi MA, Azam A, Ahmad J, Di Bari C: Some common coupled fixed point results for generalized contraction in complexvalued metric spaces. J. Appl. Math. 2013., 2013: Article ID 352927
Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81
Sintunavarat W, Petruşel A, Kumam P: Common coupled fixed point theorems for w compatible mappings without mixed monotone property. Rend. Circ. Mat. Palermo 2012, 61: 361–383. 10.1007/s1221501200960
Sintunavarat W, Kumam P, Cho YJ: Coupled fixed point theorems for nonlinear contractions without mixed monotone property. Fixed Point Theory Appl. 2012., 2012: Article ID 170
Agarwal RP, Sintunavarat W, Kumam P: Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property. Fixed Point Theory Appl. 2013., 2013: Article ID 22
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Chandok, S., Tas, K. An original coupled coincidence point result for a pair of mappings without MMP. J Inequal Appl 2014, 61 (2014). https://doi.org/10.1186/1029242X201461
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DOI: https://doi.org/10.1186/1029242X201461
Keywords
 coupled fixed point
 mixed monotone property
 ordered metric spaces