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Coupled fixed point theorems for mixed gmonotone mappings in partially ordered metric spaces
Fixed Point Theory and Applications volume 2013, Article number: 348 (2013)
Abstract
In this paper, we prove some coupled coincidence point results for mixed gmonotone mappings in partially ordered metric spaces. The main results of this paper are generalizations of the main results of Luong and Thuan (Nonlinear Anal. 74:983992, 2011).
MSC:54H25, 47H10.
1 Introduction and preliminaries
Fixed point theory plays a major role in mathematics. The Banach contraction principle [1] is the simplest one corresponding to fixed point theory. So a large number of mathematicians have extended it and have been interested in fixed point theory in some metric spaces. One of these spaces is a partially ordered metric space, that is, metric spaces endowed with a partial ordering. The first result in this direction was given by Ran and Reurings [2] who presented their applications to a matrix equation. Subsequently, the existence of solutions for matrix equations or ordinary differential equations by applying fixed point theorems were presented in [2–7].
The existence of a fixed point for contraction type mappings in partially ordered metric spaces has been considered by Ran and Reurings [2], Bhaskar and Lakshmikantham [4], Nieto and RodriquezLopez [6, 7], Lakshmikantham and Ćirić [8], Agarwal et al. [9] and Samet [10]. Bhaskar and Lakshmikantham [4] introduced the notion of coupled fixed point and proved some coupled fixed point theorems for mappings satisfying the mixed monotone property and discussed the existence and uniqueness of a solution for a periodic boundary value problem. Lakshmikantham and Ćirić [8] introduced the concept of a mixed gmonotone mapping and proved coupled coincidence and common fixed point theorems that extend theorems from [4]. Subsequently, many authors obtained several coupled coincidence and coupled fixed point theorems in some ordered metric spaces [11–27].
Definition 1 ([4])
Let (X,\le ) be a partially ordered set and F:X\times X\to X. The mapping F is said to have the mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nondecreasing in y, that is, for any x,y\in X,
and
Definition 2 ([4])
An element (x,y)\in X\times X is called a coupled fixed point of the mapping F:X\times X\to X if F(x,y)=x, F(y,x)=y.
Definition 3 ([8])
An element (x,y)\in X\times X is called a coupled coincidence point of mappings F:X\times X\to X and g:X\to X if F(x,y)=gx, F(y,x)=gy.
Definition 4 ([8])
Let X be nonempty set and F:X\times X\to X and g:X\to X. We say F and g are commutative if gF(x,y)=F(gx,gy) for all x,y\in X.
Definition 5 ([8])
Let (X,\le ) be a partially ordered set and F:X\times X\to X, g:X\to X be mappings. The mapping F is said to have the mixed gmonotone property if F is monotone gnondecreasing in its first argument and is monotone gnonincreasing in the second argument, that is, for any x,y\in X,
and
Lemma 1 ([28])
Let X be a nonempty set and F:X\times X\to X and g:X\to X be mappings. Then there exists a subset E\subseteq X such that g(E)=g(X) and g:E\to X is onetoone.
Theorem 1 ([4])
Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a continuous mapping having the mixed monotone property on X. Assume that there exists k\in [0,1) with
If there exist two elements {x}_{0},{y}_{0}\in X with
then there exist x,y\in X such that
Theorem 2 ([4])
Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Assume that X has the following property:

(1)
if a nondecreasing sequence \{{x}_{n}\}\to x, then {x}_{n}\le x for all n\in \mathbb{N},

(2)
if a nonincreasing sequence \{{y}_{n}\}\to y, then y\le {y}_{n} for all n\in \mathbb{N}.
Let F:X\times X\to X be a mapping having the mixed monotone property on X. Assume that there exists k\in [0,1) with
If there exist two elements {x}_{0},{y}_{0}\in X with
then there exist x,y\in X such that
Theorem 3 ([29])
Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a mapping having the mixed monotone property on X and there exist two elements {x}_{0},{y}_{0}\in X with {x}_{0}\le F({x}_{0},{y}_{0}) and {y}_{0}\ge F({y}_{0},{x}_{0}). Suppose that F, g satisfy
for all x,y,u,v\in X with x\ge u and y\le v. Suppose that either

(1)
F is continuous or

(2)
X has the following property:

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

(b)
if a nonincreasing sequence \{{y}_{n}\}\to y, then y\le {y}_{n} for all n\in \mathbb{N}.
Then there exist x,y\in X such that
that is, F has a coupled fixed point in X.
2 The main results
In this paper, we prove coupled coincidence and common fixed point theorems for mixed gmonotone mappings satisfying more general contractive conditions in partially ordered metric spaces. We also present results on existence and uniqueness of coupled common fixed points. Our results improve those of Luong and Thuan [29]. Our work generalizes, extends and unifies several well known comparable results in the literature.
Let Φ denote all functions \phi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) which satisfy

(1)
φ is continuous and nondecreasing,

(2)
\phi (t)=0 and only if t=0,

(3)
\phi (t+s)\le \phi (t)+\phi (s), \mathrm{\forall}t,s\in [0,\mathrm{\infty})
and Ψ denote all functions \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty}) which satisfy {lim}_{t\to r}\psi (t)>0 for all r>0 and {lim}_{t\to {0}^{+}}\psi (t)=0.
Theorem 4 Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a mapping having the mixed monotone property on X and there exist two elements {x}_{0},{y}_{0}\in X with {x}_{0}\le F({x}_{0},{y}_{0}) and {y}_{0}\ge F({y}_{0},{x}_{0}). Suppose that F, g satisfy
for all x,y,u,v\in X with gx\le gu and gy\ge gv, F(X\times X)\subseteq g(X), g(X) is complete and g is continuous.
Suppose that either

(1)
F is continuous or

(2)
X has the following property:

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

(b)
if a nonincreasing sequence \{{y}_{n}\}\to y, then y\le {y}_{n} for all n\in \mathbb{N}.
Then there exist x,y\in X such that
that is, F and g have a coupled coincidence point in X\times X.
Proof Using Lemma 1, there exists E\subseteq X such that g(E)=g(X) and g:E\to X is onetoone. We define a mapping A:g(E)\times g(E)\to X by
As g is onetoone on g(E), so A is well defined. Thus, it follows from (2.1) and (2.2) that
for all gx,gy,gu,gv\in g(E) with gx\le gu and gy\ge gv. Since F has the mixed gmonotone property, for all x,y\in X, we have
and
Thus, it follows from (2.2), (2.4) and (2.5) that, for all gx,gy\in g(E),
and
which implies that A has the mixed monotone property.
Suppose that assumption (1) holds. Since F is continuous, A is also continuous. Using Theorem 3 with the mapping A, it follows that A has a coupled fixed point (u,v)\in g(E)\times g(E).
Suppose that assumption (2) holds. We can conclude similarly in the proof of Theorem 3 that the mapping A has a coupled fixed point (u,v)\in g(X)\times g(X).
Finally, we prove that F and g have a coupled fixed point in X. Since (u,v) is a coupled fixed point of A, we get
Since (u,v)\in g(X)\times g(X), there exists a point ({u}^{\prime},{v}^{\prime})\in X\times X such that
Thus, it follows from (2.6) and (2.7) that
Also, from (2.2) and (2.8), we get
Therefore, ({u}^{\prime},{v}^{\prime}) is a coupled coincidence point of F and g. This completes the proof. □
Corollary 1 Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a mapping having the mixed monotone property on X and there exist two elements {x}_{0},{y}_{0}\in X with {x}_{0}\le F({x}_{0},{y}_{0}) and {y}_{0}\ge F({y}_{0},{x}_{0}). Suppose that F, g satisfy
for all x,y,u,v\in X with gx\le gu and gy\ge gv, F(X\times X)\subseteq g(X), g(X) is complete and g is continuous.
Suppose that either

(1)
F is continuous or

(2)
X has the following property:

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

(b)
if a nonincreasing sequence \{{y}_{n}\}\to y, then y\le {y}_{n} for all n\in \mathbb{N}.
Then there exist x,y\in X such that
that is, F and g have a coupled coincidence point in X\times X.
Proof In Theorem 4, taking \phi (t)=t, we get Corollary 1. □
Corollary 2 Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a mapping having the mixed monotone property on X, and there exist two elements {x}_{0},{y}_{0}\in X with {x}_{0}\le F({x}_{0},{y}_{0}) and {y}_{0}\ge F({y}_{0},{x}_{0}). Suppose that F, g satisfy
for all x,y,u,v\in X with gx\le gu and gy\ge gv, F(X\times X)\subseteq g(X), g(X) is complete and g is continuous.
Suppose that either

(1)
F is continuous or

(2)
X has the following property:

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

(b)
if a nonincreasing sequence \{{y}_{n}\}\to y, then y\le {y}_{n} for all n\in \mathbb{N}.
Then there exist x,y\in X such that
that is, F and g have a coupled coincidence point in X\times X.
Proof In Corollary 1, taking \psi (t)=\frac{1k}{2}t, we get Corollary 2. □
Theorem 5 Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let (X,\le ) be a partially ordered set and suppose that there exists a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a mapping having the mixed monotone property on X and there exist two elements {x}_{0},{y}_{0}\in X with {x}_{0}\le F({x}_{0},{y}_{0}) and {y}_{0}\ge F({y}_{0},{x}_{0}). Suppose that F, g satisfy
for all x,y,u,v\in X with gx\le gu and gy\ge gv, F(X\times X)\subseteq g(X), g(X) is complete and g is continuous.
Suppose that either

(1)
F is continuous or

(2)
X has the following property:

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

(b)
if a nonincreasing sequence \{{y}_{n}\}\to y, then y\le {y}_{n} for all n\in \mathbb{N}.
Then there exist x,y\in X such that
and
that is, F and g have a coupled common fixed point (x,y)\in X\times X.
Proof Following the proof of Theorem 4, F and g have a coupled coincidence point. We only have to show that x=gx and y=gy.
Now, {x}_{0} and {y}_{0} are two points in the statement of Theorem 4. Since F(X\times X)\subseteq g(X), we can choose {x}_{1},{y}_{1}\in X such that g{x}_{1}=F({x}_{0},{y}_{0}) and g{y}_{1}=F({y}_{0},{x}_{0}). In the same way, we construct g{x}_{2}=F({x}_{1},{y}_{1}) and g{y}_{2}=F({y}_{1},{x}_{1}). Continuing in this way, we can construct two sequences \{{x}_{n}\} and \{{y}_{n}\} in X such that
Since gx\ge g{x}_{n+1} and gy\le g{y}_{n+1}, from (2.1) and (2.9), we have
Similarly, since g{y}_{n+1}\ge gy and g{x}_{n+1}\le gx, from (2.1) and (2.9), we have
From (2.10) and (2.11), we have
By property (3) of φ, we have
From (2.12) and (2.13), we have
which implies
Using the fact that φ is nondecreasing, we get
Set {\delta}_{n}=d(g{x}_{n+1},gx)+d(g{y}_{n+1},gy), then sequence \{{\delta}_{n}\} is decreasing. Therefore, there is some \delta \ge 0 such that
We shall show that \delta =0. Suppose, to the contrary, that \delta >0. Then taking the limit as n\to \mathrm{\infty} (equivalently, {\delta}_{n}\to \delta) of both sides of (2.13) and having in mind that we suppose that {lim}_{t\to r}\psi (t)>0 for all r>0 and φ is continuous, we have
a contradiction. Thus \delta =0, that is,
Hence d(g{x}_{n+1},gx)=0 and d(g{y}_{n+1},gy)=0, that is, x=gx and y=gy. □
Theorem 6 In addition to the hypotheses of Theorem 4, suppose that for every (x,y), (z,t) in X\times X, there exists (u,v) in X\times X that is comparable to (x,y) and (z,t), then F and g have a unique coupled fixed point.
Proof From Theorem 4, the set of coupled fixed points of F is nonempty. Suppose that (x,y) and (z,t) are coupled coincidence points of F, that is, gx=F(x,y), gy=F(y,x), gz=F(z,t) and gt=F(t,z). We will prove that
By assumption, there exists (u,v) in X\times X such that (F(u,v),F(v,u)) is comparable with (F(x,y),F(y,x)) and (F(z,t),F(t,z)). Put {u}_{0}=u and {v}_{0}=v and choose {u}_{1},{v}_{1}\in X so that g{u}_{1}=F({u}_{0},{v}_{0}) and g{v}_{1}=F({v}_{0},{u}_{0}). Then, similarly as in the proof of Theorem 3, we can inductively define sequences \{g{u}_{n}\}, \{g{v}_{n}\} with
Further set {x}_{0}=x, {y}_{0}=y, {z}_{0}=z and {t}_{0}=t, in a similar way, define the sequences \{g{x}_{n}\}, \{g{y}_{n}\} and \{g{z}_{n}\}, \{g{t}_{n}\}. Then it is easy to show that
as n\to \mathrm{\infty}. Since
are comparable, then gx\le g{u}_{1} and gy\ge g{v}_{1}, or vice versa. It is easy to show that, similarly, (gx,gy) and (g{u}_{n},g{v}_{n}) are comparable for all n\ge 1, that is, gx\le g{u}_{n} and gy\ge g{v}_{n}, or vice versa. Thus from (2.1), we have
Similarly,
From (2.16), (2.17) and the property of φ, we have
which implies
Thus,
That is, the sequence \{d(gx,g{u}_{n})+d(gy,g{v}_{n})\} is decreasing. Therefore, there exists \alpha \ge 0 such that
We shall show that \alpha =0. Suppose, to the contrary, that \alpha >0. Taking the limit as n\to \mathrm{\infty} in (2.18), we have
a contradiction. Thus, \alpha =0, that is,
It implies
Similarly, we show that
From (2.19), (2.20) and by the uniqueness of the limit, it follows that we have gx=gz and gy=gt. Hence (gx,gy) is the unique coupled point of coincidence of F and g. □
Example 1 Let X=[0,+\mathrm{\infty}) endowed with the standard metric d(x,y)=xy for all x,y\in X. Then (X,d) is a complete metric space. Define the mapping F:X\times X\to X by
Suppose that g:X\to X is such that gx={x}^{2} for all x\in X and \phi (t):[0,+\mathrm{\infty})\to [0,+\mathrm{\infty}) is such that \phi (t)=t. Assume that \psi (t)=\frac{t}{1+t}.
It is easy to show that for all x,y,u,v\in X with gx\le gu and gy\ge gv, we have
Thus, it satisfies all the conditions of Theorem 4. So we deduce that F and g have a coupled coincidence point (x,y)\in X\times X. Here, (0,0) is a coupled coincidence point of F and g.
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002993903072204
Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory Appl. 2010., 2010: Article ID 621469
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
Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. TMA 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003
Nieto JJ, RodriquezLopez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equation. Order 2005, 22: 223–239. 10.1007/s1108300590185
Nieto JJ, RodriquezLopez R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23(12):2205–2212. 10.1007/s1011400507690
Lakshmikantham V, Ćirić LB: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2008. 10.1016/j.na.2008.09.020
Agarwal RP, ElGebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 1–8. 10.1080/00036810701714164
Samet B: Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces. Nonlinear Anal. TMA 2010. 10.1016/j.na.2010.02.026
Aydi H, Karapınar E, Shatanawi W:Coupled fixed point results for (\psi \text{}\phi )weakly contractive condition in ordered partial metric spaces. Comput. Math. Appl. 2011, 62: 4449–4460. 10.1016/j.camwa.2011.10.021
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
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
Nashine HK, Kadelburg Z, Radenović S:Coupled common fixed point theorems for {\omega}^{\ast}compatible mappings in ordered cone metric spaces. Appl. Math. Comput. 2012, 218: 5422–5432. 10.1016/j.amc.2011.11.029
Radenović S, Kadelburg Z: Generalized weak contractions in partially ordered metric spaces. Comput. Math. Appl. 2010, 60: 1776–1783. 10.1016/j.camwa.2010.07.008
Abbas M, Sintunavarat W, Kumam P: Coupled fixed point of generalized contractive mappings on partially ordered G metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31
Sintunavarat W, Cho YJ, Kumam P: Coupled fixed point theorems for weak contraction mapping under F invariant set. Abstr. Appl. Anal. 2012., 2012: Article ID 324874
Sintunavarat W, Kumam P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 93
Sintunavarat, W, Kumam, P: Coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces. Thai J. Math. (2012, in press)
Sintunavarat W, Cho YJ, Kumam P: Coupled fixed point theorems for contraction mapping induced by cone ballmetric in partially ordered spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 128
Sintunavarat W, Petrusel 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
Karapınar E, Kumam P, Sintunavarat W: Coupled fixed point theorems in cone metric spaces with a c distance and applications. Fixed Point Theory Appl. 2012., 2012: Article ID 194
Sintunavarat W, Radenović S, Golubović Z, Kumam P: Coupled fixed point theorems for F invariant set. Appl. Math. Inform. Sci. 2013, 7(1):247–255. 10.12785/amis/070131
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
Berinde V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 7347–7355. 10.1016/j.na.2011.07.053
Nashine HK, Shatanawi W: Coupled common fixed point theorems for pair of commuting mappings in partially ordered complete metric spaces. Comput. Math. Appl. 2011, 62: 1984–1993. 10.1016/j.camwa.2011.06.042
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
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
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Turkoglu, D., Sangurlu, M. Coupled fixed point theorems for mixed gmonotone mappings in partially ordered metric spaces. Fixed Point Theory Appl 2013, 348 (2013). https://doi.org/10.1186/168718122013348
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DOI: https://doi.org/10.1186/168718122013348
Keywords
 coupled fixed point
 mappings having a mixed monotone property
 partially ordered metric space