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Almost contractive coupled mapping in ordered complete metric spaces
Journal of Inequalities and Applications volume 2013, Article number: 565 (2013)
Abstract
In this paper, we introduce the notion of almost contractive mapping F:X\times X\to X with respect to the mapping g:X\to X and establish some existence and uniqueness theorems of a coupled common coincidence point in ordered complete metric spaces. Also, we introduce an example to support our main results. Our results generalize several wellknown comparable results in the literature.
MSC:54H25, 47H10, 34B15.
1 Introduction and preliminaries
The existence and uniqueness theorems of a fixed point in complete metric spaces play an important role in constructing methods for solving problems in differential equations, matrix equations, and integral equations. Furthermore, the fixed point theory is a crucial method in numerical analysis to present a way for solving and approximating the roots of many equations in real analysis. One of the main theorems on a fixed point is the Banach contraction theorem [1]. Many authors generalized the Banach contraction theorem in different metric spaces in different ways. For some works on fixed point theory, we refer the readers to [2–17]. The study of a coupled fixed point was initiated by Bhaskar and Lakshmikantham [18]. Bhaskar and Lakshmikantham [18] obtained some nice results on a coupled fixed point and applied their results to solve a pair of differential equations. For some results on a coupled fixed point in ordered metric spaces, we refer the reader to [18–26].
The following definitions will be needed in the sequel.
Definition 1.1 Let (X,\u2aaf) 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 nonincreasing in y, that is, for any
and
Definition 1.2 We call an element (x,y)\in X\times X a coupled fixed point of the mapping F:X\times X\to X if
Definition 1.3 [20]
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 is monotone gnondecreasing in its first argument and is monotone gnonincreasing in its second argument, that is, for any x,y\in X,
and
Definition 1.4 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
The main results of Bhaskar and Lakshmikantham in [18] are the following.
Theorem 1.1 [18]
Let (X,\u2aaf) be a partially ordered set and d be a metric 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 a 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 1.2 [18]
Let (X,\u2aaf) be a partially ordered set and d be a metric on X such that (X,d) is a complete metric space. Assume that X has the following property:

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

(ii)
if a nonincreasing sequence \{{y}_{n}\} in X converges to y\in X, then {y}_{n}\u2ab0y for all 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
Definition 1.5 Let (X,d) be a metric space and F:X\times X\to X and g:X\to X be mappings. We say that F and g commute if
for all x,y\in X.
Nashine and Shatanawi [22] proved the following coupled coincidence point theorems.
Theorem 1.3 [22]
Let (X,d,\u2aaf) be an ordered metric space. Let F:X\times X\to X and g:X\to X be mappings such that F has the mixed gmonotone property on X such that there exist two elements {x}_{0},{y}_{0}\in X with g({x}_{0})\u2aafF({x}_{0},{y}_{0}) and g({y}_{0})\u2ab0F({y}_{0},{x}_{0}). Suppose that there exist nonnegative real numbers α, β, L with \alpha +\beta <1 such that
for all (x,y),(u,v)\in X\times X with g(x)\u2aafg(u) and g(y)\u2ab0g(v). Further suppose that F(X\times X)\subseteq g(X) and g(X) is a complete subspace of X. Also suppose that X satisfies the following properties:

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

(ii)
if a nonincreasing sequence \{{y}_{n}\} in X converges to y\in X, then {y}_{n}\u2ab0y for all n.
Then there exist x,y\in X such that
that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Theorem 1.4 [22]
Let (X,\u2aaf) be a partially ordered set and suppose that there is a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X and g:X\to X be mappings such that F has the mixed gmonotone property on X such that there exist two elements {x}_{0},{y}_{0}\in X with g({x}_{0})\u2aafF({x}_{0},{y}_{0}) and g({y}_{0})\u2ab0F({y}_{0},{x}_{0}). Suppose that there exist nonnegative real numbers α, β, L with \alpha +\beta <1 such that
for all (x,y),(u,v)\in X\times X with g(x)\u2aafg(u) and g(y)\u2ab0g(v). Further suppose that F(X\times X)\subseteq g(X), g is continuous nondecreasing and commutes with F, and also suppose that either

(a)
F is continuous, or

(b)
X has the following property:

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

(ii)
if a nonincreasing sequence \{{y}_{n}\} in X converges to y\in X, then {y}_{n}\u2ab0y for all n.

(i)
Then there exist x,y\in X such that
that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Berinde [27–30] initiated the concept of almost contractions and studied many interesting fixed point theorems for a Ćirić strong almost contraction. So, it is fundamental to recall the following definition.
Definition 1.6 [27]
A singlevalued mapping f:X\times X is called a Ćirić strong almost contraction if there exist a constant \alpha \in [0,1) and some L\ge 0 such that
for all x,y\in X, where
The aim of this paper is to introduce the notion of almost contractive mapping F:X\times X\to X with respect to the mapping g:X\to X and present some uniqueness and existence theorems of coupled fixed and coincidence point. Our results generalize Theorems 1.11.4.
2 Main theorems
We start with the following definition.
Definition 2.1 Let (X,d,\u2aaf) be an ordered metric space. We say that the mapping F:X\times X\to X is an almost contractive mapping with respect to the mapping g:X\to X if there exist a real number \alpha \in [0,1) and a nonnegative number L such that
for all (x,y),(u,v)\in X\times X with g(x)\u2aafg(u) and g(y)\u2ab0g(v).
Theorem 2.1 Let (X,d,\u2aaf) be an ordered metric space. Let F:X\times X\to X and g:X\to X be mappings such that

(1)
F is an almost contractive mapping with respect to g.

(2)
F has the mixed gmonotone property on X.

(3)
There exist two elements {x}_{0},{y}_{0}\in X with g({x}_{0})\u2aafF({x}_{0},{y}_{0}) and g({y}_{0})\u2ab0F({y}_{0},{x}_{0}).

(4)
F(X\times X)\subseteq g(X) and g(X) is a complete subspace of X.
Also, suppose that X satisfies the following properties:

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

(ii)
if a nonincreasing sequence \{{y}_{n}\} in X converges to y\in X, then {y}_{n}\u2ab0y for all n.
Then there exist x,y\in X such that
that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Proof Let {x}_{0},{y}_{0}\in X be such that g({x}_{0})\u2aafF({x}_{0},{y}_{0}) and g({y}_{0})\u2ab0F({y}_{0},{x}_{0}). 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 construct two sequences \{{x}_{n}\} and \{{y}_{n}\} in X such that
Since F has the mixed gmonotone property, by induction we may show that
and
If (g({x}_{n+1}),g({y}_{n+1}))=(g({x}_{n}),g({y}_{n})) for some n\in \mathbb{N}, then F({x}_{n},{y}_{n})=g({x}_{n}) and F({y}_{n},{x}_{n})=g({y}_{n}), that is, ({x}_{n},{y}_{n}) is a coincidence point of F and g. So we may assume that (g({x}_{n+1}),g({y}_{n+1}))\ne (g({x}_{n}),g({y}_{n})) for all n\in \mathbb{N}. Let n\in \mathbb{N}. Since g({x}_{n})\u2ab0g({x}_{n1}) and g({y}_{n})\u2aafg({y}_{n1}), from (5) and (6), we have
If max\{d(g({x}_{n1}),g({x}_{n})),d(g({y}_{n1}),g({y}_{n})),d(g({x}_{n+1}),g({x}_{n}))\}=d(g({x}_{n+1}),g({x}_{n})), then d(g({x}_{n+1}),g({x}_{n}))\le \alpha d(g({x}_{n+1}),g({x}_{n})) and hence d(g({x}_{n+1}),g({x}_{n}))=0. Thus d(g({x}_{n1}),g({x}_{n}))=d(g({y}_{n1}),g({y}_{n}))=0. Therefore d(g({x}_{n1}),g({y}_{n1}))=d(g({x}_{n}),g({y}_{n})), a contradiction. Thus
Therefore
Similarly, we may show that
From (7) and (8), we have
Repeating (9) ntimes, we get
Now, we shall prove that \{g({x}_{n})\} and \{g({y}_{n})\} are Cauchy sequences in g(X).
For each m\ge n, we have
Letting n,m\to +\mathrm{\infty} in the above inequalities, we get that \{g({x}_{n})\} is a Cauchy sequence in g(X). Similarly, we may show that \{g({y}_{n})\} is a Cauchy sequence in g(X). Since g(X) is a complete subspace of X, there exists (x,y)\in X\times X such that g({x}_{n})\to g(x) and g({y}_{n})\to g(y). Since \{g({x}_{n})\} is a nondecreasing sequence and g({x}_{n})\to g(x) and as \{g({y}_{n})\} is a nonincreasing sequence and g({y}_{n})\to g(y), by the assumption we have g({x}_{n})\u2aafg(x) and g({y}_{n})\u2ab0g(y) for all n. Since
Letting n\to \mathrm{\infty} in the above inequality, we get d(g(x),F(x,y))=0. Hence g(x)=F(x,y). Similarly, one can show that g(y)=F(y,x). Thus we proved that F and g have a coupled coincidence point. □
Theorem 2.2 Let (X,\u2aaf) be a partially ordered set and suppose that there is a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X and g:X\to X be mappings such that

(1)
F is an almost contractive mapping with respect to g.

(2)
F has the mixed gmonotone property on X.

(3)
There exist two elements {x}_{0},{y}_{0}\in X with g({x}_{0})\u2aafF({x}_{0},{y}_{0}) and g({y}_{0})\u2ab0F({y}_{0},{x}_{0}).

(4)
F(X\times X)\subseteq g(X).

(5)
g is continuous nondecreasing and commutes with F.
Also suppose that either

(a)
F is continuous, or

(b)
X has the following property:

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

(ii)
if a nonincreasing sequence \{{y}_{n}\} in X converges to y\in X, then {y}_{n}\u2ab0y for all n.

(i)
Then there exist x,y\in X such that
that is, F and g have a coupled coincidence point (x,y)\in X\times X.
Proof As in the proof of Theorem 2.1, we construct two Cauchy sequences (g{x}_{n}) and (g{y}_{n}) in X such that (g{x}_{n}) is a nondecreasing sequence in X and (g{y}_{n}) is a nonincreasing sequence in X. Since X is a complete metric space, there is (x,y)\in X\times X such that g{x}_{n}\to x and g{y}_{n}\to y. Since g is continuous, we have g(g{x}_{n})\to gx and g(g{y}_{n})\to gy.
Suppose that (a) holds. Since F is continuous, we have F(g{x}_{n},g{y}_{n})\to F(x,y) and F(g{y}_{n},g{x}_{n})\to F(y,x). Also, since g commutes with F and g is continuous, we have F(g{x}_{n},g{y}_{n})=gF({x}_{n},{y}_{n})=g(g{x}_{n+1})\to gx and F(g{y}_{n},g{x}_{n})=gF({y}_{n},{x}_{n})=g(g{y}_{n+1})\to gy. By uniqueness of limit, we get gx=F(x,y) and gy=F(y,x).
Second, suppose that (b) holds. Since g({x}_{n}) is a nondecreasing sequence such that g({x}_{n})\to x, g({y}_{n}) is a nonincreasing sequence such that g({y}_{n})\to y, and g is a nondecreasing function, we get that g(g{x}_{n})\u2aafgx and g(g{y}_{n})\u2ab0g(y) hold for all n\in \mathbb{N}. By (5), we have
Letting n\to +\mathrm{\infty}, we get d(g(x),F(x,y))=0 and hence g(x)=F(x,y). Similarly, one can show that g(y)=F(y,x). Thus (x,y) is a coupled coincidence point of F and g. □
Corollary 2.1 Let (X,\u2aaf) be a partially ordered set and suppose that there is a metric d on X such that (X,d) is a complete metric space. Let F:X\times X\to X be a mapping such that F has the mixed monotone property on X such that there exist two elements {x}_{0},{y}_{0}\in X with {x}_{0}\u2aafF({x}_{0},{y}_{0}) and {y}_{0}\u2ab0F({y}_{0},{x}_{0}). Suppose that there exist a real number \alpha \in [0,1) and a nonnegative number L such that
for all (x,y),(u,v)\in X\times X with x\u2aafu and y\u2ab0v and also suppose that either

(a)
F is continuous, or

(b)
X has the following property:

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

(ii)
if a nonincreasing sequence \{{y}_{n}\} in X converges to y\in X, then {y}_{n}\u2ab0y for all n,

(i)
then there exist x,y\in X such that
that is, F has a coupled fixed point (x,y)\in X\times X.
Proof Follows from Theorem 2.2 by taking g=I, the identity mapping. □
Let (X,\u2aaf) be a partially ordered set. Then we define a partial order ⪯ on the product space X\times X as follows:
Now, we prove some uniqueness theorem of a coupled common fixed point of mappings F:X\times X\to X and g:X\to X.
Theorem 2.3 In addition to the hypotheses of Theorem 2.1, suppose that L=0, \alpha <\frac{1}{2}, F and g commute and for every (x,y),({y}^{\ast},{x}^{\ast})\in X\times X, there exists (u,v)\in X\times X such that (F(u,v),F(v,u)) is comparable to (F(x,y),F(y,x)) and (F({x}^{\ast},{y}^{\ast}),F({y}^{\ast},{x}^{\ast})). Then F and g have a unique coupled common fixed point, that is, there exists a unique (x,y)\in X\times X such that
Proof The existence of coupled coincidence points of F and g follows from Theorem 2.1. To prove the uniqueness, let (x,y) and ({x}^{\ast},{y}^{\ast}) be coupled coincidence points of F and g; that is, g(x)=F(x,y), g(y)=F(y,x), g({x}^{\ast})=F({x}^{\ast},{y}^{\ast}) and g({y}^{\ast})=F({y}^{\ast},{x}^{\ast}). Now, we prove that
By the hypotheses, there exists (u,v)\in X\times X such that (F(u,v),F(v,u)) is comparable to (F(x,y),F(y,x)) and (F({x}^{\ast},{y}^{\ast}),F({y}^{\ast},{x}^{\ast})). Put {u}_{0}=u, {v}_{0}=v. Let {u}_{1},{v}_{1}\in X be such that g({u}_{1})=F({u}_{0},{v}_{0}) and g({v}_{1})=F({v}_{0},{u}_{0}). Then as a similar proof of Theorem 2.1, we construct two sequences \{g({u}_{n})\}, \{g({v}_{n})\} in g(X), where g({u}_{n+1})=F({u}_{n},{v}_{n}) and g({v}_{n+1})=F({v}_{n},{u}_{n}) for all n\in \mathbb{N}. Further, set {x}_{0}=x, {y}_{0}=y, {x}_{0}^{\ast}={x}^{\ast}, {y}_{0}^{\ast}={y}^{\ast}. Define the sequences \{g({x}_{n})\}, \{g({y}_{n})\} in the following way: define g{x}_{1}=F({x}_{0},{y}_{0})=F(x,y) and g{y}_{1}=F({y}_{0},{x}_{0})=F(y,x). Also, define g{x}_{2}=F({x}_{1},{y}_{1}) and g{y}_{2}=F({y}_{1},{x}_{1}). For each n\in \mathbb{N}, define g{x}_{n+1}=F({x}_{n},{y}_{n}) and g{y}_{n+1}=F({y}_{n},{x}_{n}). In the same way, we define the sequences \{g({x}_{n}^{\ast})\}, \{g({y}_{n}^{\ast})\}. Now, we prove that
Since (x,y) is a coupled coincidence point of F and g, we have F(x,y)=g(x) and F(y,x)=g(y). Thus g({x}_{1})=F({x}_{0},{y}_{0})=F(x,y)=g(x) and g({y}_{1})=F({y}_{0},{x}_{0})=F(y,x)=g(y). Therefore g({x}_{1})\u2aafg(x), g(x)\u2aafg({x}_{1}), g({y}_{1})\u2aafg(y) and g(y)\u2aafg({y}_{1}). Since F is monotone gnondecreasing on its first argument, g({x}_{1})\u2aafg(x), and g(x)\u2aafg({x}_{1}), we have F({x}_{1},{y}_{1})\u2aafF(x,{y}_{1}) and F(x,{y}_{1})\u2aafF({x}_{1},{y}_{1}). Therefore,
Also, since F is monotone gnonincreasing on its second argument, g({y}_{1})\u2aafg(y) and g(y)\u2aafg({y}_{1}), we have F(x,y)\u2aafF(x,{y}_{1}) and F(x,{y}_{1})\u2aafF(x,y). Therefore,
From (13) and (14), we have
Similarly, we may show that
Note that g({x}_{2})\u2aafg(x), g(x)\u2aafg({x}_{2}), g({y}_{2})\u2aafg(y) and g(y)\u2aafg({y}_{2}). Since F is monotone gnondecreasing on its first argument, g({x}_{2})\u2aafg(x), and g(x)\u2aafg({x}_{2}), we have F({x}_{2},{y}_{2})\u2aafF(x,{y}_{2}) and F(x,{y}_{2})\u2aafF({x}_{2},{y}_{2}). Therefore,
Also, since F is monotone gnonincreasing on its second argument, g({y}_{2})\u2aafg(y) and g(y)\u2aafg({y}_{2}), we have F(x,y)\u2aafF(x,{y}_{2}) and F(x,{y}_{2})\u2aafF(x,y). Therefore,
From (15) and (16), we have
Similarly, we may show that
Continuing in the same way, we have that
hold for all n\in \mathbb{N}. Similarly, we can show that
hold for all n\in \mathbb{N}. Since
and
are comparable, g(x)\u2aafg({u}_{1}) and g(y)\u2ab0g({v}_{1}). Since F has the mixed gmonotone property of X, we have g(x)\u2aafg({u}_{n}) and g(y)\u2ab0g({v}_{n}) for all n\in \mathbb{N}. Also, since (g({x}^{\ast}),g({y}^{\ast})) and (F(u,v),F(v,u))=(g({u}_{1}),g({v}_{1})) are comparable, and F has the gmonotone property, then we can show that for n\in \mathbb{N}, we have that (g({x}^{\ast}),g({y}^{\ast})) and (g({u}_{n}),g({v}_{n})) are comparable. Now, if (g(x),g(y))=(g({u}_{k}),g({v}_{k})) for some k\in \mathbb{N} or (g({x}^{\ast}),g({y}^{\ast}))=(g({u}_{k}),g({v}_{k})) for some k\in \mathbb{N}, then (g(x),g(y)) and (g({x}^{\ast}),g({y}^{\ast})) are comparable, say g(x)\u2aafg({x}^{\ast}) and g(y)\u2ab0g({y}^{\ast}). Thus from (5) we have
and
From (17) and (18), we have
Since \alpha <1, we have d(g(x),g({x}^{\ast}))=0 and d(g(y),g({y}^{\ast}))=0. Therefore (12) is satisfied. Now, suppose that (g(x),g(y))\ne (g({u}_{n}),g({v}_{n})) for all n\in \mathbb{N} and (g({x}^{\ast}),g({y}^{\ast}))\ne (g({u}_{n}),g({v}_{n})) for all n\in \mathbb{N}. Let n\in \mathbb{N}. Since g(x)\u2aafg({u}_{n}) and g(y)\u2ab0g({v}_{n}), then from (5) we have
If
then d(g({u}_{n+1}),g(x))\le 2\alpha d(g({u}_{n+1}),g(x)). Since 2\alpha <1, we have d(g({u}_{n+1}),g(x))=0. Therefore d(g(x),g({u}_{n}))=0 and d(g(y),g({v}_{n}))=0 and hence (g(x),g(y))=(g({u}_{n}),g({v}_{n})), a contradiction. Thus
Similarly, we may show that
From (19) and (20), we have
By repeating (21) ntimes, we have
Letting n\to +\mathrm{\infty} in the above inequalities, we get that
Hence
and
Similarly, we may show that
and
By the triangle inequality, (22), (23), (24) and (25),
we have g(x)=g({x}^{\ast}) and g(y)=g({y}^{\ast}). Thus we have (12). This implies that (g(x),g(y))=(g({x}^{\ast}),g({y}^{\ast})).
Since g(x)=F(x,y) and g(y)=F(y,x), by commutativity of F and g, we have
Denote g(x)=z, g(y)=w. Then from (26)
Thus (z,w) is a coupled coincidence point. Then from (26) with {x}^{\ast}=z and {y}^{\ast}=w it follows g(z)=g(x) and g(w)=g(y), that is,
From (27) and (28),
Therefore, (z,w) is a coupled common fixed point of F and g. To prove the uniqueness, assume that (p,q) is another coupled common fixed point. Then by (26) we have p=g(p)=g(z)=z and q=g(q)=g(w)=w. □
Corollary 2.2 In addition to the hypotheses of Corollary 2.1, suppose that L=0, \alpha <\frac{1}{2}, and for every (x,y),({y}^{\ast},{x}^{\ast})\in X\times X, there exists (u,v)\in X\times X such that u\u2aafF(u,v), v\u2ab0F(v,u), and (F(u,v),F(v,u)) is comparable to (F(x,y),F(y,x)) and (F({x}^{\ast},{y}^{\ast}),F({y}^{\ast},{x}^{\ast})). Then F has a unique coupled fixed point, that is, there exist a unique (x,y)\in X\times X such that
Proof Follows from Theorem 2.3 by taking g=I, the identity mapping. □
Theorem 2.4 In addition to the hypotheses of Theorem 2.1, if g{x}_{0} and g{y}_{0} are comparable and L=0, then F and g have a coupled coincidence point (x,y) such that gx=F(x,y)=F(y,x)=gy.
Proof Follow the proof of Theorem 2.1 step by step until constructing two sequences \{{x}_{n}\} and \{{y}_{n}\} in X such that g{x}_{n}\to gx and g{y}_{n}\to gy, where (x,y) is a coincidence point of F and g. Suppose g{x}_{0}\u2aafg{y}_{0}, then it is an easy matter to show that
Thus, by (5) we have
On taking the limit as n\to +\mathrm{\infty}, we get d(gx,gy)=0. Hence
A similar argument can be used if g{y}_{0}\u2aafg{x}_{0}. □
Corollary 2.3 In addition to the hypotheses of Corollary 2.1, if {x}_{0} and {y}_{0} are comparable and L=0, then F has a coupled fixed point of the form (x,x).
Proof Follows from Theorem 2.4 by taking g=I, the identity mapping. □
Now, we introduce the following example to support our results.
Example 2.1 Let X=[0,1]. Then (X,\le ) is a partially ordered set with the natural ordering of real numbers. Define the metric d on X by
Define g:X\to X by g(x)={x}^{2} and F:X\times X\to X by
Then

(1)
g(X) is a complete subset of X.

(2)
F(X\times X)\subseteq g(X).

(3)
X satisfies (i) and (ii) of Theorem 2.1.

(4)
F has the mixed gmonotone property.

(5)
For any L\in [0,+\mathrm{\infty}), F and g satisfy that
\begin{array}{r}d(F(x,y),F(u,v))\\ \phantom{\rule{1em}{0ex}}\le \frac{3}{4}max\{d(g(x),g(u)),d(g(y),g(v)),d(F(x,y),g(x)),d(F(u,v),g(u))\}\\ \phantom{\rule{2em}{0ex}}+Lmin\{d(F(x,y),g(u)),d(F(u,v),g(x))\}\end{array}
for all g(x)\le g(u) and g(y)\ge g(v) holds for all x,y,u,v\in X with g(x)\le g(u) and g(y)\ge g(v).
Thus, by Theorem 2.1, F has a coupled fixed point. Moreover, (0,0) is a coupled coincidence point of F.
Proof The proof of (1)(4) is clear. We divide the proof of (5) into the following cases.
Case 1: If g(x)\le g(y) and g(u)\le g(v), then x\le y and u\le v. Hence
Case 2: If g(x)\le g(y) and g(u)>g(v), then x\le y and u>v. Hence
Case 3: If g(x)>g(y) and g(u)\le g(v), then x>y and u\le v. Hence v\le y<x\le u\le v. Therefore v<v, which is impossible.
Case 4: If g(x)>g(y) and g(u)>g(v), then x>y and u>v. Thus v\le y<x\le u.
Subcase I: x=u and y=v. Here, we have
Subcase II: x\ne u or y\ne v. Here, we have {u}^{2}{v}^{2}>{x}^{2}{y}^{2}. Therefore
□
Note that the mappings F and g satisfy all the hypotheses of Theorem 2.1 for \alpha =\frac{3}{4} and any L\ge 0. Thus F and g have a coupled coincidence point. Here (0,0) is a coupled coincidence point of F and g.
Remarks

(1)
Theorem 1.1 is a special case of Corollary 2.1.

(2)
Theorem 1.2 is a special case of Corollary 2.1.

(3)
Theorem 1.3 is a special case of Theorem 2.1.

(4)
Theorem 1.4 is a special case of Theorem 2.2.
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Shatanawi, W., Saadati, R. & Park, C. Almost contractive coupled mapping in ordered complete metric spaces. J Inequal Appl 2013, 565 (2013). https://doi.org/10.1186/1029242X2013565
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DOI: https://doi.org/10.1186/1029242X2013565
Keywords
 coupled fixed point
 partially ordered set
 mixed monotone property