Cho et al. (2012) proved some coupled fixed point theorems in partially ordered cone metric spaces by using the concept of a c-distance in cone metric spaces. In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion of a c-distance. Our results generalize several well-known comparable results in the literature. Also, we introduce an example to support the usability of our results.

1. Introduction and Preliminaries

Fixed point theory is an essential tool in functional nonlinear analysis. Consequently, fixed point theory has wide applications areas not only in the various branches of mathematics (see, e.g., [1, 2]) but also in many fields, such as, chemistry, biology, statistics, economics, computer science, and engineering (see, e.g., [3–11]). For example, fixed point results are incredibly useful when it comes to proving the existence of various types of Nash equilibria (see, e.g., [7]) in economics. On the other hand, fixed point theorems are vital for the existence and uniqueness of differential equations, matrix equations, integral equations (see, e.g., [1, 2]). Banach contraction mapping principle (Banach fixed point theorem) is one of the most powerful theorems of mathematics and hence fixed point theory. Huang and Zhang [12] generalized the Banach contraction principle by replacing the notion of usual metric spaces by the notion of cone metric spaces. Then many authors obtained many fixed and common fixed point theorems in cone metric spaces. For some works in cone metric spaces, we may refer the reader (as examples) to [13–24]. The concept of a coupled fixed point of a mapping F:X×X→X was initiated by Bhaskar and Lakshmikantham [25], while Lakshmikantham and Ćirić [26] initiated the notion of coupled coincidence point of mappings F:X×X→X and g:X→X and studied some coupled coincidence point theorems in partially ordered metric spaces. For some coupled fixed point and coupled coincidence point theorems, we refer the reader to [27–34].

In the present paper, N* is the set of positive integers and E stands for a real Banach space. Let P be a subset of E. We will always assume that the cone P has a nonempty interior Int(P) (such cones are called solid). Then P is called a cone if the following conditions are satisfied:

Pis closed and P≠{θ},

a,b∈R+, x,y∈P implies ax+by∈P,

x∈P∩-P implies x=θ.

For a cone P, define a partial ordering ⪯ with respect to P by x⪯y if and only if y-x∈P. We will write x≺y to indicate that x⪯y but x≠y, while x≪y will stand for y-x∈Int(P). It can be easily shown that λInt(P)⊆Int(P) for all positive scalar λ.
Definition 1.1 (see [<xref ref-type="bibr" rid="B1">12</xref>]).

Let X be a nonempty set. Suppose the mapping d:X×X→E satisfies

θ⪯d(x,y) for all x,y∈X and d(x,y)=θ if and only if x=y,

d(x,y)=d(y,x) for all x,y∈X,

d(x,y)⪯d(x,z)+d(y,z) for all x,y,z∈X.

Then d is called a cone metric on X, and (X,d) is called a cone metric space.

Bhaskar and Lakshmikantham [25] introduced the notion of mixed monotone property of the mapping F:X×X→X.

Definition 1.2 (see [<xref ref-type="bibr" rid="B14">25</xref>]).

Let (X,≤) be a partially ordered set and F:X×X→X be a mapping. Then the mapping F is said to have mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y; that is, for any x,y∈X,
(1.1)x1≤x2impliesF(x1,y)≤F(x2,y),∀y∈X,y1≤y2impliesF(x,y2)≤F(x,y1),∀x∈X.

Inspired by Definition 1.2, Lakshmikantham and Ćirić in [26] introduced the concept of a g-mixed monotone mapping.

Definition 1.3 (see [<xref ref-type="bibr" rid="B15">26</xref>]).

Let (X,≤) be a partially ordered set and F:X×X→X. Then the mapping F is said to have mixed gg-monotone property if F(x,y) is monotone g-nondecreasing in x and is monotone g-nonincreasing in y; that is:
(1.2)gx1≤gx2impliesF(x1,y)≤F(x2,y),∀y∈X,gy1≤gy2impliesF(x,y2)≤F(x,y1),∀x∈X.

Definition 1.4 (see [<xref ref-type="bibr" rid="B14">25</xref>]).

An element (x,y)∈X×X is called a coupled fixed point of a mapping F:X×X→X if
(1.3)F(x,y)=x,F(y,x)=y.

Definition 1.5 (see [<xref ref-type="bibr" rid="B15">26</xref>]).

An element (x,y)∈X×X is called a coupled coincidence point of the mappings F:X×X→X and g:X→X if
(1.4)F(x,y)=gx,F(y,x)=gy.

Recently, Cho et al. [35] introduced the concept of c-distance on cone metric space (X,d) which is a generalization of w-distance of Kada et al. [36] (see also [37, 38]).

Definition 1.6 (see [<xref ref-type="bibr" rid="B24">35</xref>]).

Let (X,d) be a cone metric space. Then a function q:X×X→E is called a c-distance on X if the following is satisfied:

θ⪯q(x,y) for all x,y∈X,

q(x,z)⪯q(x,y)+q(y,z) for all x,y,z∈X,

for each x∈X and n≥1, if q(x,yn)⪯u for some u=ux∈P, then q(x,y)⪯u whenever (yn) is a sequence in X converging to a point y∈X,

for all c∈E with θ≪c, there exists e∈E with 0≤e such that q(z,x)≪e, and q(z,y)≪e implies d(x,y)≪c.

Cho et al. [35] noticed the following important remark in the concept of c-distance on cone metric spaces.

Remark 1.7 (see [<xref ref-type="bibr" rid="B24">35</xref>]).

Let q be a c-distance on a cone metric space (X,d). Then

q(x,y)=q(y,x) does not necessarily hold for all x,y∈X,

q(x,y)=θ is not necessarily equivalent to x=y for all x,y∈X.

Very recently, Cho et al. [39] proved the following existence theorems.
Theorem 1.8 (see [<xref ref-type="bibr" rid="B27">39</xref>]).

Let (X,⊑) be a partially ordered set and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X and let F:X×X→X be a continuous function having the mixed monotone property such that
(1.5)q(F(x,y),F(x*,y*))⪯k2(q(x,x*)+q(y,y*)),
for some k∈[0,1) and all x,y,x*,y*∈X with (x⊑x*)∧(y⊒y*) or (x⊒x*)∧(y⊑y*). If there exist x0,y0∈X such that x0⊑F(x0,y0) and y0⊒F(y0,x0), then F has a coupled fixed point (u,v). Moreover, one has q(v,v)=q(u,u)=θ.

Theorem 1.9 (see [<xref ref-type="bibr" rid="B27">39</xref>]).

Let (X,⊑) be a partially ordered set and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X, and let F:X×X→X be a function having the mixed monotone property such that
(1.6)q(F(x,y),F(x*,y*))⪯k4(q(x,x*)+q(y,y*)),
for some k∈[0,1) and all x,y,x*,y*∈X with (x⊑x*)∧(y⊒y*) or (x⊒x*)∧(y⊑y*). Also, suppose that X has the following properties:

if (xn) is a nondecreasing sequence in X with xn→x, then xn⊑x for all n≥1,

if (xn) is a nonincreasing sequence in X with xn→x, then x⊑xn for all n≥1.

If there exist x0,y0∈X such that x0⊑F(x0,y0) and y0⊒F(y0,x0), then F has a coupled fixed point (u,v). Moreover, one has q(v,v)=q(u,u)=θ.

For other fixed point results using a c-distance, see [40].

In this paper, we prove some coincidence point theorems in partially ordered cone metric spaces by using the notion of c-distance. Our results generalize Theorems 1.8 and 1.9. We consider an application to illustrate our result is useful (see Section 3).

2. Main Results

The following lemma is essential in proving our results.

Lemma 2.1 (see [<xref ref-type="bibr" rid="B24">35</xref>]).

Let (X,d) be a cone metric space, and let q be a cone distance on X. Let (xn) and (yn) be sequences in X and x,y,z∈X. Suppose that (un) is a sequence in P converging to θ. Then the following holds.

If q(xn,y)⪯un and q(xn,z)⪯un, then y=z.

If q(xn,yn)⪯un and q(xn,z)⪯un, then (yn) converges to z.

If q(xn,xm)⪯un for m>n, then (xn) is a Cauchy sequence in X.

If q(y,xn)⪯un, then (xn) is a Cauchy sequence in X.

In this section, we prove some coupled fixed point theorems by using c-distance in partially partially ordered cone metric spaces.

Theorem 2.2.

Let (X,≤) be a partially ordered set and suppose that (X,d) is a cone metric space. Let q be a c-distance on X. Let F:X×X→X and g:X→X be two mappings such that
(2.1)q(F(x,y),F(x*,y*))+q(F(y,x),F(y*,x*))⪯k(q(gx,gx*)+q(gy,gy*)),
for some k∈[0,1) and for all x,y,x*,y*∈X with (gx≤gx*)∧(gy≥gy*) or (gx≥gx*)∧(gy≤gy*). Assume that F and g satisfy the following conditions:

F is continuous,

g is continuous and commutes with F,

F(X×X)⊆gX,

(X,d) is complete,

F has the mixed g-monotone property.

If there exist x0,y0∈X such that gx0≤F(x0,y0) and F(y0,x0)≤gy0, then F and g have a coupled coincidence point (u,v). Moreover, one has q(gu,gu)=θ and q(gv,gv)=θ.
Proof.

Let x0,y0∈X be such that gx0≤F(x0,y0) and F(y0,x0)≤gy0. Since F(X×X)⊆g(X), we can choose x1,y1∈X such that gx1=F(x0,y0) and gy1=F(y0,x0). Again since F(X×X)⊆g(X), we can choose x2,y2∈X such that gx2=F(x1,y1) and gy2=F(y1,x1). Since F has the mixed g-monotone property, we have gx0≤gx1≤gx2 and gy2≤gy1≤gy0. Continuing this process, we can construct two sequences (xn) and (yn) in X such that
(2.2)gxn=F(xn-1,yn-1)≤gxn+1=F(xn,yn),gyn+1=F(yn,xn)≤gyn=F(yn-1,xn-1).
Let n∈N*. Then by (2.1), we have
(2.3)q(gxn,gxn+1)+q(gyn,gyn+1)=q(F(xn-1,yn-1),F(xn,yn))+q(F(yn-1,xn-1),F(yn,xn))⪯k(q(gxn-1,gxn)+q(gyn-1,gyn)).

Repeating (2.3) n-times, we get
(2.4)q(gxn,gxn+1)+q(gyn,gyn+1)⪯kn(q(gx0,gx1)+q(gy0,gy1)).
Thus, we have
(2.5)q(gxn,gxn+1)⪯kn(q(gx0,gx1)+q(gy0,gy1)),(2.6)q(gyn,gyn+1)⪯kn(q(gx0,gx1)+q(gy0,gy1)).
Let m,n∈N* with m>n. Then by (q2) and (2.5), we have
(2.7)q(gxn,gxm)⪯∑i=nm-1q(gxi,gxi+1)⪯∑i=nm-1ki(q(gx0,gx1)+q(gy0,gy1))⪯kn1-k(q(gx0,gx1)+q(gy0,gy1)).
Similarly, we have
(2.8)q(gyn,gym)⪯kn1-k(q(gx0,gx1)+q(gy0,gy1)).
From part (3) of Lemma 2.1, we conclude that (gxn) and (gyn) are Cauchy sequences in in (X,d). Since X is complete, there are u,v∈X such that gxn→u and gyn→v. Using the continuity of g, we get g(gxn)→gu and g(gyn)→gv. Also, by continuity of F and commutativity of F and g, we have
(2.9)gu=limn→∞g(gxn+1)=limn→∞g(F(xn,yn))=limn→∞F(gxn,gyn)=F(u,v),gv=limn→∞g(gyn+1)=limn→∞g(F(yn,xn))=limn→∞F(gyn,gxn)=F(v,u).
Hence, (u,v) is a coupled coincidence point of F and g. Moreover, by (2.1) we have
(2.10)q(gu,gu)+q(gv,gv)=q(F(u,v),F(u,v))+q(F(v,u),F(v,u))⪯k(q(gu,gu)+q(gv,gv)).
Since k<1, we conclude that q(gu,gu)+q(gv,gv)=θ, and hence q(gu,gu)=θ and q(gv,gv)=θ.

The continuity of F in Theorem 2.2 can be dropped. For this, we present the following useful lemma which is a variant of Lemma 2.1, (1).

Lemma 2.3.

Let (X,d) be a cone metric space, and let q be a c-distance on X. Let (xn) be a sequence in X. Suppose that (αn) and (βn) are sequences in P converging to θ. If q(xn,y)⪯αn and q(xn,z)⪯βn, then y=z.

Proof.

Let c≫θ be arbitrary. Since αn→θ, so there exists N1∈ℕ such that αn≪c/2 for all n≥N1. Similarly, there exists N2∈ℕ such that βn≪c/2 for all n≥N2. Thus, for all N≥max{N1,N2}, we have
(2.11)q(xn,y)≪c2,q(xn,z)≪c2.
Take e=c/2, so by (q4), we get that d(y,z)≪c for each c≫θ; hence y=z.

Theorem 2.4.

Let (X,≤) be a partially ordered set and suppose that (X,d) is a cone metric space. Let q be a c-distance on X. Let F:X×X→X and let g:X→X be two mappings such that
(2.12)q(F(x,y),F(x*,y*))+q(F(y,x),F(y*,x*))⪯k(q(gx,gx*)+q(gy,gy*)),
for some k∈[0,1) and for all x,y,x*,y*∈X with (gx≤gx*)∧(gy≥gy*) or (gx≥gx*)∧(gy≤gy*). Assume that F and g satisfy the following conditions:

F(X×X)⊆gX,

gX is a complete subspace of X,

F has the mixed g-monotone property.

Suppose that X has the following properties:

if a nondecreasing sequence xn→x, then xn≤x for all n,

if a nonincreasing sequence xn→x, then x≤xn for all n.

Assume there exist x0,y0∈X such that gx0≤F(x0,y0) and F(y0,x0)≤gy0. Then F and g have a coupled coincidence point, say (u,v)∈X×X. Also, q(gu,gu)=q(gv,gv)=θ.
Proof.

As in the proof of Theorem 2.2, we can construct two Cauchy sequences (gxn) and (gyn) in the complete cone metric space (gX,d). Then, there exist u,v∈X such that gxn→gu and gyn→gv. Similarly we have for all m>n≥1(2.13)q(gxn,gxm)⪯kn1-k[q(gx0,gx1)+q(gy0,gy1)],q(gyn,gym)⪯kn1-k[q(gx0,gx1)+q(gy0,gy1)].
By (q3), we get that
(2.14)q(gxn,gu)⪯kn1-k[q(gx0,gx1)+q(gy0,gy1)],(2.15)q(gyn,gv)⪯kn1-k[q(gx0,gx1)+q(gy0,gy1)].
By summation, we get that
(2.16)q(gxn,gu)+q(gyn,gv)⪯2kn1-k[q(gx0,gx1)+q(gy0,gy1)].
Since (gxn) is nondecreasing and (gyn) is nonincreasing, using the properties (i), (ii) of X, we have
(2.17)gxn≤gu,gv≤gyn,∀n≥0.
From this and (2.14), we have
(2.18)q(gxn,F(u,v))+q(gyn,F(v,u))=q(F(xn-1,yn-1),F(u,v))+q(F(yn-1,xn-1),F(v,u))⪯k(q(gxn-1,gu)+q(gyn-1,gv)).
Therefore
(2.19)q(gxn,F(u,v))+q(gyn,F(v,u))⪯k[q(gxn-1,gu)+q(gyn-1,gv)].
By (2.16), we have
(2.20)q(gxn,F(u,v))+q(gyn,F(v,u))⪯k[q(gxn-1,gu)+q(gyn-1,gv)]⪯k2kn-11-k[q(gx0,gx1)+q(gy0,gy1)]=2kn1-k[q(gx0,gx1)+q(gy0,gy1)].
This implies that
(2.21)q(gxn,F(u,v))⪯2kn1-k[q(gx0,gx1)+q(gy0,gy1)],(2.22)q(gyn,F(v,u))⪯2kn1-k[q(gx0,gx1)+q(gy0,gy1)].
By (2.14), (2.21) and Lemma 2.3, we obtain gu=F(u,v). Similarly, by (2.15), (2.22), and Lemma 2.3, we obtain gv=F(v,u). Also, adjusting as the proof of Theorem 2.2, we get that
(2.23)q(gu,gu)=q(gv,gv)=θ.

Corollary 2.5.

Let (X,≤) be a partially ordered set and suppose that (X,d) is a cone metric space. Let q be a c-distance on X. Let F:X×X→X, and let g:X→X be two mappings such that
(2.24)q(F(x,y),F(x*,y*))⪯aq(gx,gx*)+bq(gy,gy*),
for some a,b∈[0,1) with a+b<1 and for all x,y,x*,y*∈X with (gx≤gx*)∧(gy≥gy*) or (gx≥gx*)∧(gy≤gy*). Assume that F and g satisfy the following conditions:

F is continuous,

g is continuous and commutes with F,

F(X×X)⊆gX,

(X,d) is complete,

F has the mixed g-monotone property.

If there exist x0,y0∈X such that gx0≤F(x0,y0) and F(y0,x0)≤gy0, then F and g have a coupled coincidence point (u,v). Moreover, one has q(gu,gu)=θ and q(gv,gv)=θ.
Proof.

Given x,x*,y,y*∈X such that (gx≤gx*)∧(gy≥gy*). By (2.24), we have
(2.25)q(F(x,y),F(x*,y*))⪯aq(gx,gx*)+bq(gy,gy*),q(F(y,x),F(y*,x*))⪯aq(gy,gy*)+bq(gx,gx*).
Thus
(2.26)q(F(x,y),F(x*,y*))+q(F(y,x),F(y*,x*))⪯(a+b)(q(gx,gx*)+q(gy,gy*)).
Since a+b<1, the result follows from Theorem 2.2.

Corollary 2.6.

Let (X,≤) be a partially ordered set and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X. Let F:X×X→X be a continuous mapping having the mixed monotone property such that
(2.27)q(F(x,y),F(x*,y*))⪯aq(x,x*)+bq(y,y*),
for some a,b∈[0,1) with a+b<1 and for all x,y,x*,y*∈X with (x≤x*)∧(y≥y*) or (x≥x*)∧(y≤y*). If there exist x0,y0∈X such that x0≤F(x0,y0) and F(y0,x0)≤y0, then F has a coupled fixed point (x,y). Moreover, one has q(x,x)=θ and q(y,y)=θ.

Proof.

It follows from Corollary 2.5 by taking g=IX (the identity map).

Corollary 2.7.

Let (X,≤) be a partially ordered set and suppose that (X,d) is a cone metric space. Let q be a c-distance on X. Let F:X×X→X and g:X→X be two mappings such that
(2.28)q(F(x,y),F(x*,y*))⪯aq((gx,gx*)+bq(gy,gy*)),
for some a,b∈[0,1) with a+b<1 and for all x,y,x*,y*∈X with (gx≤gx*)∧(gy≥gy*) or (gx≥gx*)∧(gy≤gy*). Assume that F and g satisfy the following conditions:

F(X×X)⊆gX,

gX is a complete subspace of X,

F has the mixed g-monotone property.

Suppose that X has the following properties:

if a nondecreasing sequence xn→x, then xn≤x for all n,

if a nonincreasing sequence xn→x, then x≤xn for all n.

Assume there exist x0,y0∈X such that gx0≤F(x0,y0) and F(y0,x0)≤gy0. Then F and g have a coupled coincidence point.
Proof.

It follows from Theorem 2.4 by similar arguments to those given in proof of Corollary 2.5.

Corollary 2.8.

Let (X,≤) be a partially ordered set and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X. Let F:X×X→X be a mapping having the mixed monotone property such that
(2.29)q(F(x,y),F(x*,y*))⪯aq((x,x*)+bq(y,y*)),
for some a,b∈[0,1) with a+b<1 and for all x,y,x*,y*∈X with (x≤x*)∧(y≥y*) or (x≥x*)∧(y≤y*). Suppose that X has the following properties:

if a nondecreasing sequence xn→x, then xn≤x for all n,

if a nonincreasing sequence xn→x, then x≤xn for all n.

Assume there exist x0,y0∈X such that x0≤F(x0,y0) and F(y0,x0)≤y0. Then F has a coupled fixed point.
Proof.

It follows from Corollary 2.7 by taking g=IX (the identity map).

Corollary 2.9.

Let (X,≤) be a partially ordered set and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X, and let F:X×X→X be a continuous mapping having the mixed monotone property such that
(2.30)q(F(x,y),F(x*,y*))+q(F(y,x),F(y*,x*))⪯k(q(x,x*)+q(y,y*)),
for some k∈[0,1) and for all x,y,x*,y*∈X with (x≤x*)∧(y≥y*) or (x≥x*)∧(y≤y*).

If there exist x0,y0∈X such that x0≤F(x0,y0) and F(y0,x0)≤y0, then F has a coupled fixed point (x,y). Moreover, we have q(x,x)=θ and q(y,y)=θ.

Proof.

It follows from Theorem 2.2 by taking g=IX.

Corollary 2.10.

Let (X,≤) be a partially ordered set and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X. Let F:X×X→X be a mapping having the mixed monotone property such that
(2.31)q(F(x,y),F(x*,y*))+q(F(y,x),F(y*,x*))⪯k(q(x,x*)+q(y,y*)),
for some k∈[0,1) and for all x,y,x*,y*∈X with (x≤x*)∧(y≥y*) or (x≥x*)∧(y≤y*). Suppose that X has the following properties:

if a nondecreasing sequence xn→x, then xn≤x for all n,

if a nonincreasing sequence xn→x, then x≤xn for all n.

Assume there exist x0,y0∈X such that x0≤F(x0,y0) and F(y0,x0)≤y0. Then F has a coupled fixed point.
Proof.

It follows from Theorem 2.4 by taking g=IX.

Example 2.11.

Let E=CR1([0,1]) with ||x||=||x||∞+||x′||∞ and P={x∈E:x(t)≥0,t∈[0,1]}. Let X=[0,1] with usual order ≤. Define d:X×X→X by d(x,y)(t)=|x-y|et for all x,y∈X. Then (X,d) is a partially ordered cone metric space. Define q:X×X→E by q(x,y)(t)=yet for all x,y∈X. Then q is a c-distance. Define F:X×X→X by
(2.32)F(x,y)={x-y2,x≥y,0,x<y.
Then,

q(F(x,y),F(x*,y*))+q(F(y,x),F(y*,x*))≤1/2(q(x,x*)+q(y,y*)), for all x≤x* and y≥y*,

there is no k∈[0,1) such that q(F(x,y),F(x*,y*))≤(k/2)(q(x,x*)+q(y,y*)) for all x≤x* and y≥y*,

there is no k∈[0,1) such that q(F(x,y),F(x*,y*))≤(k/4)(q(x,x*)+q(y,y*)) for all x≤x* and y≥y*.

Note that 0≤F(0,0) and 0≥F(0,0). Thus by Corollary 2.10, we have F which has a coupled fixed point. Here (0,0) is a coupled fixed point of F.
Proof.

The proof of (2.1) is easy. To prove (2.3), suppose the contrary; that is, there is k∈[0,1) such that q(F(x,y),F(x*,y*))≤k/2(q(x,x*)+q(y,y*)) for all x≤x* and y≥y*. Take x=0,y=1,x*=1 and y*=0. Then
(2.33)q(F(0,1),F(1,0))(t)≤k2(q(0,1)+q(1,0))(t).
Thus
(2.34)q(0,12)(t)=12et≤k2et.
Hence k≥1 is a contradiction. The proof of (2.5) is similar to proof of (2.3).

Remark 2.12.

Note that Theorems 3.1 and 3.2 of [39] are not applicable to Example 2.11.

Remark 2.13.

Theorem 3.1 of [39] is a special case of Corollary 2.6 and Corollary 2.9.

Remark 2.14.

Theorem 3.3 of [39] is a special case of Corollary 2.8 and Corollary 2.10.

3. Application

Consider the integral equations
(3.1)x(t)=∫0Tf(t,x(s),y(s))ds,t∈[0,T],y(t)=∫0Tf(t,y(s),x(s))ds,t∈[0,T],
where T>0 and f:[0,T]×ℝ×ℝ→ℝ. Let X=C([0,T],ℝ) denote the space of ℝ-valued continuous functions on I=[0,T]. The purpose of this section is to give an existence theorem for a solution (x,y) to (3.1) that belongs to X, by using the obtained result given by Corollary 2.10. Let E=ℝ2, and let P⊂E be the cone defined by
(3.2)P={(x,y)∈R2∣x≥0,y≥0}.
We endow X with the cone metric d:X×X→E defined by
(3.3)d(u,v)=(supt∈I|u(t)-v(t)|,supt∈I|u(t)-v(t)|),∀u,v∈X.
It is clear that (X,d) is a complete cone metric space. Let q(x,y)=d(x,y) for all x,y∈X. Then, q is a c-distance.

Now, we endow X with the partial order ≤ given by
(3.4)u,v∈X,u≤v⟺u(x)≤v(x),∀x∈I.
Also, the product space X×X can be equipped with the partial order (still denoted ≤) given as follows:
(3.5)(x,y)≤(u,v)⟺x≤u,y≥v.
It is easy that (i) and (ii) given in Corollary 2.10 are satisfied.

Now, we consider the following assumptions:

f:[0,T]×ℝ×ℝ→ℝ is continuous,

for all t∈[0,T], the function f(t,·,·):ℝ→ℝ has the mixed monotone property,

for all t∈[0,T], for all p,q,p′,q′∈ℝ with p≤q and p′≥q′, we have
(3.6)f(t,q,q′)-f(t,p,p′)≤1Tφ(q-p+p′-q′2),
where φ:[0,∞)→[0,∞) is continuous nondecreasing an satisfies the following condition: There exists 0<k<1 such that
(3.7)φ(r)≤kr∀r≥0,

there exists x0,y0∈C([0,T],ℝ) such that
(3.8)x0(t)≤∫0Tf(t,x0(s),y0(s))ds,∫0Tf(t,y0(s),x0(s))ds≤y0(t),∀t∈[0,T].

We have the following result.
Theorem 3.1.

Suppose that (a)–(d) hold. Then, (3.1) has at least one solution (x*,y*)∈C([0,T],ℝ)×C([0,T],ℝ).

Proof.

Define the mapping A:C([0,T],ℝ)×C([0,T],ℝ)→C([0,T],ℝ) by
(3.9)A(x,y)(t)=∫0Tf(t,x(s),y(s))ds,x,y∈C([0,T],R),t∈[0,T].
We have to prove that A has at least one coupled fixed point (x*,y*)∈C([0,T],ℝ)×C([0,T],ℝ).

From (b), it is clear that A has the mixed monotone property.

Now, let x,y,u,v∈C([0,T],ℝ) such that (x≤u and y≥v) or (x≥u and y≤v). Using (c), for all t∈[0,T], we have
(3.10)|A(u,v)(t)-A(x,y)(t)|≤∫0T[f(t,u(s),v(s))-f(t,x(s),y(s))]ds≤1T∫0Tφ(u(s)-x(s)+y(s)-v(s)2)ds≤1T∫0Tφ(supz∈[0,T]|u(z)-x(z)|+supz∈[0,T]|y(z)-v(z)|2)ds=φ(supz∈[0,T]|u(z)-x(z)|+supz∈[0,T]|y(z)-v(z)|2)≤k[supz∈[0,T]|u(z)-x(z)|+supz∈[0,T]|y(z)-v(z)|2],
which implies that
(3.11)supz∈[0,T]|A(u,v)(t)-A(x,y)(t)|≤k[supz∈[0,T]|u(z)-x(z)|+supz∈[0,T]|y(z)-v(z)|2].
Similarly, one can get
(3.12)supz∈[0,T]|A(v,u)(t)-A(y,x)(t)|≤k[supz∈[0,T]|u(z)-x(z)|+supz∈[0,T]|y(z)-v(z)|2].
We deduce
(3.13)supz∈[0,T]|A(u,v)(t)-A(x,y)(t)|+supz∈[0,T]|A(v,u)(t)-A(y,x)(t)|≤k[supz∈[0,T]|u(z)-x(z)|+supz∈[0,T]|y(z)-v(z)|].
Thus
(3.14)d(A(u,v),A(x,y))+d(A(v,u),A(y,x))⪯k[d(x,u)+d(y,v)].
Thus, we proved that condition (2.31) of Corollary 2.10 is satisfied. Moreover, from (d), we have x0≤A(x0,y0) and A(y0,x0)≤y0. Finally, applying our Corollary 2.10, we get the desired result.

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