This document presents a new coupled fixed point theorem for mappings having the mixed monotone property in partially ordered metric spaces. Specifically: 1) The theorem establishes the existence of a coupled fixed point for a mapping F that satisfies a contraction-type condition and has the mixed monotone property in a partially ordered, complete metric space. 2) It is shown that the coupled fixed point can be unique under additional conditions involving midpoint lower or upper bound properties. 3) An estimate is provided for the convergence rate as the iterates of the mapping F converge to the coupled fixed point.

1. Eshaghi et al. Fixed Point Theory and Applications 2014, 2014:130
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R ES EARCH Open Access
Pata contractions and coupled type fixed
points
Madjid Eshaghi1
, Samad Mohseni1
, Mohsen Rostamian Delavar2
, Manuel De La Sen3
,
Gwang Hui Kim4*
and Asma Arian1
*
Correspondence:
ghkim@kangnam.ac.kr
4
Department of Mathematics,
Kangnam University, Kangnam,
Korea
Full list of author information is
available at the end of the article
Abstract
A new coupled fixed point theorem related to the Pata contraction for mappings
having the mixed monotone property in partially ordered complete metric spaces is
established. It is shown that the coupled fixed point can be unique under some extra
suitable conditions involving mid point lower or upper bound properties. Also the
corresponding convergence rate is estimated when the iterates of our function
converge to its coupled fixed point.
MSC: 47H10; 34B15; 54H25
Keywords: partial ordered metric space; mixed monotone property; coupled fixed
point
1 Introduction and preliminaries
The well-known Banach contraction principle, which guarantees the existence of a unique
fixed point for a mapping defined on a complete metric space satisfying the contraction
condition, was introduced in  by Banach []. After this a great deal of effort has gone
into the theory and application of the Banach contraction theorem. Some authors gener-
alized this theorem from the single-valued case to the multivalued [–]. Some extensions
are to fixed point theorems for contraction mappings in generalized form of metric spaces,
especially a metric space endowed with a partial order. Single, coupled, tripled and other
types of fixed point theorems are investigated in many works, for instance, [–] and refer-
ences cited therein. Such fixed point theorems are applied to establishing the existence of
a unique solution to periodic boundary value problems, matrix equations, ordinary differ-
ential equations, and integral equations [–]. Recently Pata in [] introduced a fixed
point theorem with weaker hypotheses than those of the Banach contraction principle
with an explicit estimate of the convergence rate (see also []). Motivated by [], we es-
tablish a new coupled fixed point theorem related to the Pata contraction for mappings
having the mixed monotone property in partially ordered metric spaces. We prove that
the coupled fixed point can be unique under some suitable conditions. Also the corre-
sponding convergence rate is estimated when the iterates of our function converge to its
coupled fixed point.
Definition . [] Let (X,≤) be a partially ordered set. The product space X × X can
be endowed with a partial order such that for (x,y),(u,v) ∈ X × X we can define (u,v) ≤
(x,y) ⇔ x ≥ u,y ≤ v.
©2014 Eshaghi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
in any medium, provided the original work is properly cited.

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Definition . [] Let (X,≤) be a partially ordered set and F : X × X → X. We say that
F has the mixed monotone property on X if F(x,y) is monotone non-decreasing in x and
is monotone non-increasing in y, that is, for any x,y ∈ X,
x,x ∈ X, x ≤ x ⇒ F(x,y) ≤ F(x,y) and
y,y ∈ X, y ≤ y ⇒ F(x,y) ≥ F(x,y).
Definition . [] An element (x,y) ∈ X × X is called a coupled fixed point of the map-
ping F : X × X → X if
x = F(x,y), y = F(y,x).
2 Main results
In this section we prove a coupled type fixed point theorem with the convergence rate
estimation. Also an example is given as an application of the main theorem.
Definition . Let (X,d) be a metric space. The mapping d : X
× X
→ [,∞) given by
d (x,y),(u,v) = d(x,u) + d(y,v),
for each pair ((x,y),(u,v)) ∈ X
× X
, defines a metric on X
× X
, which will be denoted
for convenience by d, too.
For a metric space (X,d), selecting an arbitrary (x,y) ∈ X × X, we denote
x,y = d (x,y),(x,y) for all (x,y) ∈ X × X.
Let ψ : [,] → [,∞) be an increasing function vanishing with continuity at zero. Also
consider the vanishing sequence depending on α ≥ , wn(α) = (α
n
)α n
k= ψ(α
k
).
Theorem . Let (X,≤) 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 × X → X be a continuous mapping
having the mixed monotone property on X and let ≥ , α ≥ , and β ∈ [,α] be fixed
constants. Suppose that there exist x,y ∈ X such that
x ≤ F(x,y) and y ≥ F(y,x).
If the inequality
d F(x,y),F(u,v) ≤
( – ε)

d (x,y),(u,v) + εα
ψ(ε)  + x,y + u,v
β
(.)
is satisfied for every ε ∈ [,] and (x,y),(u,v) ∈ X×X with u ≤ x, y ≤ v, then F has a coupled
fixed point (x∗
,y∗
).
Furthermore, we denote Fn
= F ◦ ··· ◦ F (n times),
d x∗
,y∗
, Fn
(x,y),Fn
(y,x) ≤ Kwn(α) (.)
for some positive constant K ≤  ( +  x∗
,y∗
+ d(x,y))β
.

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Proof Set F(x,y) = x and F(y,x) = y. So
x ≥ x, y ≤ y.
Now from letting x = F(x,y), y = F(y,x) and choosing the notations
F
(x,y) = F F(x,y),F(y,x) = F(x,y) = x,
F
(y,x) = F F(y,x),F(x,y) = F(y,x) = y,
with the mixed monotone property of F we have
x = F
(x,y) = F(x,y) ≥ F(x,y) = x,
y = F
(y,x) = F(y,x) ≤ F(y,x) = y.
So by continuing this argument we obtain two sequences {xn}∞
n= and {yn}∞
n= such that
x ≤ x ≤ ··· ≤ xn ≤ ··· and y ≥ y ≥ ··· ≥ yn ≥ ··· .
Furthermore for n = ,,... we set
xn = F(xn–,yn–) = Fn
(x,y), yn = F(yn–,xn–) = Fn
(y,x) and
Cn = Fn
(x,y),Fn
(y,x) .
Since (.) is true for every ε ∈ [,], setting ε =  we have the following relations:
d Fn+
(x,y),Fn+
(y,x) , Fn
(x,y),Fn
(y,x) ≤ C, (.)
d Fn+
(x,y),Fn+
(y,x) ,(x,y)
≤ d Fn+
(x,y),Fn+
(y,x) , F(x,y),F(y,x)
+ d F(x,y),F(y,x) ,(x,y) , (.)
d Fn
(x,y),Fn
(y,x) ,(x,y)
≤ d Fn
(x,y),Fn
(y,x) , Fn+
(x,y),Fn+
(y,x)
+ d Fn+
(x,y),Fn+
(y,x) ,(x,y) . (.)
Using (.), (.), (.), and (.) we have
Cn ≤ d Fn+
(x,y),Fn+
(y,x) , F(x,y),F(y,x) + C
= d Fn+
(x,y),F(x,y) + d Fn+
(y,x),F(y,x) + C
= d F Fn
(x,y),Fn
(y,x) ,F(x,y)
+ d F Fn
(y,x),Fn
(x,y) ,F(y,x) + C
≤
( – ε)

d Fn
(x,y),Fn
(y,x) ,(x,y)
+ εα
ψ(ε)  + Fn
(x,y),Fn
(y,x) + x,y
β

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+
( – ε)

d Fn
(y,x),Fn
(x,y) ,(y,x)
+ εα
ψ(ε)  + Fn
(y,x),Fn
(x,y) + y,x
β
+ C.
But from the definition of · we have
 + Fn
(x,y),Fn
(y,x) + x,y ≤  + Cn ≤  + Cn + d(x,y),
 + Fn
(y,x),Fn
(x,y) + y,x ≤  + Cn + d(x,y),
so for α ≥ β there are some E,D >  such that
Cn ≤
( – ε)

Cn + εα
ψ(ε)  + Cn + d(x,y)
β
+
( – ε)

Cn + εα
ψ(ε)  + Cn + d(x,y)
β
+ C
= ( – ε)Cn +  εα
ψ(ε)  + Cn + d(x,y)
β
+ C ≤ ( – ε)Cn + Eεα
ψ(ε)Cα
n + D.
Accordingly,
εCn ≤ Eεα
ψ(ε)Cα
n + D,
which holds by hypothesis for any ε ∈ [,] taken for each n ∈ N. If there is a subsequence
Cnk
→ ∞, then the choice εnk
= min(, +D
Cnk
) leads to the following contradiction:
 ≤ E( + D)α
ψ(εnk
) →  as nk → ∞.
Then the sequence {Cn}∞
n= is bounded. Also the sequences {Fn
(x,y)} and {Fn
(y,x)} are
Cauchy sequences:
d Fn+m+
(x,y),Fn+
(x,y) + d Fn+m+
(y,x),Fn+
(y,x)
= d F Fn+m
(x,y),Fn+m
(y,x) ,F Fn
(x,y),Fn
(y,x)
+ d F Fn+m
(y,x),Fn+m
(x,y) ,F Fn
(y,x),Fn
(x,y)
≤
( – ε)

d Fn+m
(x,y),Fn+m
(y,x) , Fn
(x,y),Fn
(y,x)
+ εα
ψ(ε)  + Fn+m
(x,y),Fn+m
(y,x) + Fn
(x,y),Fn
(y,x)
β
+
( – ε)

d Fn+m
(y,x),Fn+m
(x,y) , Fn
(y,x),Fn
(x,y)
+ εα
ψ(ε)  + Fn+m
(y,x),Fn+m
(x,y) + Fn
(y,x),Fn
(x,y)
β
.
But since
Fn+m
(x,y),Fn+m
(y,x) + Fn
(x,y),Fn
(y,x)
≤ cn+m + cn ≤ cn+m + cn + d(x,y),

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Fn+m
(y,x),Fn+m
(x,y) + Fn
(y,x),Fn
(x,y) ≤ cn+m + cn + d(x,y),
we have
d Fn+m+
(x,y),Fn+
(x,y) + d Fn+m+
(y,x),Fn+
(y,x)
≤ ( – ε) d Fn+m
(x,y),Fn
(x,y) + d Fn+m
(y,x),Fn
(y,x)
+  εα
ψ(ε)  + cn+m + cn + d(x,y)
β
.
For fixed m, set
K = sup
n∈N
  + cn + d(x,y)
β
, (.)
and ε =  – ( n
n+
)α
≤ α
n+
. So
(n + )α
d Fn+m+
(x,y),Fn+
(x,y) + d Fn+m+
(y,x),Fn+
(y,x)
≤ nα
d Fn+m
(x,y),Fn
(x,y) + d Fn+m
(y,x),Fn
(y,x) + Kαα
ψ
α
n + 
.
Setting rn := nα
[d(Fn+m
(x,y),Fn
(x,y)) + d(Fn+m
(y,x),Fn
(y,x))] we have
rn+ ≤ rn + Kαα
ψ
α
n + 
≤ rn– + Kαα
ψ
α
n
+ Kαα
ψ
α
n + 
≤ ···
≤ r + Kαα
n+
k=
ψ
α
k
= Kαα
n+
k=
ψ
α
k
.
Therefore
d Fn+m
(x,y),Fn
(x,y) + d Fn+m
(y,x),Fn
(y,x)
≤ K
α
n
α n
k=
ψ
α
k
= Kwn(α). (.)
Taking limits as n → ∞, we get d(Fn+m
(x,y),Fn
(x,y)) + d(Fn+m
(y,x),Fn
(y,x)) → .
This implies that {Fn
(x,y)} and {Fn
(y,x)} are Cauchy sequences in X. Since X is a com-
plete metric space, there are x∗
,y∗
∈ X such that
lim
n→∞
Fn
(x,y) = lim
n→∞
xn = x∗
and lim
m→∞
Fm
(y,x) = lim
m→∞
ym = y∗
.
Using (.) and Definition . we have
lim
n→∞
d x∗
,y∗
, F(xn–,yn–),F(yn–,xn–)
= lim
n→∞
d x∗
,y∗
, Fn
(x,y),Fn
(y,x)

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= lim
n,m→∞
d Fn+m
(x,y),Fn+m
(y,x) , Fn
(x,y),Fn
(y,x)
≤ Kwn(α)
and from the continuity of F we have F(x∗
,y∗
) = x∗
and F(y∗
,x∗
) = y∗
. Also, the convergence
rate estimate stated in (.) is obtained from the following relations:
d x∗
,y∗
, Fn
(x,y),Fn
(y,x)
= d F x∗
,y∗
,F y∗
,x∗
, Fn
(x,y),Fn
(y,x)
= d F x∗
,y∗
,Fn
(x,y) + d F y∗
,x∗
,Fn
(y,x)
≤ /d x∗
,y∗
, Fn–
(x,y),Fn–
(y,x)
+ /d y∗
,x∗
, Fn–
(y,x),Fn–
(x,y)
= d x∗
,y∗
, Fn–
(x,y),Fn–
(y,x)
...
≤ d x∗
,y∗
,(x,y) = x∗
,y∗
,
which implies that
cn = d Fn
(x,y),Fn
(y,x) ,(x,y)
≤ d Fn
(x,y),Fn
(y,x) , x∗
,y∗
+ d x∗
,y∗
,(x,y)
≤  x∗
,y∗
.
From the last inequality and (.) we have K ≤  [ +  x∗
,y∗
+ d(x,y)]β
.
Example . Let X = [–,] and d : X × X → [,∞) be a metric on X defined as d(x,y) =
|x – y| for all x,y ∈ X which (X,d) is a complete metric space. Now consider X × X with
the partial order defined in Definition .. The metric on the product metric space X × X
is defined by
d (x,y),(x,y) = |x – x| + |y – y|, for all xi,yi ∈ X (i = ,).
Consider the mapping F defined as
F(x,y) = ,
for all fixed constants ≥ , α ≥ , and β ∈ [,α], and by using (.) we have
 ≤
( – ε)

d (x,y),(u,v) + εα
ψ(ε)  + x,y + u,v
β
.
Since d[(x,y),(u,v)] and εα
ψ(ε)[+ x,y + u,v ]β
are positive, we have (–ε)

≥ , which
forces ε ∈ [,]. Also there exists (,) ∈ X ×X such that  ≤ F(,) and  ≥ F(,). Thus
F satisfies all conditions of Theorem .. The pair (,) is the coupled fixed point for the
mapping F.

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3 Supplementary results
In this section we prove some supplementary results. The first theorem is about replacing
the continuity condition of the function F in Theorem . with a new one. The second is
about the uniqueness of the coupled fixed point. At last we show that the hypothesis of
Theorem . is weaker than those of Theorem . in [].
Theorem . Let (X,≤) 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 × X → X be a mapping having the
mixed monotone property on X such that there are two elements x,y ∈ X with
x ≤ F(x,y) and y ≥ F(y,x).
Let ≥ , α ≥ , and β ∈ [,α] be fixed constants. Suppose that the inequality
d F(x,y),F(u,v) ≤
( – ε)

d (x,y),(u,v) + εα
ψ(ε)  + x,y + u,v
β
(.)
is satisfied for every ε ∈ [,] and (x,y),(u,v) ∈ X × X with u ≤ x, y ≤ v. Furthermore,
suppose that X has the following properties:
(i) for a non-decreasing sequence xn → x, then xn ≤ x for all n ∈ N,
(ii) for a non-increasing sequence yn → y, then yn ≥ y for all n ∈ N.
Then F has a coupled fixed point (x∗
,y∗
).
Furthermore, calling Fn
= F ◦ ··· ◦ F (n times), we have
d x∗
,y∗
, Fn
(x,y),Fn
(y,x) ≤ Kwn(α) (.)
for some positive constant K ≤  ( +  x∗
,y∗
+ d(x,y))β
.
Proof We only need to prove that x∗
= F(x∗
,y∗
) and y∗
= F(y∗
,x∗
). Consider the following
relations:
d x∗
,y∗
, F x∗
,y∗
,F y∗
,x∗
≤ d x∗
,y∗
, Fn+
(x,y),Fn+
(y,x)
+ d Fn+
(x,y),Fn+
(y,x) , F x∗
,y∗
,F y∗
,x∗
= d x∗
,y∗
, Fn+
(x,y),Fn+
(y,x)
+ d F Fn
(x,y),Fn
(y,x) ,F x∗
,y∗
+ d F Fn
(y,x),Fn
(x,y) ,F y∗
,x∗
≤ d x∗
,y∗
, Fn+
(x,y),Fn+
(y,x)
+
( – ε)

d Fn
(x,y),Fn
(y,x) , x∗
,y∗
+ εα
ψ(ε)  + Fn
(x,y),Fn
(y,x) + x∗
,y∗ β
+
( – ε)

d Fn
(y,x),Fn
(x,y) , y∗
,x∗
+ εα
ψ(ε)  + Fn
(y,x),Fn
(x,y) + y∗
,x∗ β
.

8. Eshaghi et al. Fixed Point Theory and Applications 2014, 2014:130 Page 8 of 10
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Now since
lim
n→∞
Fn
(x,y) = lim
n→∞
xn = x∗
and lim
m→∞
Fm
(y,x) = lim
m→∞
ym = y∗
,
and since the contractive condition (.) holds for any real constant ε ∈ [,], we can re-
place ε, for each n ∈ N, by a sequence [,] εn →  as n → ∞. Then by letting ε = εn → 
as n → ∞, we have d[(x∗
,y∗
),(F(x∗
,y∗
),F(y∗
,x∗
))] = .
Definition . Suppose that X is a partially ordered metric space with a metric d. A pair
of (x,y),(x∗
,y∗
) ∈ X × X has either a mid point lower bound or a mid point upper bound
if there are (z,z) ∈ X × X comparable to (x,y) and (u,v) such that d[(x,y),(z,z)] +
d[(z,z),(x∗
,y∗
)] = d[(x,y),(x∗
,y∗
)]. The space (X,d) has the mid point lower bound or
the mid point upper bound property if any pair in X × X has a mid point lower bound or
a mid point upper bound.
Theorem . Adding the condition of the above definition to the space (X,d) in the hy-
pothesis of Theorem ., we obtain the uniqueness of the coupled fixed point of F.
Proof If (x,y) ∈ X × X is another coupled fixed point of F where x = F(x,y), y = F(y,x),
then we have two cases.
Case (): If (x∗
,y∗
) is comparable to (x,y) then
d (x,y), x∗
,y∗
= d F(x,y),F(y,x) , F x∗
,y∗
,F y∗
,x∗
= d F(x,y),F x∗
,y∗
+ d F(y,x),F y∗
,x∗
≤
( – ε)

d (x,y), x∗
,y∗
+ tεψ(ε)
+
( – ε)

d (y,x), y∗
,x∗
+ tεψ(ε),
for some t,t > . Putting t = t + t, we have
εd (x,y), x∗
,y∗
≤ εtψ(ε),
which is valid for every ε ∈ [,]. We can replace ε, for each n ∈ N, by a sequence [,]
εn →  as n → ∞ which forces d[(x,y),(x∗
,y∗
)] = .
Case (): If (x∗
,y∗
) is not comparable to (x,y) then there exists a mid upper bound or
mid lower bound (u,v) ∈ X × X for (x∗
,y∗
) and (x,y). So (F(u,v),F(v,u)) is comparable to
(x∗
,y∗
) = (F(x∗
,y∗
),F(y∗
,x∗
)) and (x,y) = (F(x,y),F(y,x)). It follows that
d (x,y), x∗
,y∗
= d F(x,y),F(y,x) , F x∗
,y∗
,F y∗
,x∗
≤ d F(x,y),F(y,x) , F(u,v),F(v,u) + d F(u,v),F(v,u) , F x∗
,y∗
,F y∗
,x∗
= d F(x,y),F(u,v) + d F(y,x),F(v,u) + d F(u,v),F x∗
,y∗
+ d F(v,u),F y∗
,x∗

9. Eshaghi et al. Fixed Point Theory and Applications 2014, 2014:130 Page 9 of 10
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≤
( – ε)

d (x,y),(u,v) + tεψ(ε) +
( – ε)

d (y,x),(v,u) + tεψ(ε)
+
( – ε)

d (u,v), x∗
,y∗
+ tεψ(ε) +
( – ε)

d (v,u), y∗
,x∗
+ tεψ(ε),
for some t,t,t,t > . Set t = t + t + t + t. Hence,
d (x,y), x∗
,y∗
≤ ( – ε) d(x,u) + d(y,v) + d u,x∗
+ d v,y∗
+ tεψ(ε). (.)
Now by Definition . we have
εd (x,y), x∗
,y∗
≤ εtψ(ε)
for every ε ∈ [,]. We can replace ε, for each n ∈ N, by the sequence [,] εn →  as
n → ∞, which forces d[(x,y),(x∗
,y∗
)] = .
Remark . Note that Theorem . is stronger than Theorem . in []. Indeed with the
hypothesis of Theorem . in [], for each (x,y),(u,v) ∈ X × X, we have
d F(x,y),F(u,v) ≤
λ

d (x,y),(u,v) for all x ≥ u,y ≤ v, ≤ λ < .
Thus for α = β = , ψ(ε) = ε, and = (ν,λ) = νν
(+ν)+ν(–λ)ν , for arbitrary ν > , we get
d F(x,y),F(u,v) ≤
( – ε)

d (x,y),(u,v) + ε+ν
 + x,y + u,v ,
for every ε ∈ [,].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1
Department of Mathematics, Faculty of Basic Sciences, Semnan University, P.O. Box 35195-363, Semnan, Iran. 2
Young
Researchers Club, Semnan Branch, Islamic Azad University, Semnan, Iran. 3
Institute of Research and Development of
Processes, University of Basque Country, Campus of Leioa (Bizkaia) - Aptdo. 644- Bilbao, Bilbao, 48080, Spain.
4
Department of Mathematics, Kangnam University, Kangnam, Korea.
Received: 22 November 2013 Accepted: 8 May 2014 Published: 02 Jun 2014
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10.1186/1687-1812-2014-130
Cite this article as: Eshaghi et al.: Pata contractions and coupled type fixed points. Fixed Point Theory and Applications
2014, 2014:130