Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
DOI 10.1186/s13663-015-0278-5
RESEARCH
Open Access
Common coupled fixed point theorems for
Geraghty-type contraction mappings using
monotone property
Zoran Kadelburg1 , Poom Kumam2,3* , Stojan Radenović4 and Wutiphol Sintunavarat5*
*
Correspondence:
poom.kum@kmutt.ac.th;
wutiphol@mathstat.sci.tu.ac.th;
poom_teun@hotmail.com
2
Department of Mathematics,
Faculty of Science, King Mongkut’s
University of Technology Thonburi,
126 Pracha Uthit Road, Bang Mod,
Thung Khru, Bangkok, 10140,
Thailand
5
Department of Mathematics and
Statistics, Faculty of Science and
Technology, Thammasat University,
Rangsit Center, Pathumthani, 12121,
Thailand
Full list of author information is
available at the end of the article
Abstract
In this paper, we prove common coupled fixed point theorems under
a Geraghty-type condition using monotone instead of the often used mixed
monotone property. Also we give some sufficient conditions for the uniqueness of
a common coupled fixed point. An example illustrating our results is provided.
MSC: 47H10; 54H25
Keywords: partially ordered metric space; coupled fixed point; mixed monotone
property; Geraghty-type condition
1 Introduction
As the Banach contraction principle (BCP) is a power tool for solving many problems in
applied mathematics and sciences, it has been improved and extended in many ways. In
particular, Geraghty proved in  [] an interesting generalization of BCP which has had
a lot of applications.
The notion of a coupled fixed point was firstly introduced and studied by Opoitsev [,
] and then by Guo and Lakshmikantham in []. In , Bhaskar and Lakshmikantham
[] were the first to introduce the notion of mixed monotone property. They also studied
and proved the following classical coupled fixed point theorems for mappings by using
this property under contractive type conditions.
Theorem . ([]) Let (X, d, ) be a partially ordered complete metric space and let F :
X × X → X be a continuous mapping having the mixed monotone property, i.e., for any
x, y ∈ X,
x , x ∈ X,
x x
⇒
F(x , y) F(x , y)
y , y ∈ X,
y y
⇒
F(x, y ) F(x, y )
and
hold. Assume that there exists k ∈ [, ) such that
k
d F(x, y), F(u, v) ≤ d(x, u) + d(y, v)

(.)
© 2015 Kadelburg et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
Page 2 of 14
holds for all x, y, u, v ∈ X with x u and y v. If there exist x , y ∈ X such that
x F(x , y ),
y F(y , x ),
then there exist x, y ∈ X such that x = F(x, y) and y = F(y, x), i.e., F has a coupled fixed point.
Theorem . ([]) If the continuity assumption for the mapping F is replaced by the assumption that the space (X, d, ) has the following properties:
. if {xn } is a non-decreasing sequence with {xn } → x, then xn x for all n ≥ ,
. if {yn } is a non-increasing sequence with {yn } → y, then yn y for all n ≥ .
Then the same conclusion as in the previous theorem holds.
Due to the important role of Theorems . and . for the investigation of solutions of
nonlinear differential and integral equations, several authors have studied various generalizations of these results (see, e.g., papers [–] and the references cited therein). Almost all of them used the mixed monotone property or, in the case of additional mapping
g : X → X, the so-called g-mixed monotone property.
Recently, in [–], the author established common coupled fixed point theorems by
using (g-)monotone property instead of (g-)mixed monotone property. These kinds of
results can be applied in another type of situations, so they give an opportunity to widen
the field of applications. In particular the so-called tripled fixed point results (and, more
generally, n-tupled results) can be more easily handled using monotone property instead
of mixed monotone property (see, e.g., [–]).
The aim of this work is to prove some common coupled fixed point theorems for
Geraghty-type contraction mappings by using monotone and g-monotone property instead of mixed monotone and g-mixed monotone property. An illustrative example is
presented in this work showing how our results can be used in proving the existence of a
common coupled fixed point, while the results of many other papers cannot.
2 Preliminaries
In this section, we give some definitions that are useful for our main results in this paper.
Throughout this paper (X, ) denotes a partially ordered set. By x y, we mean y x.
Let f , g : X → X be mappings. A mapping f is said to be g-non-decreasing (resp., g-nonincreasing) if, for all x, y ∈ X, gx gy implies fx fy (resp., fy fx). If g is an identity
mapping, then f is said to be non-decreasing (resp., non-increasing).
Definition . Let (X, ) be a partially ordered set and let F : X × X → X and g : X → X
be two mappings. The mapping F is said to have the g-monotone property if F is monotone
g-non-decreasing in both of its arguments, that is, for any x, y ∈ X,
x , x ∈ X,
gx gx
⇒
F(x , y) F(x , y)
y , y ∈ X,
gy gy
⇒
F(x, y ) F(x, y )
and
hold. If in the previous relations g is the identity mapping, then F is said to have the monotone property.
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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Definition . ([, ]) Let X be a nonempty set and F : X × X → X, g : X → X be two
mappings. An element (x, y) ∈ X × X is called:
(C ) a coupled fixed point of F if x = F(x, y) and y = F(y, x);
(C ) a coupled coincidence point of mappings g and F if
gx = F(x, y) and gy = F(y, x),
and in this case (gx, gy) is called a coupled point of coincidence;
(C ) a common coupled fixed point of mappings g and F if
x = gx = F(x, y) and y = gy = F(y, x).
Definition . ([]) Let (X, d) be a metric space and let g : X → X, F : X × X → X. The
mappings g and F are said to be compatible if
lim d gF(xn , yn ), F(gxn , gyn ) = 
n→∞
and
lim d gF(yn , xn ), F(gyn , gxn ) = 
n→∞
hold whenever {xn } and {yn } are sequences in X such that limn→∞ F(xn , yn ) = limn→∞ gxn
and limn→∞ F(yn , xn ) = limn→∞ gyn .
3 Main results
Let denote the class of all functions θ : [, ∞) × [, ∞) → [, ) which satisfy the following conditions:
(θ ) θ (s, t) = θ (t, s) for all s, t ∈ [, ∞);
(θ ) for any two sequences {sn } and {tn } of nonnegative real numbers,
θ (sn , tn ) → 
⇒
sn , tn → .
The following are examples of some functions belonging to .
() θ (s, t) = k for s, t ∈ [, ∞), where k ∈ [, ).
ln(+ks+lt) , s >  or t > ,
() θ (s, t) = r ∈ ks+lt
where k, l ∈ (, ).
[, ),
s = , t = ,
ln(+max{s,t}) , s >  or t > ,
() θ (s, t) = r ∈ max{s,t}
[, ),
s = , t = .
Now, we will prove our main result.
Theorem . Let (X, d, ) be a complete partially ordered metric space and let g : X → X
and F : X × X → X be such that F has the g-monotone property. Suppose that the following
hold:
(i) g is continuous and g(X) is closed;
(ii) F(X × X) ⊂ g(X) and g and F are compatible;
(iii) there exist x , y ∈ X such that gx F(x , y ) and gy F(y , x );
(iv) there exists θ ∈ such that for all x, y, u, v ∈ X satisfying gx gu and gy gv or
gx gu and gy gv,
d F(x, y), F(u, v) ≤ θ d(gx, gu), d(gy, gv) max d(gx, gu), d(gy, gv)
holds true;
(.)
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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(v) (a) F is continuous or (b) if, for an increasing sequence {xn } in X, xn → x ∈ X as
n → ∞, then xn x for all n ∈ N.
Then there exist u, v ∈ X such that gu = F(u, v) and gv = F(v, u), i.e., g and F have a coupled coincidence point.
Proof Starting from x , y (condition (iii)) and using that F(X × X) ⊂ g(X) (condition (ii)),
we can construct sequences {xn } and {yn } in X such that
gxn = F(xn– , yn– )
and gyn = F(yn– , xn– )
for n = , , . . . .
If gxn = gxn – and gyn = gyn – for some n ∈ N, then (gxn – , gyn – ) is a coupled point
of coincidence for g and F. Therefore, in what follows, we will assume that for each n ∈ N,
gxn = gxn– or gyn = gyn– holds.
By (iii), gx F(x , y ) = gx and gy F(y , x ) = gy , hence the g-monotone property
of F implies that gx = F(x , y ) F(x , y ) = gx and gy = F(y , x ) F(y , x ) = gy . Proceeding by induction we get that gxn– gxn and gyn– gyn hold for each n ∈ N. Hence,
the contractive condition (.) can be used to conclude that
d(gxn , gxn+ ) = d F(xn– , yn– ), F(xn , yn )
≤ θ d(gxn– , gxn ), d(gyn– , gyn ) max d(gxn– , gxn ), d(gyn– , gyn )
(.)
d(gyn , gyn+ ) = d F(yn– , xn– ), F(yn , xn )
≤ θ d(gyn– , gyn ), d(gxn– , gxn ) max d(gyn– , gyn ), d(gxn– , gxn )
= θ d(gxn– , gxn ), d(gyn– , gyn ) max d(gxn– , gxn ), d(gyn– , gyn )
(.)
and
for all n ∈ N. From (.) and (.), we get
max d(gxn , gxn+ ), d(gyn , gyn+ )
≤ θ d(gxn– , gxn ), d(gyn– , gyn ) max d(gxn– , gxn ), d(gyn– , gyn )
≤ max d(gxn– , gxn ), d(gyn– , gyn )
(.)
for all n ∈ N. Thus the sequence dn := max{d(gxn– , gxn ), d(gyn– , gyn )} is decreasing. It follows that dn → d as n → ∞ for some d ≥ . Next, we claim that d = .
Assume on the contrary that d > ; then from (.) we obtain that
max{d(gxn , gxn+ ), d(gyn , gyn+ )}
≤ θ d(gxn– , gxn ), d(gyn– , gyn ) < .
max{d(gxn– , gxn ), d(gyn– , gyn )}
On taking limit as n → ∞, we get
θ d(gxn– , gxn ), d(gyn– , gyn ) →  as n → ∞.
Since θ ∈ , we have
d(gxn– , gxn ) → 
and d(gyn– , gyn ) → 
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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as n → ∞ and hence
dn = max d(gxn– , gxn ), d(gyn– , gyn ) → 
as n → ∞,
(.)
which contradicts the assumption d > . Therefore, we can conclude that dn =max{d(gxn– ,
gxn ), d(gyn– , gyn )} →  as n → ∞.
Next, we show that {gxn } and {gyn } are Cauchy sequences. On the contrary, assume that
at least one of {gxn } or {gyn } is not a Cauchy sequence. Then there is >  for which we
can find subsequences {gxnk }, {gxmk } of {gxn } and {gynk }, {gymk } of {gyn } with nk > mk ≥ k
such that
max d(gxnk , gxmk ), d(gynk , gymk ) ≥ (.)
max d(gxnk – , gxmk ), d(gynk – , gymk ) < .
(.)
and
Using (.), (.) and the triangle inequality, we have
≤ rk := max d(gxnk , gxmk ), d(gynk , gymk )
≤ max d(gxnk , gxnk – ), d(gynk , gynk – ) + max d(gxnk – , gxmk ), d(gynk – , gymk )
< max d(gxnk , gxnk – ), d(gynk , gynk – ) + .
On taking limit as k → ∞, we have
rk = max d(gxnk , gxmk ), d(gynk , gymk ) → .
(.)
By the triangle inequality, we get
rk = max d(gxnk , gxmk ), d(gynk , gymk )
≤ max d(gxnk , gxnk + ), d(gynk , gynk + ) + max d(gxnk + , gxmk + ), d(gynk + , gymk + )
+ max d(gxmk + , gxmk ), d(gymk + , gymk )
= max d(gxnk , gxnk + ), d(gynk , gynk + ) + max d(gxmk + , gxmk ), d(gymk + , gymk )
+ max d(gxnk + , gxmk + ), d(gynk + , gymk + )
≤ max d(gxnk , gxnk + ), d(gynk , gynk + ) + max d(gxmk + , gxmk ), d(gymk + , gymk )
+ θ d(gxnk , gxmk ), d(gynk , gymk ) max d(gxnk , gxmk ), d(gynk , gymk )
= dnk + + dmk + + θ d(gxnk , gxmk ), d(gynk , gymk ) rk
≤ dnk + + dmk + + rk
(the usage of contractive condition (.) was possible since the sequences {gxn } and {gyn }
are increasing). Now, we have
rk ≤ dnk + + dmk + + θ d(gxnk , gxmk ), d(gynk , gymk ) rk ≤ dnk + + dmk + + rk .
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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On taking limit as k → ∞ and using (.) and (.), we get
θ d(gxnk , gxmk ), d(gynk , gymk ) → .
Using the properties of function θ , we obtain
d(gxnk , gxmk ) →  and
d(gynk , gymk ) → 
as k → ∞, which imply that
lim rk = lim max d(gxnk , gxmk ), d(gynk , gymk ) = ,
k→∞
k→∞
which contradicts with > .
Therefore, we get that {gxn } and {gyn } are Cauchy sequences. Since g(X) is a closed subset
of a complete metric space, there exist u, v ∈ g(X) such that
lim gxn = lim F(xn , yn ) = u
n→∞
n→∞
and
lim gyn = lim F(yn , xn ) = v.
n→∞
n→∞
By condition (ii), the compatibility of g and F implies that
lim d gF(xn , yn ), F(gxn , gyn ) = 
n→∞
and
lim d gF(yn , xn ), F(gyn , gxn ) = .
n→∞
(.)
Consider the two possibilities given in condition (v).
(a) Suppose that F is continuous. Using the triangle inequality we get that
d gu, F(gxn , gyn ) ≤ d gu, gF(xn , yn ) + d gF(xn , yn ), F(gxn , gyn ) .
Passing to the limit as n → ∞ and using (.) and the continuity of g and F, we get that
d(gu, F(u, v)) = , i.e., gu = F(u, v). In a similar way, gv = F(v, u) is obtained.
(b) In this case gxn u = gx and gyn v = gy for some x, y ∈ X and each n ∈ N. Using
(.) we get
d F(x, y), gx ≤ d F(x, y), gxn+ + d(gxn+ , gx)
= d F(x, y), F(xn , yn ) + d(gxn+ , gx)
≤ θ d(gx, gxn ), d(gy, gyn ) max d(gx, gxn ), d(gy, gyn ) + d(gxn+ , gx)
→
as n → ∞. Hence, gx = F(x, y) and similarly gy = F(y, x).
Note that in this case continuity and compatibility assumptions were not needed in the
proof.
Remark . In Theorem ., the condition that F has the g-monotone property is a substitution for the g-mixed monotone property that was used in most of the coupled fixed point
results so far. Note that this condition is maybe more natural than the mixed g-monotone
property and can be used in various examples.
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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Putting g = IX , where IX is an identity mapping on X in Theorem ., we obtain the
following.
Corollary . Let (X, d, ) be a partially ordered complete metric space and let F : X ×
X → X have the monotone property. Suppose that the following hold:
(i) there exist x , y ∈ X such that x F(x , y ) and y F(y , x );
(ii) there exists θ ∈ such that for all x, y, u, v ∈ X satisfying (x u and y v) or (u x
and v y),
d F(x, y), F(u, v) ≤ θ d(x, u), d(y, v) max d(x, u), d(y, v)
holds true;
(iii) (a) F is continuous or (b) if {xn } is an increasing sequence in X and xn → x as
n → ∞, then xn x for all n.
Then there exist x, y ∈ X such that x = F(x, y) and y = F(y, x), i.e., F has a coupled fixed
point.
Taking θ (t , t ) = k with k ∈ [, ) for all t , t ∈ [, ∞) in Theorem . and Corollary .,
we obtain the following corollary.
Corollary . Let (X, d, ) be a complete partially ordered metric space and let g : X → X
and F : X × X → X be such that F has the g-monotone property. Suppose that the following
hold:
(i) g is continuous and g(X) is closed;
(ii) F(X × X) ⊂ g(X) and g and F are compatible;
(iii) there exist x , y ∈ X such that gx F(x , y ) and gy F(y , x );
(iv) there exists k ∈ [, ) such that for all x, y, u, v ∈ X satisfying (gx gu and gy gv) or
(gu gx and gv gy),
d F(x, y), F(u, v) ≤ k max d(gx, gu), d(gy, gv)
(.)
holds true;
(v) (a) F is continuous or (b) if {xn } is an increasing sequence in X and xn → x as
n → ∞, then xn x for all n.
Then there exist u, v ∈ X such that gu = F(u, v) and gv = F(v, u), i.e., g and F have a coupled
coincidence point.
Corollary . Let (X, d, ) be a partially ordered complete metric space and let F : X ×
X → X. Suppose that the following hold:
(i) F satisfies the monotone property;
(ii) there exist x , y ∈ X such that x F(x , y ) and y F(y , x );
(iii) there exists k ∈ [, ) such that for all x, y, u, v ∈ X satisfying (x u and y v) or
(u x and v y),
d F(x, y), F(u, v) ≤ k max d(x, u), d(y, v)
holds true;
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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(iv) (a) F is continuous or (b) if {xn } is an increasing sequence in X and xn → x as
n → ∞, then xn x for all n.
Then there exist x, y ∈ X such that x = F(x, y) and y = F(y, x), i.e., F has a coupled fixed
point.
Remark . Since, for k, l ≥ , k + l ≤ ,
kd(gx, gu) + ld(gy, gv) ≤ max d(gx, gu), d(gy, gv) ,
Corollary . remains valid if the right-hand side of condition (.) is replaced by kd(gx,
gu) + ld(gy, gv) for some k, l ≥ , k + l < .
Now, reasoning on Theorem ., some questions arise naturally. To be precise, one can
ask the following.
Question Is it possible to find a sufficient condition for the existence and uniqueness of
the coupled common fixed point of g and F?
Motivated by the interest in this research, we give a positive answer to this question
adding to Theorem . some hypotheses.
For the given partial order on the set X, we shall denote also by the order on X × X
given by
(x , y ) (x , y )
⇔
x x
and
y y .
Theorem . In addition to the hypotheses of Theorem . assume that
(vi) for any two elements (x, y), (u, v) ∈ X × X, there exists (w, z) ∈ X × 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)).
Then g and F have a unique common coupled fixed point, i.e., there exists unique (p, q) ∈
X × X such that p = gp = F(p, q) and q = gq = F(q, p).
Proof Theorem . implies that there exists a coupled coincidence point (x, y) ∈ X × X,
that is, gx = F(x, y) and gy = F(y, x). Suppose that there exists another coupled coincidence
point (u, v) ∈ X × X and hence gu = F(u, v) and gv = F(v, u). We will prove that gx = gu and
gy = gv.
From condition (vi) we get that there exists (w, z) ∈ X × 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 = w, z = z and, analogously to the proof of Theorem ., choose sequences {wn }, {zn } in X satisfying
gwn = F(wn– , zn– ) and
gzn = F(zn– , wn– )
for n ∈ N. Starting from x = x, y = y and u = u, v = v, choose sequences {xn }, {yn }
and {un }, {vn } satisfying gxn = F(xn– , yn– ), gyn = F(yn– , xn– ) and gun = F(un– , vn– ), gvn =
F(vn– , un– ) for n ∈ N. Taking into account the properties of coincidence points, it is easy
to see that it can be done so that xn = x, yn = y and un = u, vn = v, i.e.,
gxn = F(x, y),
gyn = F(y, x) and gun = F(u, v),
gvn = F(v, u) for all n ∈ N.
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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Since (F(x, y), F(y, x)) = (gx, gy) and (F(w, z), F(z, v)) = (gw , gz ) are comparable, then, for
example, gx gw , gy gz and in a similar way, gx gwn , gy gzn . Thus, we can apply
the contractive condition (.) to obtain
d(gx, gwn+ ) = d F(x, y), F(wn , zn )
≤ θ d(gx, gwn ), d(gy, gzn ) max d(gx, gwn ), d(gy, gzn )
and
d(gy, gzn+ ) = d F(zn , wn ), F(y, x)
≤ θ d(gy, gzn ), d(gx, gwn ) max d(gy, gzn ), d(gx, gwn )
= θ d(gx, gwn ), d(gy, gzn ) max d(gx, gwn ), d(gy, gzn ) .
This implies that
max d(gx, gwn+ ), d(gy, gzn+ )
≤ θ d(gx, gwn ), d(gy, gzn ) max d(gx, gwn ), d(gy, gzn )
< max d(gx, gwn ), d(gy, gzn ) .
(.)
Therefore, we get that the sequence dn := max{d(gx, gwn ), d(gy, gzn )} is decreasing and
hence dn → d as n → ∞ for some d ≥ . Now, we prove that d = . Assume to the contrary
that d > ; then from (.) we have
max{d(gx, gwn+ ), d(gy, gzn+ )}
≤ θ d(gx, gwn ), d(gy, gzn ) < .
max{d(gx, gwn ), d(gy, gzn )}
Taking the limit as n → ∞ in the above inequality, we have
θ d(gx, gwn ), d(gy, gzn ) → 
as n → ∞.
By the property (θ ) of θ ∈ , we get
d(gx, gwn ) → 
and d(gy, gzn ) → 
as n → ∞. Now we have
dn := max d(gx, gwn ), d(gy, gzn ) →  as n → ∞,
which contradicts with d > . Therefore, we conclude that dn = max{d(gx, gwn ), d(gy,
gzn )} →  as n → ∞ and then
lim d(gx, gwn ) =  and
n→∞
lim d(gy, gzn ) = .
n→∞
In a similar way, we have
lim d(gu, gwn ) =  and
n→∞
lim d(gv, gzn ) = .
n→∞
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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By the triangle inequality, we have
d(gx, gu) ≤ d(gx, gwn ) + d(gwn , gu)
and d(gy, gv) ≤ d(gy, gzn ) + d(gzn , gv)
for all n ∈ N. Taking n → ∞ in the above two inequalities, we get that d(gx, gu) =  and
d(gy, gv) = . Therefore, we have gx = gu and gy = gv.
Now we let p := gx and q := gy. Hence we have
gp = g(gx) = gF(x, y) and
gq = g(gy) = gF(y, x).
By the definition of sequences {xn } and {yn } we have
gxn = F(x, y) = F(xn– , yn– ) and
gyn = F(y, x) = F(yn– , xn– )
for all n ∈ N. So we have
lim F(xn , yn ) = lim gxn = F(x, y) and
n→∞
n→∞
lim F(yn , xn ) = lim gyn = F(y, x).
n→∞
n→∞
Since g and F are compatible, we have
lim d gF(xn , yn ), F(gxn , gyn ) = ,
n→∞
that is, gF(x, y) = F(gx, gy). Therefore, we get
gp = gF(x, y) = F(gx, gy) = F(p, q)
and in a similar way
gq = gF(y, x) = F(gy, gx) = F(q, p).
This implies that (p, q) is another coincidence point. By the property we have just proved,
it follows that gp = gx = p and gq = gy = q. So,
p = gp = F(p, q) and
q = gq = F(q, p),
and (p, q) is a common coupled fixed point of g and F. For the uniqueness of the common
coupled fixed point, we can proceed easily.
Corollary . In addition to the hypotheses of Corollary . assume that
(vi) for any two elements (x, y), (u, v) ∈ X × X, there exists (w, z) ∈ X × 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)).
Then g and F have a unique common coupled fixed point.
Theorem . In addition to the hypotheses of Corollary ., let the condition (vi) of Theorem . be satisfied. Then the coupled fixed point of F is unique. Moreover, if for the terms
of sequences {xn }, {yn } defined by xn = F(xn– , yn– ) and yn = F(xn– , yn– ), xn yn holds for
n sufficiently large, then the coupled fixed point of F has the form (x, x).
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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Proof We have only to prove the last statement. Suppose that for n sufficiently large,
xn yn . Then, by (.) (with g = IX ), we get
d(xn+ , yn+ ) = d F(xn , yn ), F(yn , xn )
≤ θ d(xn , yn ), d(xn , yn ) d(xn , yn ).
(.)
This implies that
d(xn+ , yn+ ) < d(xn , yn ),
that is, the sequence dn := d(xn , yn ) is decreasing. Hence dn → d as n → ∞ for some d ≥ .
Next, we prove that d = . Assume to the contrary that d > ; then from (.) we have
d(xn+ , yn+ )
≤ θ d(xn , yn ), d(xn , yn ) < .
d(xn , yn )
Letting n → ∞, we get θ (d(xn , yn ), d(xn , yn )) → . Since θ ∈ , we have d(xn , yn ) →  as
n → ∞, which contradicts with d > . Therefore, we have d(xn , yn ) →  as n → ∞.
By the triangle inequality, we have
d(x, y) ≤ d(x, xn+ ) + d(xn+ , yn+ ) + d(yn+ , y)
= d(x, xn+ ) + d F(xn , yn ), F(yn , xn ) + d(yn+ , y)
≤ d(x, xn+ ) + θ d(xn , yn ), d(xn , yn ) d(xn , yn ) + d(yn+ , y)
< d(x, xn+ ) + d(xn , yn ) + d(yn+ , y).
Passing to the limit as n → ∞, since xn → x, yn → y and d(xn , yn ) → , we get that d(x, y) ≤
 and thus x = y. This completes the proof.
Corollary . In addition to the hypotheses of Corollary ., let the condition (vi) of Theorem . be satisfied. Then the coupled fixed point of F is unique. Moreover, if for the terms
of sequences {xn }, {yn } defined by xn = F(xn– , yn– ) and yn = F(xn– , yn– ), xn yn holds for
all n, then the coupled fixed point of F has the form (x, x).
Finally, we give an example showing that our theorem can be used when many results
in this field cannot.
Example . Let X = [, ] ⊆ R with the usual metric and order. Consider the mappings
g : X → X and F : X × X → X defined by

gx = x ,
x y
+
.
F(x, y) = ln  +


Note that F does not satisfy the mixed g-monotone property of [] and []. Indeed, for
y =  ∈ X and y =  ∈ X, we have gy gy . But for x =  ∈ X, we get
F(x, y ) = ln


ln = F(x, y ).


Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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Hence a lot of results in literature that used the g-mixed monotone property cannot be
applied in this example. Moreover, g and F do not commute as in [].
Next, we show that Theorem . can be used in this example.
Let θ ∈ be defined by
θ (s, t) =
⎧
s t
⎨ ln(+s  t+  ) ,
s >  or t > ,
⎩,
s =  and t = .
+
We will show that condition (.) holds with the function θ . Let x, y, u, v be arbitrary points
in X. We obtain that
u v
x y
+
, ln  +
+
d F(x, y), F(u, v) = d ln  +








y
u
v x
+
– ln  +
+
= ln  +



  
 + x + y = ln

  + u + v



 ( x + y ) – ( u + v ) = ln  +


 + u + v


x
y
u
v +
–
+
≤ ln  + 


 

≤ ln  + x – u + y – v 

 

ln( +  |x – u | +  |y – v |)  
x – u +  y – v =


(  |x – u | +  |y – v |)
ln( +  d(gx, gu) +  d(gy, gv)) 

d(gx, gu) + d(gy, gv)
=


(  d(gx, gu) +  d(gy, gv))


d(gx, gu) + d(gy, gv)
= θ d(gx, gu), d(gy, gv)


≤ θ d(gx, gu), d(gy, gv) max d(gx, gu), d(gy, gv) .
This shows that condition (.) holds with the function θ .
Next, we check that g and F are compatible. Let {xn } and {yn } be two sequences in X
such that
lim gxn = lim F(xn , yn ) = a
n→∞
Then ln( +
and
n→∞
a

+ b ) = a and ln( +
a

lim gyn = lim F(yn , xn ) = b.
n→∞
n→∞
+ b ) = b, wherefrom it follows that a = b = . Then

x y
x y d gF(xn , yn ), F(gxn , gyn ) = ln  + n + n
– ln  + n + n →  (n → ∞),




and similarly d(gF(yn , xn ), F(gyn , gxn )) → .
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
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We can easily check that all other conditions of Theorem . hold. Therefore, g and
F have a unique common coupled fixed point. In this example, we can see that a point
(, ) ∈ X × X is a unique common coupled fixed point of g and F.
Remark . We conclude remarking that the approach used in this paper (i.e., using
g-monotone instead of g-mixed monotone mappings) has the advantage when considering
n-tupled common fixed points for odd n. For example, one can consider mappings F :
X  → X and g : X → X and say that the mapping F is g-monotone if F(x, y, z) is g-nondecreasing in all three variables, that is, for all x, y, z ∈ X,
x , x ∈ X,
g(x ) g(x )
⇒
F(x , y, z) F(x , y, z),
y , y ∈ X,
g(y ) g(y )
⇒
F(x, y , z) F(x, y , z),
z , z ∈ X,
g(z ) g(z )
⇒
F(x, y, z ) F(x, y, z ).
In particular, if g = IX , F is said to be monotone. An element (x, y, z) ∈ X  is called a tripled
coincidence point of F and g if F(x, y, z) = gx, F(y, z, x) = gy and F(z, x, y) = gz. Moreover,
(x, y, z) is called a tripled common fixed point of F and g if
F(x, y, z) = gx = x,
F(y, z, x) = gy = y and
F(z, x, y) = gz = z.
In particular, for g = IX , the element (x, y, z) is called a tripled fixed point of F.
Results concerning these notions can be found, e.g., in []. Such an approach is not
possible for mixed monotone mappings when n =  (see []).
Open problems
• In Theorem ., can the set {d(gx, gu), d(gy, gv)} be replaced by other sets?
• Can Theorem . be improved by replacing the control function θ by other functions?
• Can Theorem . be extended and generalized replacing condition (.) by other
conditions?
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The main idea of this paper was proposed by ZK, PK, SR and WS prepared the manuscript initially and performed all the
steps of the proofs in this research. All authors read and approved the final manuscript.
Author details
1
Faculty of Mathematics, University of Belgrade, Beograd, 11000, Serbia. 2 Department of Mathematics, Faculty of Science,
King Mongkut’s University of Technology Thonburi, 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140,
Thailand. 3 Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King
Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140,
Thailand. 4 Faculty of Mathematics and Information Technology, Teacher Education, Dong Thap University, Cao Lanh City,
Dong Thap Province, Vietnam. 5 Department of Mathematics and Statistics, Faculty of Science and Technology,
Thammasat University, Rangsit Center, Pathumthani, 12121, Thailand.
Acknowledgements
The first author is thankful to the Ministry of Education, Science and Technological Development of Serbia, Project
No. 174002. The second author was supported by the Higher Education Research Promotion and National Research
University Project of Thailand, Office of the Higher Education Commission (NRU) and Theoretical and Computational
Science Center (TaCS). The fourth author would like to thank the Thailand Research Fund and Thammasat University
under Grant No. TRG5780013 for financial support during the preparation of this manuscript.
Received: 15 November 2014 Accepted: 2 February 2015
Kadelburg et al. Fixed Point Theory and Applications (2015) 2015:27
Page 14 of 14
References
1. Geraghty, M: On contractive mappings. Proc. Am. Math. Soc. 40, 604-608 (1973)
2. Opoitsev, VI: Heterogenic and combined-concave operators. Sib. Math. J. 16, 781-792 (1975) (in Russian)
3. Opoitsev, VI: Dynamics of collective behavior. III. Heterogenic systems. Autom. Remote Control 36, 111-124 (1975)
4. Guo, D, Lakshmikantham, V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. TMA 11,
623-632 (1987)
5. Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear
Anal. TMA 65, 1379-1393 (2006)
6. Abbas, M, Ali Khan, M, Radenović, S: Common coupled point theorems in cone metric spaces for w-compatible
mappings. Appl. Math. Comput. 217, 195-202 (2010)
7. Abbas, M, Sintunavarat, W, Kumam, P: Coupled fixed point in partially ordered G-metric spaces. Fixed Point Theory
Appl. 2012, 31 (2012)
8. Berinde, V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric
spaces. Nonlinear Anal. TMA 74, 7347-7355 (2011)
9. Ðorić, D, Kadelburg, Z: Coupled fixed point results for mappings without mixed monotone property. Appl. Math. Lett.
25, 1803-1808 (2012)
10. Harjani, J, López, B, Sadarangani, K: Fixed point theorems for mixed monotone operators and applications to integral
equations. Nonlinear Anal. TMA 74, 1749-1760 (2011)
11. Hussain, N, Latif, A, Shah, M: Coupled and tripled coincidence point results without compatibility. Fixed Point Theory
Appl. 2012, 77 (2012)
12. Lakshmikantham, V, Ćirić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric
spaces. Nonlinear Anal. TMA 70, 4341-4349 (2009)
13. Petrusel, A, Petrusel, G, Urs, C: Vector-valued metrics, fixed points and coupled fixed points for nonlinear operators.
Fixed Point Theory Appl. 2013, 218 (2013)
14. Roldán, A, Martínez-Moreno, J, Roldán, C, Karapinar, E: Some remarks on multidimensional fixed point theorems.
Fixed Point Theory 15, 545-558 (2014)
15. Samet, B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces.
Nonlinear Anal. TMA 72, 4508-4517 (2010)
16. Shatanawi, W: Partially ordered metric spaces and coupled fixed point results. Comput. Math. Appl. 60, 2508-2515
(2010)
17. Sintunavarat, W, Cho, YJ, Kumam, P: Coupled fixed point theorems for weak contraction mapping under F-invariant
set. Abstr. Appl. Anal. 2012, Article ID 324874 (2012)
18. Sintunavarat, W, Kumam, P: Coupled coincidence and coupled common fixed point theorems in partially ordered
metric spaces. Thai J. Math. 10, 551-563 (2012)
19. Radenović, S: Bhaskar-Lakshmikantham type results for monotone mappings in partially ordered metric spaces. Int.
J. Nonlinear Anal. Appl. 5(2), 96-103 (2014)
20. Radenović, S: Some coupled coincidence points results of monotone mappings in partially ordered metric spaces.
Int. J. Anal. Appl. 5(2), 174-184 (2014)
21. Radenović, S: Coupled fixed point theorems for monotone mappings in partially ordered metric spaces. Kragujev.
J. Math. 38, 249-257 (2014)
22. Borcut, M: Tripled fixed point theorems for monotone mappings in partially ordered metric spaces. Carpath. J. Math.
28(2), 207-214 (2012)
23. Borcut, M: Tripled coincidence theorems for monotone mappings in partially ordered metric spaces. Creative Math.
Inform. 21(2), 135-142 (2012)
24. Kadelburg, Z, Radenović, S: Remarks on some M. Borcut’s results in partially ordered metric spaces. Vietnam J. Math.
(2014, submitted)
25. Choudhury, BS, Kundu, A: A coupled coincidence point result in partially ordered metric spaces for compatible
mappings. Nonlinear Anal. TMA 73, 2524-2531 (2010)
26. Roldán, A, Martínez-Moreno, J, Roldán, C: Multidimensional fixed point theorems in partially ordered complete metric
spaces. J. Math. Anal. Appl. 396, 536-545 (2012)