coupled fixed point theorems for mixed g monotone mappings in partially ordered metric spaces

R E S E A R C H Open AccessCoupled ﬁxed point theorems for mixed g-monotone mappings in partially ordered metric spaces Duran Turkoglu1,2and Muzeyyen Sangurlu1,3* * Correspondence: msang

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R E S E A R C H Open Access

Coupled ﬁxed point theorems for mixed

g-monotone mappings in partially ordered

metric spaces

Duran Turkoglu1,2and Muzeyyen Sangurlu1,3*

* Correspondence:

msangurlu@gazi.edu.tr

1 Department of Mathematics,

Faculty of Science, University of

Gazi, Teknikokullar, Ankara, 06500,

Turkey

3 Department of Mathematics,

Faculty of Science and Arts,

University of Giresun, Gazipa¸sa,

Giresun, Turkey

Full list of author information is

available at the end of the article

Abstract

In this paper, we prove some coupled coincidence point results for mixed

g-monotone mappings in partially ordered metric spaces The main results of this

paper are generalizations of the main results of Luong and Thuan (Nonlinear Anal 74:983-992, 2011)

MSC: Primary 54H25; secondary 47H10 Keywords: coupled ﬁxed point; mappings having a mixed monotone property;

partially ordered metric space

1 Introduction and preliminaries

Fixed point theory plays a major role in mathematics The Banach contraction principle [] is the simplest one corresponding to fixed point theory So a large number of mathe-maticians have extended it and have been interested in fixed point theory in some met-ric spaces One of these spaces is a partially ordered metmet-ric space, that is, metmet-ric spaces endowed with a partial ordering The first result in this direction was given by Ran and Reurings [] who presented their applications to a matrix equation Subsequently, the ex-istence of solutions for matrix equations or ordinary differential equations by applying fixed point theorems were presented in [–]

The existence of a ﬁxed point for contraction type mappings in partially ordered met-ric spaces has been considered by Ran and Reurings [], Bhaskar and Lakshmikantham

[], Nieto and Rodriquez-Lopez [, ], Lakshmikantham and Ćirić [], Agarwal et al []

and Samet [] Bhaskar and Lakshmikantham [] introduced the notion of coupled ﬁxed point and proved some coupled ﬁxed point theorems for mappings satisfying the mixed monotone property and discussed the existence and uniqueness of a solution for a peri-odic boundary value problem Lakshmikantham and Ćirić [] introduced the concept of

a mixed g-monotone mapping and proved coupled coincidence and common ﬁxed point

theorems that extend theorems from [] Subsequently, many authors obtained several coupled coincidence and coupled ﬁxed point theorems in some ordered metric spaces [–]

Deﬁnition  ([]) Let (X, ≤) be a partially ordered set and F : X × X → X The mapping

F is said to have the mixed monotone property if F(x, y) is monotone non-decreasing in x

©2013 Turkoglu and Sangurlu; licensee Springer This is an Open Access article distributed under the terms of the Creative Com-mons Attribution License (http://creativecomCom-mons.org/licenses/by/2.0), which permits unrestricted use, distribution, and

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repro-and is monotone non-decreasing 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 mapping

F : X × X → X if F(x, y) = x, F(y, x) = y.

Deﬁnition  ([]) An element (x, y) ∈ X ×X is called a coupled coincidence point of

map-pings F : X × X → X and g : X → X if F(x, y) = gx, F(y, x) = gy.

Deﬁnition  ([]) Let X be non-empty set and F : X × X → X and g : X → X We say F

and g are commutative if gF(x, y) = F(gx, gy) for all x, y ∈ X.

Deﬁnition  ([]) Let (X, ≤) be a partially ordered set and F : X × X → X, g : X → X be

mappings The mapping F is said to have the mixed g-monotone property if F is monotone

g-non-decreasing in its ﬁrst argument and is monotone g-non-increasing in the second

argument, that is, for any x, y ∈ X,

x, x∈ X, gx≤ gx ⇒ F(x, y) ≤ F(x, y)

and

y, y∈ X, gy≤ gy ⇒ F(x, y)≥ F(x, y)

Lemma  ([]) Let X be a non-empty set and F : X × X → X and g : X → X be mappings.

Then there exists a subset E ⊆ X such that g(E) = g(X) and g : E → X is one-to-one.

Theorem  ([]) Let (X, ≤) be a partially ordered set and suppose that there exists a metric

d on X such that (X, d) is a complete metric space Let F : X × X → X be a continuous

mapping having the mixed monotone property on X Assume that there exists k ∈ [, ) with

d

F(x, y), F(u, v)

≤k



d(x, u) + d(y, v)

for all x ≥ u and y ≤ v.

If there exist two elements x, y∈ X with

x≤ F(x, y) and y≥ F(y, x),

then there exist x, y ∈ X such that

x = F(x, y) and y = F(y, x).

Theorem  ([]) Let (X, ≤) be a partially ordered set and suppose that there exists a

met-ric d on X such that (X, d) is a complete metmet-ric space Assume that X has the following

property:

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() if a non-decreasing sequence {x n } → x, then x n ≤ x for all n ∈ N, () if a non-increasing sequence {y n } → y, then y ≤ y n for all n∈ N

Let F : X × X → X be a mapping having the mixed monotone property on X Assume that

there exists k ∈ [, ) with

d

F(x, y), F(u, v)

≤k



d(x, u) + d(y, v)

for all x ≥ u and y ≤ v.

If there exist two elements x, y∈ X with

x≤ F(x, y) and y≥ F(y, x),

then there exist x, y ∈ X such that

x = F(x, y) and y = F(y, x).

Theorem  ([]) Let (X, ≤) be a partially ordered set and suppose that there exists a

metric d on X such that (X, d) is a complete metric space Let (X, ≤) be a partially ordered

set and suppose that there exists a metric d on X such that (X, d) is a complete metric space.

Let F : X × X → X be a mapping having the mixed monotone property on X and there exist

two elements x, y∈ X with x≤ F(x, y) and y≥ F(y, x) Suppose that F, g satisfy

ϕd

F(x, y), F(u, v)

≤

ϕd(x, u) + d(y, v)

–ψ

d(x, u) + d(y, v)



for all x, y, u, v ∈ X with x ≥ u and y ≤ v Suppose that either

() F is continuous or () X has the following property:

(a) if a non-decreasing sequence {x n } → x, then x n ≤ x for all n ∈ N, (b) if a non-increasing sequence {y n } → y, then y ≤ y n for all n∈ N

Then there exist x, y ∈ X such that

x = F(x, y) and y = F(y, x), that is, F has a coupled ﬁxed point in X.

2 The main results

In this paper, we prove coupled coincidence and common ﬁxed point theorems for mixed

g-monotone mappings satisfying more general contractive conditions in partially ordered

metric spaces We also present results on existence and uniqueness of coupled common

ﬁxed points Our results improve those of Luong and Thuan [] Our work generalizes,

extends and uniﬁes several well known comparable results in the literature

Let denote all functions ϕ : [, ∞) → [, ∞) which satisfy

() ϕ is continuous and non-decreasing,

() ϕ(t) =  and only if t = ,

() ϕ(t + s) ≤ ϕ(t) + ϕ(s), ∀t, s ∈ [, ∞)

and denote all functions ψ : [, ∞) → [, ∞) which satisfy lim t →r ψ(t) >  for all r > 

and limt→ +ψ(t) = .

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Theorem  Let (X, ≤) be a partially ordered set and suppose that there exists a metric d

on X such that (X, d) is a complete metric space Let (X, ≤) be a partially ordered set and

suppose that there exists a metric d on X such that (X, d) is a complete metric space Let

F : X × X → X be a mapping having the mixed monotone property on X and there exist

ϕd

F(x, y), F(u, v)

≤

ϕd(gx, gu) + d(gy, gv)

–ψ

d(gx, gu) + d(gy, gv)



(.)

for all x, y, u, v ∈ X with gx ≤ gu and gy ≥ gv, F(X × X) ⊆ g(X), g(X) is complete and g is

continuous.

Suppose that either

gx = F(x, y) and gy = F(y, x), that is, F and g have a coupled coincidence point in X × X.

Proof Using Lemma , there exists E ⊆ X such that g(E) = g(X) and g : E → X is

one-to-one We deﬁne a mapping A : g(E) × g(E) → X by

As g is one-to-one on g(E), so A is well deﬁned Thus, it follows from (.) and (.) that

ϕA(x, y), A(u, v)

≤

–ψ



(.)

for all gx, gy, gu, gv ∈ g(E) with gx ≤ gu and gy ≥ gv Since F has the mixed g-monotone

property, for all x, y ∈ X, we have

x, x∈ X, gx≤ gx ⇒ F(x, y) ≤ F(x, y) (.) and

y, y∈ X, gy≥ gy ⇒ F(x, y)≤ F(x, y) (.)

Thus, it follows from (.), (.) and (.) that, for all gx, gy ∈ g(E),

gx, gx∈ g(X), gx≤ gx ⇒ A(gx, gy) ≤ A(gx, gy)

and

gy, gy∈ g(X), gy≥ gy ⇒ A(gx, gy)≤ A(gx, gy),

which implies that A has the mixed monotone property.

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Suppose that assumption () holds Since F is continuous, A is also continuous Using Theorem  with the mapping A, it follows that A has a coupled ﬁxed point (u, v) ∈ g(E) ×

g(E).

Suppose that assumption () holds We can conclude similarly in the proof of Theorem 

that the mapping A has a coupled ﬁxed point (u, v) ∈ g(X) × g(X).

Finally, we prove that F and g have a coupled ﬁxed point in X Since (u, v) is a coupled ﬁxed point of A, we get

Since (u, v) ∈ g(X) × g(X), there exists a point (u, v)∈ X × X such that

Thus, it follows from (.) and (.) that

gu= A

gu, gv , gv= A

gv, gu

Also, from (.) and (.), we get

gu= F

u, v , gv= F

v, u

Therefore, (u, v) is a coupled coincidence point of F and g This completes the proof.

Corollary  Let (X, ≤) be a partially ordered set and suppose that there exists a metric d

ϕd

F(x, y), F(u, v)

≤k

d(gx, gu) + d(gy, gv)

continuous.

Proof In Theorem , taking ϕ(t) = t, we get Corollary .

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Corollary  Let (X, ≤) be a partially ordered set and suppose that there exists a metric d

F : X × X → X be a mapping having the mixed monotone property on X, and there exist

d

F(x, y), F(u, v)

≤



–ψ



continuous.

Proof In Corollary , taking ψ(t) = –k

Theorem  Let (X, ≤) be a partially ordered set and suppose that there exists a metric d

ϕd

F(x, y), F(u, v)

≤

–ψ



continuous.

gx = F(x, y), gy = F(y, x) and

x = gx = F(x, y), y = gy = F(y, x), that is, F and g have a coupled common ﬁxed point (x, y) ∈ X × X.

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Proof Following the proof of Theorem , F and g have a coupled coincidence point We

only have to show that x = gx and y = gy.

Now, xand yare two points in the statement of Theorem  Since F(X × X) ⊆ g(X),

we can choose x, y∈ X such that gx= F(x, y) and gy= F(y, x) In the same way, we

construct gx= F(x, y) and gy= F(y, x) Continuing in this way, we can construct two

sequences{x n } and {y n } in X such that

gx n+ = F(x n , y n) and gy n+ = F(y n , x n), ∀n ≥ . (.)

Since gx ≥ gx n+ and gy ≤ gy n+, from (.) and (.), we have

ϕd(gx n+ , gx)

=ϕd

F(x n , y n ), F(x, y)

≤ 

ϕd(gx n , gx) + d(gy n , gy)

–ψ

d(gx n , gx) + d(gy n , gy)



(.)

Similarly, since gy n+ ≥ gy and gx n+ ≤ gx, from (.) and (.), we have

ϕd(gy, gy n+)

=ϕd

F(y, x), F(y n , x n)

≤ 

ϕd(gy, gy n ) + d(gx, gx n)

–ψ

d(gy, gy n ) + d(gx, gx n)



(.) From (.) and (.), we have

ϕd(gx n+ , gx)

+ϕd(gy, gy n+)

≤ ϕd(gx n , gx) + d(gy n , gy)

– ψ



(.)

By property () ofϕ, we have

ϕd(gx n+ , gx) + d(gy, gy n+)

≤ ϕd(gx n+ , gx)

+ϕd(gy, gy n+)

(.) From (.) and (.), we have

ϕd(gx n+ , gx)+d(gy, gy n+)

≤ ϕd(gx n , gx)+d(gy n , gy)

–ψ



, which implies

ϕd(gx n+ , gx) + d(gy, gy n+)

≤ ϕd(gx n , gx) + d(gy n , gy)

Using the fact thatϕ is non-decreasing, we get

d(gx n+ , gx) + d(gy, gy n+)≤ d(gx n , gx) + d(gy n , gy). (.) Set δ n = d(gx n+ , gx) + d(gy n+ , gy), then sequence {δ n} is decreasing Therefore, there is

someδ ≥  such that

lim

n→∞δ n= lim

n→∞

d(gx n+ , gx) + d(gy n+ , gy)

=δ.

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We shall show thatδ =  Suppose, to the contrary, that δ >  Then taking the limit as

n → ∞ (equivalently, δ n → δ) of both sides of (.) and having in mind that we suppose

that limt →r ψ(t) >  for all r >  and ϕ is continuous, we have

ϕ(δ) = lim

n→∞ϕ(δ n)≤ lim

n→∞

ϕ(δ n–) – ψ

δ n–



=ϕ(δ) –  lim

δ n– →δ ψ

δ n–



<ϕ(δ),

a contradiction Thusδ = , that is,

lim

n→∞δ n= lim

n→∞

d(gx n+ , gx) + d(gy n+ , gy)

=  (.)

Hence d(gx n+ , gx) =  and d(gy n+ , gy) = , that is, x = gx and y = gy.

Theorem  In addition to the hypotheses of Theorem , suppose that for every (x, y), (z, t)

in X × X, there exists (u, v) in X × X that is comparable to (x, y) and (z, t), then F and g

have a unique coupled ﬁxed point.

Proof From Theorem , the set of coupled ﬁxed points of F is non-empty Suppose that

(x, y) and (z, t) are coupled coincidence points of F, that is, gx = F(x, y), gy = F(y, x), gz =

F(z, t) and gt = F(t, z) We will prove that

gx = gz and gy = gt.

By assumption, there exists (u, v) in X × X such that (F(u, v), F(v, u)) is comparable with

(F(x, y), F(y, x)) and (F(z, t), F(t, z)) Put u= u and v= v and choose u, v∈ X so that

gu= F(u, v) and gv= F(v, u) Then, similarly as in the proof of Theorem , we can

inductively deﬁne sequences{gu n }, {gv n} with

gu n+ = F(u n , v n) and gv n+ = F(v n , u n) for all n.

Further set x= x, y= y, z= z and t= t, in a similar way, deﬁne the sequences {gx n},

{gy n } and {gz n }, {gt n} Then it is easy to show that

gx n → F(x, y), gy n → F(y, x) and gz n → F(z, t), gt n → F(t, z)

as n→ ∞ Since

F(x, y), F(y, x)

= (gx, gy) = (gx, gy) and

F(u, v), F(v, u)

= (gu, gv)

are comparable, then gx ≤ guand gy ≥ gv, or vice versa It is easy to show that, similarly,

(gx, gy) and (gu n , gv n ) are comparable for all n ≥ , that is, gx ≤ gu n and gy ≥ gv n, or vice

versa Thus from (.), we have

ϕd(gx, gu n+)

=ϕF(x, y), F(u n , v n)

≤ 

ϕd(gx, gu n ) + d(gy, gv n)

–ψ

d(gx, gu n ) + d(gy, gv n)



(.)

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ϕd(gv n+ , gy)

=ϕF(v n , u n ), F(y, x)

≤ 

ϕd(gv n , gy) + d(gu n , gx)

–ψ

d(gv n , gy) + d(gu n , gx)



(.) From (.), (.) and the property ofϕ, we have

ϕd(gx, gu n+ ) + d(gv n+ , gy)

≤ ϕd(gx, gu n+)

+ϕd(gv n+ , gy)

≤ ϕd(gx, gu n ) + d(gy, gv n)

– ψ



, (.) which implies

ϕd(gx, gu n+ ) + d(gv n+ , gy)

≤ ϕd(gx, gu n ) + d(gy, gv n)

Thus,

d(gx, gu n+ ) + d(gv n+ , gy) ≤ d(gx, gu n ) + d(gy, gv n)

That is, the sequence{d(gx, gu n ) + d(gy, gv n)} is decreasing Therefore, there exists α ≥ 

such that

lim

n→∞

=α.

We shall show thatα =  Suppose, to the contrary, that α >  Taking the limit as n → ∞

in (.), we have

ϕ(α) ≤ ϕ(α) –  lim

n→∞ψ



<ϕ(α),

a contradiction Thus,α = , that is,

lim

n→∞

= 

It implies

lim

n→∞d(gx, gu n) = lim

Similarly, we show that

lim

n→∞d(gz, gu n) = lim

From (.), (.) and by the uniqueness of the limit, it follows that we have gx = gz and

gy = gt Hence (gx, gy) is the unique coupled point of coincidence of F and g.

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Example  Let X = [, +∞) endowed with the standard metric d(x, y) = |x – y| for all

x, y ∈ X Then (X, d) is a complete metric space Deﬁne the mapping F : X × X → X by

F(x, y) = y if x ≥ y,

x if x < y.

Suppose that g : X → X is such that gx = xfor all x ∈ X and ϕ(t) : [, +∞) → [, +∞) is

such thatϕ(t) = t Assume that ψ(t) = t

+t

It is easy to show that for all x, y, u, v ∈ X with gx ≤ gu and gy ≥ gv, we have

ϕd

F(x, y), F(u, v)

≤

–ψ



Thus, it satisﬁes all the conditions of Theorem  So we deduce that F and g have a coupled

coincidence point (x, y) ∈ X × X Here, (, ) is a coupled coincidence point of F and g.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally and signiﬁcantly in writing this paper Both authors read and approved the ﬁnal

manuscript.

Author details

1 Department of Mathematics, Faculty of Science, University of Gazi, Teknikokullar, Ankara, 06500, Turkey 2 Department of

Mathematics, Faculty of Science and Arts, University of Amasya, Amasya, 05100, Turkey 3 Department of Mathematics,

Faculty of Science and Arts, University of Giresun, Gazipa¸sa, Giresun, Turkey.

Received: 11 June 2013 Accepted: 1 December 2013 Published: 20 Dec 2013

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