Sintunavarat et al. Fixed Point Theory and Applications 2012, 2012:128http://www.fixedpointtheoryandapplications.com/content/2012/1/128

RESEARCH Open Access

Coupled fixed-point theorems for contractionmapping induced by cone ball-metric inpartially ordered spacesWutiphol Sintunavarat1, Yeol Je Cho2* and Poom Kumam1*

*Correspondence: yjcho@gnu.ac.kr;poom.kum@kmutt.ac.th1Department of Mathematics,Faculty of Science, King Mongkut’sUniversity of Technology Thonburi(KMUTT), Bangkok, 10140, Thailand2Department of MathematicsEducation and the RINS,Gyeongsang National University,Chinju, 660-701, Korea

AbstractRecently, Chen et al. (Appl. Math. Lett. 25:692-697, 2012) introduced the concept ofthe cone ball-metric and studied the common fixed-point theorems for the strongerMeir-Keeler cone-type function in cone ball-metric spaces. The purpose of this paperis to establish the coupled fixed-point theorems for nonlinear contraction mappings,which have a mixed monotone property by using the cone ball-metric. Also, we givesome examples to validate our main results. At the end of this paper, we give an openproblem for further investigation.MSC: 47H10; 54H25

Keywords: coupled fixed point; mixed monotone property; cone metric space; coneball-metric space

1 IntroductionFixed-point theory has been the most attractive topic to hundreds of researchers since with the celebrated Banach’s contraction principle []. This principle provides atechnique for solving a variety of applied problems in various branches of mathematics.Moreover, it provides the applications in many fields such as chemistry, biology, statistics,economics, computer science, and engineering. The Banach’s contraction principle hasbeen extended and improved by many mathematicians (see [, , , , , –] andothers).In , the Banach’s contraction principle was extended to metric spaces endowed

with a partial ordering by Ran and Reurings []. Afterward, many generalizations andapplications of the work of Ran and Reurings exist in the literature (see in [, , ]). Forexample, Nieto and Rodríguez-López [] extended results of Ran and Reurings for non-decreasing mappings and studied a unique solution for a first-order ordinary differentialequation with periodic boundary conditions.In , Bhaskar and Lakshmikantham [] first introduced the concept of the mixed

monotone property. Furthermore, they proved some coupled fixed-point theorems formapping that satisfy the mixed monotone property and give some applications in the ex-istence and uniqueness of a solution for a periodic boundary value problem

u′(t) = f(t,u(t)

), t ∈ [,T], u() = u(T) (.)

© 2012 Sintunavarat et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

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where the function f satisfies certain conditions. Afterward, several authors studied andextended coupled fixed-point theorems of Bhaskar and Lakshmikantham [] to differentgeneralized condition (see, e.g., [, , , , , , , , ]).On the other hand, the concept of conemetric spaces is a generalization ofmetric spaces,

where each pair of points is assigned to a member of a real Banach space with a cone. Thiscone naturally induces a partial order in the Banach spaces. The concept of the conemetricspace was reintroduced in the work of Huang and Zhang [] where they also establishedthe Banach’s contraction mapping principle in this space. Afterward, several authors havestudied fixed point and coupled fixed-point problems in conemetric spaces. Some of theseworks are noted in [–, , , , , ]. Recently, Chen et al. [] introduced the con-cept of cone ball-metric spaces and proved some fixed- point theorems in these spaces formappings satisfying a contraction involving a stronger Meir-Keeler cone-type function.Motivated by the interesting concept of cone ball-metric spaces of Chen et al. [], in

this paper, we establish some coupled fixed-point theorems for a contraction mappinginduced by the cone ball-metric in partially ordered spaces and also study the conditionclaim of the uniqueness of a coupled fixed point. An open problem is also given at the endfor further investigation.

2 PreliminariesIn this section, we shall recall some definitions and mathematical preliminaries.

Definition . Recall that a binary relation � on a nonempty set X is said to be an orderrelation (and X equipped with� is called a partially ordered set) if it satisfies the followingthree properties:

(i) reflexivity: x� x for all x ∈ X ,(ii) antisymmetry: x � y and y� x imply x = y,(iii) transitivity: x � y and y� z imply x� z.

Throughout this paper (X,�) denotes a partially ordered set. By x � y holds, we meanthat y � x holds and by x ≺ y holds we mean that x � y holds, but x �= y. If (X,�) is apartially ordered set and f : X → X is such that, for all x, y ∈ X, x � y implies f (x) � f (y),then a mapping f is said to be nondecreasing. Similarly, a nonincreasing mapping is alsodefined.

Definition . ([]) Let (X,�) be a partial ordered set and F : X ×X → X be a mapping.Themapping F is said to has themixedmonotone property if F ismonotone nondecreasingin its first argument and is monotone nonincreasing in its second argument, that is, forany 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 . ([]) Let X be a nonempty set. An element (x, y) ∈ X × X is called a cou-pled fixed point of the mapping F : X ×X → X if x = F(x, y) and y = F(y,x).

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Next, we give some notations and lemmas of conemetric spaces which are reintroducedby Huang and Zhang [].Let E be a real Banach space and E denote the zero element in E. A cone P is a subset

of E such that

(C) P is nonempty closed and P �= {E};(C) if a, b are nonnegative real numbers and x, y ∈ P, then ax + by ∈ P;(C) P ∩ (–P) = {E}.For any cone P ⊂ E, the partial ordering ≤E with respect to P defined by x ≤E y if and

only if y – x ∈ P. We write x <E y to indicate that x ≤E y, but x �= y, while x y stands fory – x ∈ int(P), where int(P) denotes the interior of P.A cone P is said to be normal if there is a number K > such that, for all x, y ∈ E,

E ≤E x≤E y =⇒ ‖x‖ ≤ K‖y‖.

The least positive number satisfying above is called the normal constant of P.The cone P is said to be regular if every increasing sequence which is bounded from

above is convergent, that is, if {xn} is a sequence in E such that

x ≤E x ≤E · · · ≤E y

for some y ∈ E, then there is x ∈ E such that limn→∞ ‖xn–x‖ = . Equivalently, the cone P issaid to be regular if every decreasing sequencewhich is bounded frombelow is convergent.It is well known that a regular cone is a normal cone (see also []).

Remark . ([]) () If E be a real Banach space with a cone P in E and a ≤E ka, wherea ∈ E and < k < , then a = E .() If c ∈ int(P), E ≤E an and an → E , then there exists a positive integer N such that

an c for all n≥ N .

Lemma . ([]) If E be a real Banach space with a cone P in E, then we have the follow-ing:() If E ≤E x≤E y and k is a nonnegative real number, then E ≤E kx ≤E ky.() If E ≤E xn ≤E yn for all n ∈N and xn → x, yn → y as n→ ∞, then E ≤E x ≤E y.

Lemma. ([]) If E be a real Banach spacewith a cone P in E, thenwe have the following:for all a,b, c ∈ E,() If a ≤E b and b c, then a c.() If a b and b c, then a c.

Using the notation of a cone, we have following definitions of cone metric space.

Definition . ([]) Let X be a nonempty set and E be a real Banach space equippedwith the partial ordering ≤E with respect to the cone P ⊂ E. Suppose that the mappingd : X ×X → E satisfies the following conditions:

(CM) E <E d(x, y) for all x, y ∈ X with x �= y and d(x, y) = E if and only if x = y;

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(CM) d(x, y) = d(y,x) for all x, y ∈ X ;(CM) d(x, y)≤E d(x, z) + d(z, y) for all x, y, z ∈ X .

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

For other basic properties on a cone metric space, the reader can refer to [].Next, we give the concept of a cone ball-metric space introduced by Chen et al. [] and

its properties.In the following, we always suppose that E is a real Banach space endowed with a cone

P with the apex at the origin E , int(P) �= ∅ and a linear ordering ≤E with respect to P.

Definition . ([]) Let (X,d) be a cone metric space. A cone ball-metricwith respect tothe cone metric d is a function B : X ×X ×X → E defined by

B(x, y, z) = inf{γ : {x, y, z} ⊂ Bγ

},

where

Bγ (x) :={y ∈ X : d(x, y) < γ

}

for x ∈ X is a ball in X with the center x and radius γ � E . The ordered pair (X,B) iscalled a cone ball-metric space.

Proposition . ([]) If (X,B) is a cone ball-metric space, then the following statementshold:

(B) B(x, y, z) = E if x = y = z;(B) B(x,x, y) > E for all x, y ∈ X with x �= y;(B) B(x,x, y)≤ B(x, y, z) for all x, y, z ∈ X ;(B) B(x, y, z) = B(x, z, y) = B(y, z,x) = · · · (symmetry in all three variables);(B) B(x, y, z)≤ B(x,w,w) +B(w, y, z) for all x, y, z,w ∈ X (rectangle inequality);(B) B(x,x, y) = B(x, y, y) for all x, y ∈ X .

Definition . ([]) Let (X,B) be a cone ball-metric space and {xn} be a sequence in X.We say that {xn} is called:() A Cauchy sequence if, for any ε ∈ E with E ε, there exists a positive integer N

such that B(xn,xm,xl) ε for all n,m, l >N .() A convergent sequence if, for any ε ∈ E with E ε, there exists a positive integer N

such that, for all n,m >N , B(x,xn,xm) ε for some x ∈ X . Here, x is called the limitof the sequence {xn} and is denoted by lim

n→∞xn or xn → x.

Remark . We can prove easily that {xn} is a Cauchy sequence if and only if, for anyε ∈ E with E ε, there exists a positive integer N such that B(xn,xm,xm) ε for alln,m >N .

Definition . ([]) A cone ball-metric space (X,B) is said to be complete if everyCauchy sequence is convergent in X.

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Proposition . ([]) Let (X,B) be a cone ball-metric space and {xn} be a sequence ofpoints of X. Then the following are equivalent:() xn → x as n → ∞.() B(xn,xn,x)→ E as n→ ∞.() B(xn,x,x)→ E as n→ ∞.() B(xn,xm,x)→ E as n,m → ∞.

Proposition . ([]) Let (X,B) be a cone ball-metric space, {xn} be a sequence of pointsof X and x, y ∈ X. If xn → x and xn → y as n → ∞, then x = y.

Proposition . ([]) Let (X,B) be a cone ball-metric space and {xn}, {ym} and {zl} betree sequences in X. If xn → x, ym → y, and zl → z as n,m, l → ∞, then B(xn, ym, zl) →B(x, y, z) as n,m, l → ∞.

Definition . Let (X,B) be a cone ball-metric space. A mapping F : X ×X → X is saidto be continuous if for any two convergent sequences {xn} and {yn} converging to x and yin X, respectively, then {F(xn, yn)} is convergent to F(x, y).

3 Existence of coupled fixed point in cone ball-metric spacesLet � denote the class of all functions δ : (int(P) ∪ {E}) × (int(P) ∪ {E}) → [, ) whichsatisfies the following condition.For any sequences {an} and {bn} in int(P)∪ {E},

δ(an,bn) → =⇒ an,bn → E .

Let E = R be a usual norm space with a cone P = {z ∈ R|z ≥ } and X = R be a conemetric space with a metric d(x, y) = |x – y| for all x, y ∈ X. The following are examples ofthe functions in � under above setting:() δ(a,b) = k, where k ∈ [, ).

() δ(a,b) =

⎧⎨⎩

ln(+ka+lb)ka+lb ; a > or b > ,

r ∈ [, ); a = b = ,where k, l ∈ [, ).

() δ(a,b) =

⎧⎨⎩

ln(+max{a,b})max{a,b} ; a > or b > ,

r ∈ [, ); a = b = .Next, we prove our main theorems.

Theorem . Let (X,�) be a partially ordered set and B be a cone ball-metric induced bythe cone metric d on X with a regular cone P such that (X,B) is a complete cone ball-metricspace. Let F : X × X → X be a continuous mapping having the mixed monotone propertyon X. Suppose that there exists δ ∈ � such that

B(F(x, y),F(u, v),F(w, z)

)+B(

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

≤E δ(B(x,u,w),B(y, v, z))(B(x,u,w) +B(y, v, z)) (.)

for all x, y, z,u, v,w ∈ X for which x � u � w and y � v � z. If there exists x, y ∈ X suchthat

x � F(x, y) and y � F(y,x),

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then F has a coupled fixed point, that is, there exist x, y ∈ X such that

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

Proof We construct two sequences {xn}, {yn} in X such that, for all n≥ ,

xn+ = F(xn, yn) and yn+ = F(yn,xn). (.)

First, by induction, we show that, for all n≥ ,

xn � xn+ and yn � yn+. (.)

From x � F(x, y) = x and y � F(y,x) � y, in case n = , (.) holds. Assume that(.) holds for some n≥ . Then we get

xn � xn+ and yn � yn+. (.)

Since F has the mixed monotone property, it follows from (.) and (.) that

F(xn, y) � F(xn+, y) and F(yn+,x)� F(yn,x) (.)

for all x, y ∈ X. From (.) and (.), we have

F(y,xn)� F(y,xn+) and F(x, yn+) � F(x, yn), (.)

for all x, y ∈ X.If we take y = yn and x = xn in (.), we get

xn+ = F(xn, yn) � F(xn+, yn) and F(yn+,xn) � F(yn,xn) = yn+. (.)

If we take y = yn+ and x = xn+ in (.), we get

F(yn+,xn) � F(yn+,xn+) = yn+ and xn+ = F(xn+, yn+) � F(xn+, yn). (.)

From (.) and (.), we also have

xn+ � xn+ and yn+ � yn+. (.)

Consequently, by induction, we have (.) holds for all n≥ . This implies

x � x � x � · · · � xn � xn+ � · · · (.)

and

y � y � y � · · · � yn � yn+ � · · · . (.)

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If there exists an integer number n� ≥ such that

B(xn�+,xn�+,xn� ) +B(yn�+, yn�+, yn� ) = E ,

then we have

B(xn�+,xn�+,xn� ) = B(yn�+, yn�+, yn� ) = E ,

which implies xn� = xn�+ and yn� = yn�+. Therefore, xn� = F(xn� , yn� ) and yn� = F(yn� ,xn� )and so (xn� , yn� ) is a coupled fixed point of F .Now, we assume thatB(xn+,xn+,xn)+B(yn+, yn+, yn) >E E for all n≥ . Since xn– � xn

and yn– � yn for all n ∈N, by (.) and (.), we have

B(xn+,xn+,xn) +B(yn+, yn+, yn)= B(

F(xn, yn),F(xn, yn),F(xn–, yn–))+B(

F(yn,xn),F(yn,xn),F(yn–,xn–))

≤E δ(B(xn,xn,xn–, ),B(yn, yn, yn–)

)[B(xn,xn,xn–, ) +B(yn, yn, yn–)]

<E B(xn,xn,xn–, ) +B(yn, yn, yn–). (.)

So, we have the sequence {Bn} defined by Bn := B(xn,xn,xn–) +B(yn, yn, yn–) is a decreas-ing sequence. Since P is regular, there exists B ≥E E such that Bn → B as n→ ∞.Next, we prove that B = E . Suppose that B >E E . From (.), letting n→ ∞, we have

δ(B(xn,xn,xn–),B(yn, yn, yn–)

) → .

Since δ ∈ �, we have B(xn,xn,xn–) → E and B(yn, yn, yn–) → E and then

B(xn,xn,xn–) +B(yn, yn, yn–) → E , (.)

which contradictions with B >E E . Consequently, we must get

B(xn,xn,xn–) +B(yn, yn, yn–) → E .

By the property (B), we have

B′n := B(xn,xn–,xn–) +B(yn, yn–, yn–) → E . (.)

Now, we show that {xn} and {yn} are Cauchy sequences in cone ball-metric space (X,B).Suppose on the contrary that at least one of {xn} and {yn} are not a Cauchy sequence in(X,B). Then there exists ε ∈ E such that E ε and sequences of positive integers {n(k)}and {m(k)} such that for all positive integers k,

n(k) >m(k)≥ k

and

B(xn(k),xm(k),xm(k)) +B(yn(k), ym(k), ym(k)) >E ε. (.)

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Further, for the integer m(k), we can choose n(k) is the smallest integer for which (.)holds. Then we have

B(xn(k)–,xm(k),xm(k)) +B(yn(k)–, ym(k), ym(k)) ≤E ε. (.)

Using (.) and (.) and the rectangle inequality, we have

ε <E rk := B(xn(k),xm(k),xm(k)) +B(yn(k), ym(k), ym(k))

≤E B(xn(k),xn(k)–,xn(k)–) +B(xn(k)–,xm(k),xm(k))

+B(yn(k), yn(k)–, yn(k)–) +B(yn(k)–, ym(k), ym(k))

≤E B(xn(k),xn(k)–,xn(k)–) +B(yn(k), yn(k)–, yn(k)–) + ε.

Letting k → ∞, we have

rk = B(xn(k),xm(k),xm(k)) +B(yn(k), ym(k), ym(k)) → ε. (.)

By the rectangle inequality, we get

rk = B(xn(k),xm(k),xm(k)) +B(yn(k), ym(k), ym(k))

≤E B(xn(k),xn(k)+,xn(k)+) +B(xn(k)+,xm(k)+,xm(k)+) +B(xm(k)+,xm(k),xm(k))

+B(yn(k), yn(k)+, yn(k)+) +B(yn(k)+, ym(k)+, ym(k)+) +B(ym(k)+, ym(k), ym(k))

=[B(xn(k)+,xm(k)+,xm(k)+) +B(yn(k)+, ym(k)+, ym(k)+)

]

+[B(xn(k),xn(k)+,xn(k)+) +B(yn(k), yn(k)+, yn(k)+)

]

+[B(xm(k)+,xm(k),xm(k)) +B(ym(k)+, ym(k), ym(k))

]

=[B(xn(k)+,xm(k)+,xm(k)+) +B(yn(k)+, ym(k)+, ym(k)+)

]+Bn(k)+ +B′

m(k)+

=[B(xm(k)+,xm(k)+,xn(k)+) +B(ym(k)+, ym(k)+, yn(k)+)

]+Bn(k)+ +B′

m(k)+

=[B(

F(xm(k), ym(k)),F(xm(k), ym(k)),F(xn(k), yn(k)))

+B(F(ym(k),xm(k)),F(ym(k),xm(k)),F(yn(k),xn(k))

)]+Bn(k)+ +B′

m(k)+

≤E δ(B(xm(k),xm(k),xn(k)),B(ym(k), ym(k), yn(k))

)

× (B(xm(k),xm(k),xn(k)) +B(ym(k), ym(k), yn(k)))+Bn(k)+ +B′

m(k)+

= δ(B(xn(k),xm(k),xm(k)),B(yn(k), ym(k), ym(k))

)rk +Bn(k)+ +B′

m(k)+

<E rk +Bn(k)+ +B′m(k)+.

Taking k → ∞ and using (.), (.), and (.), we get

δ(B(xn(k),xm(k),xm(k)),B(yn(k), ym(k), ym(k))

) → .

Since δ ∈ �, we have B(xn(k),xm(k),xm(k)) → E and B(yn(k), ym(k), ym(k))→ E , that is,

B(xn(k),xm(k),xm(k)) +B(yn(k), ym(k), ym(k))→ E,

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which contradictions with (.). Therefore, {xn} and {yn} are Cauchy sequences in (X,B).Since X complete, we get xn → x and yn → y as n→ ∞ for some x, y ∈ X.Finally, we prove that (x, y) is a coupled fixed point of F . Since F is a continuous, taking

n→ ∞ in (.), we get

x = limn→∞xn+ = lim

n→∞F(xn, yn) = F(

limn→∞xn, lim

n→∞ yn)= F(x, y)

and

y = limn→∞ yn+ = lim

n→∞F(yn,xn) = F(

limn→∞ yn, lim

n→∞xn)= F(y,x).

Therefore, x = F(x, y) and y = F(y,x), that is, F has a coupled fixed point. This completesthe proof. �

Corollary . Let (X,�) be a partially ordered set and B be a cone ball-metric induced bythe cone metric d on X with a regular cone P such that (X,B) is a complete cone ball-metricspace. Let F : X × X → X be a continuous mapping having the mixed monotone propertyon X. Suppose that there exists k ∈ [, ) such that

B(F(x, y),F(u, v),F(w, z)

)+B(

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

≤E k(B(x,u,w) +B(y, v, z)) (.)

for all x, y, z,u, v,w ∈ X for which x � u � w and y � v � z. If there exists x, y ∈ X suchthat

x � F(x, y) and y � F(y,x),

then F has a coupled fixed point, that is, there exist x, y ∈ X such that

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

In the next theorem, we omit the continuity hypothesis of F .

Theorem . Let (X,�) be a partially ordered set and B be a cone ball-metric induced bythe cone metric d on X with a regular cone P such that (X,B) is a complete cone ball-metricspace. Let F : X×X → X be amapping having themixedmonotone property on X. Supposethat there exists δ ∈ � such that

B(F(x, y),F(u, v),F(w, z)

)+B(

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

≤E δ(B(x,u,w),B(y, v, z))(B(x,u,w) +B(y, v, z)) (.)

for all x, y, z,u, v,w ∈ X for which x � u � w and y � v � z. If there exists x, y ∈ X suchthat

x � F(x, y) and y � F(y,x)

and X has the following property:

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(i) if a nondecreasing sequence {xn} converges to x, then xn � x for all n ∈N,(ii) if a nonincreasing sequence {yn} converges to y, then y � yn for all n ∈N,

then F has a coupled fixed point, that is, there exist x, y ∈ X such that

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

Proof By the similar the proof as in Theorem ., we have the nondecreasing sequence{xn} converges to x and the nonincreasing sequence {yn} converges to y for some x, y ∈ X.By (i), (ii), we get xn � x and yn � y for all n ∈N. Thus, by the rectangle inequality of B, weget

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

)+B(

F(y,x), y, y)

≤E B(F(x, y),F(xn, yn),F(xn, yn)

)+B(

F(xn, yn),x,x)

+B(F(y,x),F(yn,xn),F(yn,xn)

)+B(

F(yn,xn), y, y)

= B(F(x, y),F(xn, yn),F(xn, yn)

)+B(

F(y,x),F(yn,xn),F(yn,xn))

+B(F(xn, yn),x,x

)+B(

F(yn,xn), y, y)

≤E δ(B(x,xn,xn),B(y, yn, yn)

)[B(x,xn,xn) +B(y, yn, yn)]

+B(xn+,x,x) +B(yn+, y, y)<E

[B(x,xn,xn) +B(y, yn, yn)]+B(xn+,x,x) +B(yn+, y, y).

Taking the limit as n→ ∞, we haveB(F(x, y),x,x)+B(F(y,x), y, y) = E , and thus x = F(x, y)and y = F(y,x). Therefore, F has a coupled fixed point (x, y) in X × X. This completes theproof. �

Corollary . Let (X,�) be a partially ordered set and B be a cone ball-metric induced bythe cone metric d on X with a regular cone P such that (X,B) is a complete cone ball-metricspace. Let F : X×X → X be amapping having themixedmonotone property on X. Supposethat there exists k ∈ [, ) such that

B(F(x, y),F(u, v),F(w, z)

)+B(

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

≤E k(B(x,u,w) +B(y, v, z)) (.)

for all x, y, z,u, v,w ∈ X for which x � u � w and y � v � z. If there exists x, y ∈ X suchthat

x � F(x, y) and y � F(y,x)

and X has the following property:(i) if a nondecreasing sequence {xn} converges to x, then xn � x for all n ∈N,(ii) if a nonincreasing sequence {yn} converges to y, then y � yn for all n ∈N,

Sintunavarat et al. Fixed Point Theory and Applications 2012, 2012:128 Page 11 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/128

then F has a coupled fixed point, that is, there exist x, y ∈ X such that

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

Theorem . In addition to the hypotheses in Theorem ., suppose that x, y are com-parable then x = y, that is, x = F(x,x).

Proof From Theorem ., we have two sequences {xn}, {yn} in X such that, for all n≥ ,

xn+ = F(xn, yn) and yn+ = F(yn,xn)

and also xn → x and yn → y as n → ∞. Now, we assume that x � y. Since F has themixed monotone property, we have xn � yn for all n ∈N. From (.) and property of cone-ball metric B, we have

B(yn+,xn+,xn+)= B(yn+,xn+,xn+) +B(xn+, yn+, yn+)= B(

F(yn,xn),F(xn, yn),F(xn, yn))+B(

F(xn, yn),F(yn,xn),F(yn,xn))

≤E δ(B(yn,xn,xn),B(xn, yn, yn)

)[B(yn,xn,xn) +B(xn, yn, yn)]

= δ(B(yn,xn,xn),B(yn,xn,xn)

)[B(yn,xn,xn)

].

This implies

B(yn+,xn+,xn+) ≤E δ(B(yn,xn,xn),B(yn,xn,xn)

)[B(yn,xn,xn)]

<E B(yn,xn,xn). (.)

So we have {B(yn,xn,xn)} is a decreasing sequence. Similar to the prove in Theorem .,we get B(yn,xn,xn) → E as n→ ∞.By the rectangular inequality and (.), we have

B(y,x,x) ≤E B(y, yn+, yn+) +B(yn+,xn+,xn+) +B(xn+,x,x)<E B(y, yn+, yn+) +B(yn,xn,xn) +B(xn+,x,x).

From above inequality, taking n → ∞, we obtain that B(y,x,x) = E and then x = y. Thiscompletes the proof. �

Theorem . In addition to the hypotheses in Theorem ., suppose that x, y are com-parable then x = y, that is, x = F(x,x).

Proof By the similar method as in the prove of Theorem . and by applying Theorem .,we can get the conclusion. �

Theorem . Let (X,�) be a partially ordered set and B be a cone ball-metric induced bythe cone metric d on X with a regular cone P such that (X,B) is a complete cone ball-metric

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space. Let F : X×X → X be amapping having themixedmonotone property on X. Supposethat there exists ϑ ∈ � such that

B(F(x, y),F(u, v),F(w, z)

) ≤Eϑ

(B(x,u,w),B(y, v, z))(B(x,u,w) +B(y, v, z)) (.)

for all x, y, z,u, v,w ∈ X for which x � u � w and y � v � z. If there exists x, y ∈ X suchthat

x � F(x, y) and y � F(y,x)

and either(a) F is continuous or(b) X has the following property:

(i) if a nondecreasing sequence {xn} converges to x, then xn � x for all n ∈ N,(ii) if a nonincreasing sequence {yn} converges to y, then y� yn for all n ∈N,

then F has a coupled fixed point.

Proof For any x, y, z,u, v,w ∈ X with x� u � w and y � v� z, it follows from (.) that

B(F(x, y),F(u, v),F(w, z)

) ≤Eϑ

(B(x,u,w),B(y, v, z))(B(x,u,w) +B(y, v, z)) (.)

and

B(F(y,x),F(v,u),F(z,w)

)= B(

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

≤Eϑ

(B(z, v, y),B(w,u,x))(B(z, v, y) +B(w,u,x))

=ϑ

(B(z, v, y),B(w,u,x))(B(x,u,w),B(y, v, z)). (.)

From (.) and (.), we have

B(F(x, y),F(u, v),F(w, z)

)+B(

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

≤E[ϑ

(B(x,u,w),B(y, v, z)) + ϑ(B(z, v, y),B(w,u,x))](B(x,u,w) +B(y, v, z))

= δ(B(x,u,w),B(y, v, z)))(B(x,u,w) +B(y, v, z)) (.)

for all x, y, z,u, v,w ∈ X with x � u� w and y� v � z, where

δ(a,b) = [ϑ(a,b) + ϑ(b,a)

]

for all a,b ∈ int(P)∪ {E}. It is easy to verify that δ ∈ �. If we apply Theorems . and .,we know that F has a coupled fixed point. �

Corollary . Let (X,�) be a partially ordered set and B be a cone ball-metric induced bythe cone metric d on X with a regular cone P such that (X,B) is a complete cone ball-metric

Sintunavarat et al. Fixed Point Theory and Applications 2012, 2012:128 Page 13 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/128

space. Let F : X×X → X be amapping having themixedmonotone property on X. Supposethat there exists k ∈ [, ) such that

B(F(x, y),F(u, v),F(w, z)

) ≤Ek(B(x,u,w) +B(y, v, z)) (.)

for all x, y, z,u, v,w ∈ X for which x � u � w and y � v � z. If there exists x, y ∈ X suchthat

x � F(x, y) and y � F(y,x)

and either(a) F is continuous or(b) X has the following property:

(i) if a nondecreasing sequence {xn} converges to x, then xn � x for all n ∈N,(ii) if a nonincreasing sequence {yn} converges to y, then y � yn for all n ∈N,

then F has a coupled fixed point.

Let � denote the class of functions ξ : int(P)∪{E} → [, ) which satisfies the followingcondition:For any sequence {an} in int(P)∪ {E},

ξ (an) → =⇒ an → E .

Theorem . Let (X,�) be a partially ordered set and B be a cone ball-metric induced bythe cone metric d on X with a regular cone P such that (X,B) is a complete cone ball-metricspace. Let F : X×X → X be amapping having themixedmonotone property on X. Supposethat there exists ξ ∈ � such that

B(F(x, y),F(u, v),F(w, z)

)+B(

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

≤E ξ(B(x,u,w) +B(y, v, z))(B(x,u,w) +B(y, v, z)) (.)

for all x, y, z,u, v,w ∈ X for which x� u� w and y� v � z If there exists x, y ∈ X such that

x � F(x, y) and y � F(y,x)

and either(a) F is continuous or(b) X has the following property:

(i) if a nondecreasing sequence {xn} converges to x, then xn � x for all n ∈N,(ii) if a nonincreasing sequence {yn} converges to y, then y � yn for all n ∈N,

then F has a coupled fixed point.

Proof If we taking δ(a,b) = ξ (a+b) for all a,b ∈ int(P)∪{E} in Theorem., then, from (a),we get the conclusion. Also, if we take δ(a,b) = ξ (a + b) for all a,b ∈ int(P) ∪ {E} in The-orem ., then, from (b), we obtain the conclusion. �

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Theorem . Let (X,�) be a partially ordered set and B be a cone ball-metric inducedby the cone metric d on X with a regular cone P such that (X,B) is a complete cone ball-metric space. Let F : X × X → X be a mapping having the mixed monotone property on Xand such that F(x, y)� F(y,x), whenever x � y . Suppose that there exists δ ∈ � such that

B(F(x, y),F(u, v),F(w, z)

)+B(

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

≤E δ(B(x,u,w),B(y, v, z))(B(x,u,w) +B(y, v, z)) (.)

for all x, y, z,u, v,w ∈ X for which w� u� x ≺ y� v � z. If there exists x, y ∈ X such that

x � y, x � F(x, y) and y � F(y,x)

and either(a) F is continuous or(b) X has the following property:

(i) if a nondecreasing sequence {xn} converges to x, then xn � x for all n ∈N,(ii) if a nonincreasing sequence {yn} converges to y, then y � yn for all n ∈N,

then F has a coupled fixed point.

Proof From the assumption, there exist x, y ∈ X such that x � F(x, y) and y �F(y,x). Now, we define x, y ∈ X such that x = F(x, y) � x and y = F(y,x) � y.Further, the fact x � y, we get F(x, y) � F(y,x). Thus, we now have

x � x = F(x, y) � F(y,x) = y � y.

Let x = F(x, y) and y = F(y,x). From the fact that x � y and the mixed monotoneproperty, we have

x � x � y � y.

Continuing this procedure, we have two sequences {xn} and {yn} such that

xn+ = F(xn, yn), yn+ = F(yn,xn)

and

xn � xn+ = F(xn, yn)� F(yn,xn) = yn+ � yn

for all n ≥ . If there exists a nonnegative integer k such that xk = yk = c (say), then we have

c� F(c, c) � F(c, c) � c,

that is, c = F(c, c). Therefore, (c, c) is a coupled fixed point of F .Therefore, we assume that

xn ≺ yn (.)

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for all n ≥ . In view of (.), for all n≥ , the inequality (.) holds with

w = xn, u = xn+, x = xn+,

y = yn+, v = yn+ and z = yn.

The rest of the proof can be completed by repeating the same steps given in Theorem .and Theorem .. This completes the proof. �

Example . Let E =R be a usual norm space with a regular cone P = {z ∈R|z ≥ } andX =N∪{} be a conemetric space with ametric d(x, y) = |x– y| for all x, y ∈ X. Then (X,d)is a complete cone metric space. Therefore, we get a cone ball metric B : X × X × X → Esuch that

B(x, y, z) = max{x, y, z} – min{x, y, z}

for all x, y, z ∈ X and so (X,B) is a complete cone ball-metric space.Let a partial order � on X be defined as follows: For x, y ∈ X,

x � y holds if x ≥E y.

Let F : X ×X → X be defined by

F(x, y) =

⎧⎨⎩; if x≺ y,; if otherwise.

Let w � u � x ≺ y � v � z hold, then we have w ≥E u ≥E x >E y ≥E v ≥E z. Therefore, wehave

F(x, y) = F(u, v) = F(w, z) =

and

F(y,x) = F(v,u) = F(z,w) = .

So the left side of (.) is

B(, , ) +B(, , ) =

and then (.) is satisfied for all δ ∈ �. Thus, Theorem . is applicable to this examplewith x = and y = . Therefore, F has a coupled fixed points that is a point (, ) and(, ).

Remark . Example . is not applied by Theorems ., ., and .. This is evident bythe fact that the inequality (.), (.), and (.) are not satisfied when w = u = x = y = ,v = and z = . Moreover, the coupled fixed point is not unique.

Sintunavarat et al. Fixed Point Theory and Applications 2012, 2012:128 Page 16 of 18http://www.fixedpointtheoryandapplications.com/content/2012/1/128

4 Uniqueness of coupled fixed point in cone ball-metric spacesIn this section, we study the necessary condition for the uniqueness of a coupled fixedpoint. If (X,�) is a partially ordered set, then we endow the product of X × X with thefollowing partial order relation: for any (x, y), (u, v) ∈ X ×X,

(x, y)� (u, v) ⇐⇒ x� u and y� v.

Theorem . In addition to the hypotheses in Theorem ., suppose that, for any (x, y),(z, t) ∈ X × X, there exists a point (u, v) ∈ X × X which is comparable to (x, y) and (z, t).Then F has a unique coupled fixed point.

Proof By Theorem ., we get F has a coupled fixed point (x, y), that is,

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

We may assume that (z, t) are another coupled fixed points of F and so

z = F(z, t) and t = F(t, z).

Next, we prove that x = z and y = t. By assumption, there exists (u, v) ∈ X × X which iscomparable to (x, y) and (z, t). We put u = u and v = v and construct two sequences {un}and {vn} by

un = F(un–, vn–) and vn = F(vn–,un–)

for all n ∈ N. Since (u, v) is comparable with (x, y), we may assume that (u, v) = (u, v)�(x, y). It easy to see that (un, vn)� (x, y) for all n ∈N. From (.), we have

B(x,x,un) +B(y, y, vn)= B(

F(x, y),F(x, y),F(un–, vn–))+B(

F(y,x),F(y,x),F(vn–,un–))

≤E δ(B(x,x,un–),B(vn–, y, y)

)[B(x,x,un–, ) +B(vn–, y, y)]

<E B(x,x,un–) +B(vn–, y, y). (.)

This implies that {B(x,x,un) +B(y, y, vn)} is a decreasing sequence and so

B(x,x,un) +B(y, y, vn) → B

for some B ≥E E .Now, we show that B = E . We may assume that B >E E . By the similar method as in

the proof of Theorem ., we can conclude that

δ(B(x,x,un–),B(vn–, y, y)

) → .

Since δ ∈ �, we get B(x,x,un–) → E and B(vn–, y, y) → E . Therefore, we have

B(x,x,un–) +B(vn–, y, y) → E ,

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which is a contradiction. Thus, we have B(x,x,un) +B(vn, y, y) → E as n → ∞. Similarly,one can prove B(z, z,un) +B(vn, t, t)→ E as n→ ∞.Finally, we have

B(z,x,x) +B(y, t, t) ≤E B(z,un,un) +B(un,x,x) +B(y, vn, vn) +B(vn, t, t)= B(z, z,un) +B(x,x,un) +B(y, y, vn) +B(vn, t, t).

Taking n → ∞ in above inequalities, we have B(z,x,x) + B(y, t, t) = E , that is, z = x andy = t.For the case when (u, v) = (u, v)� (x, y) is similar. This completes the proof. �

Theorem . In addition to the hypotheses in Theorem ., suppose that, for any(x, y), (z, t) ∈ X × X, there exists a point (u, v) ∈ X × X which is comparable to (x, y) and(z, t). Then F has a unique coupled fixed point.

Proof By the similar method given in the prove of Theorem . and by applying Theo-rem ., we can get the conclusion. �

Open problems:• In our theorems, can the mixed monotone property be replaced by a more generalproperty (see the work of Sintunavarat et al. [])?

• In our theorems, can the mixed monotone property be replaced by another property(see the work of Ðorić et al. [])?

• Can the coupled fixed-point theorems in this paper be extended to coupled bestproximity point theorems (see the work of Sintunavarat et al. [])?

• Can the main results in this paper be extended to multivalued case of coupled fixedpoint?

• Can the concept of cone ball-metric be extended to another distance?

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors read and approved the final manuscript.

AcknowledgementsThe first author would like to thank the Research Professional Development Project under the Science AchievementScholarship of Thailand (SAST), the second author was supported by the Basic Science Research Program through theNational Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (GrantNo. 2011-0021821), and the third author was supported by the Commission on Higher Education, the Thailand ResearchFund and the King Mongkut’s University of Technology Thonburi (Grant No. MRG5580213) for financial support duringthe preparation of this manuscript.

Received: 11 April 2012 Accepted: 19 July 2012 Published: 3 August 2012

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doi:10.1186/1687-1812-2012-128Cite this article as: Sintunavarat et al.: Coupled fixed-point theorems for contraction mapping induced by coneball-metric in partially ordered spaces. Fixed Point Theory and Applications 2012 2012:128.