Coupled fixed-point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces

Recently, Chen et al. (Appl. Math. Lett. 25:692-697, 2012) introduced the concept of the cone ball-metric and studied the common fixed-point theorems for the stronger Meir-Keeler cone-type function in cone ball-metric spaces. The purpose of this paper is 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 give some examples to validate our main results. At the end of this paper, we give an open problem for further investigation.

MSC: 47H10, 54H25.

Fixed-point theory has been the most attractive topic to hundreds of researchers since 1922 with the celebrated Banach’s contraction principle [11]. This principle provides a technique 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 has been extended and improved by many mathematicians (see [7,9,13,15,24,31-34] and others).

In 2004, the Banach’s contraction principle was extended to metric spaces endowed with a partial ordering by Ran and Reurings [26]. Afterward, many generalizations and applications of the work of Ran and Reurings exist in the literature (see in [6,17,25]). For example, Nieto and Rodríguez-López [25] extended results of Ran and Reurings for nondecreasing mappings and studied a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

In 2006, Bhaskar and Lakshmikantham [12] first introduced the concept of the mixed monotone property. Furthermore, they proved some coupled fixed-point theorems for mapping that satisfy the mixed monotone property and give some applications in the existence and uniqueness of a solution for a periodic boundary value problem

where the function f satisfies certain conditions. Afterward, several authors studied and extended coupled fixed-point theorems of Bhaskar and Lakshmikantham [12] to different generalized condition (see, e.g., [4,5,8,22,23,28,30,35,36]).

On the other hand, the concept of cone metric spaces is a generalization of metric spaces, where each pair of points is assigned to a member of a real Banach space with a cone. This cone naturally induces a partial order in the Banach spaces. The concept of the cone metric space was reintroduced in the work of Huang and Zhang [18] where they also established the Banach’s contraction mapping principle in this space. Afterward, several authors have studied fixed point and coupled fixed-point problems in cone metric spaces. Some of these works are noted in [1-3,10,20,21,29,38]. Recently, Chen et al.[14] introduced the concept of cone ball-metric spaces and proved some fixed- point theorems in these spaces for mappings 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.[14], in this paper, we establish some coupled fixed-point theorems for a contraction mapping induced by the cone ball-metric in partially ordered spaces and also study the condition claim of the uniqueness of a coupled fixed point. An open problem is also given at the end for further investigation.

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

Definition 2.1 Recall that a binary relation ⪯ on a nonempty set X is said to be an order relation (and X equipped with ⪯ is called a partially ordered set) if it satisfies the following three properties:

(i) reflexivity: for all ,

(ii) antisymmetry: and imply ,

(iii) transitivity: and imply .

Throughout this paper denotes a partially ordered set. By holds, we mean that holds and by holds we mean that holds, but . If is a partially ordered set and is such that, for all , implies , then a mapping f is said to be nondecreasing. Similarly, a nonincreasing mapping is also defined.

Definition 2.2 ([12])

Let be a partial ordered set and be a mapping. The mapping F is said to has the mixed monotone property if F is monotone nondecreasing in its first argument and is monotone nonincreasing in its second argument, that is, for any ,

and

Definition 2.3 ([12])

Let X be a nonempty set. An element is called a coupled fixed point of the mapping if and .

Next, we give some notations and lemmas of cone metric spaces which are reintroduced by Huang and Zhang [18].

Let E be a real Banach space and denote the zero element in E. A coneP is a subset of E such that

() P is nonempty closed and ;

() if a, b are nonnegative real numbers and , then ;

() .

For any cone , the partial ordering with respect to P defined by if and only if . We write to indicate that , but , while stands for , where denotes the interior of P.

A cone P is said to be normal if there is a number such that, for all ,

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 is a sequence in E such that

for some , then there is such that . Equivalently, the cone P is said to be regular if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone (see also [27]).

Remark 2.4 ([18])

(1) If E be a real Banach space with a cone P in E and , where and , then .

(2) If , and , then there exists a positive integer N such that for all .

Lemma 2.5 ([21])

IfEbe a real Banach space with a conePinE, then we have the following:

(1) Ifandkis a nonnegative real number, then.

(2) Iffor alland, as, then.

Lemma 2.6 ([19])

IfEbe a real Banach space with a conePinE, then we have the following: for all,

(1) Ifand, then.

(2) Ifand, then.

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

Definition 2.7 ([18])

Let X be a nonempty set and E be a real Banach space equipped with the partial ordering with respect to the cone . Suppose that the mapping satisfies the following conditions:

() for all with and if and only if ;

() for all ;

() for all .

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

For other basic properties on a cone metric space, the reader can refer to [18].

Next, we give the concept of a cone ball-metric space introduced by Chen et al.[14] 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 , and a linear ordering with respect to P.

Definition 2.8 ([14])

Let be a cone metric space. A cone ball-metric with respect to the cone metric d is a function defined by

where

for is a ball in X with the center x and radius . The ordered pair is called a cone ball-metric space.

Proposition 2.9 ([14])

Ifis a cone ball-metric space, then the following statements hold:

() if;

() for allwith;

() for all;

() (symmetry in all three variables);

() for all (rectangle inequality);

() for all.

Definition 2.10 ([14])

Let be a cone ball-metric space and be a sequence in X. We say that is called:

(1) A Cauchy sequence if, for any with , there exists a positive integer N such that for all .

(2) A convergent sequence if, for any with , there exists a positive integer N such that, for all , for some . Here, x is called the limit of the sequence and is denoted by or .

Remark 2.11 We can prove easily that is a Cauchy sequence if and only if, for any with , there exists a positive integer N such that for all .

Definition 2.12 ([14])

A cone ball-metric space is said to be complete if every Cauchy sequence is convergent in X.

Proposition 2.13 ([14])

Letbe a cone ball-metric space andbe a sequence of points ofX. Then the following are equivalent:

(1) as.

(2) as.

(3) as.

(4) as.

Proposition 2.14 ([14])

Letbe a cone ball-metric space, be a sequence of points ofXand. Ifandas, then.

Proposition 2.15 ([14])

Letbe a cone ball-metric space and, andbe tree sequences inX. If, , andas, thenas.

Definition 2.16 Let be a cone ball-metric space. A mapping is said to be continuous if for any two convergent sequences and converging to x and y in X, respectively, then is convergent to .

Let Δ denote the class of all functions which satisfies the following condition.

For any sequences and in ,

Let be a usual norm space with a cone and be a cone metric space with a metric for all . The following are examples of the functions in Δ under above setting:

(1) , where .

(2) .

(3)

Next, we prove our main theorems.

Theorem 3.1Letbe a partially ordered set andbe a cone ball-metric induced by the cone metricdonXwith a regular conePsuch thatis a complete cone ball-metric space. Letbe a continuous mapping having the mixed monotone property onX. Suppose that there existssuch that

for allfor whichand. If there existssuch that

thenFhas a coupled fixed point, that is, there existsuch that

Proof We construct two sequences , in X such that, for all ,

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

From and , in case , (3.3) holds. Assume that (3.3) holds for some . Then we get

Since F has the mixed monotone property, it follows from (3.4) and (2.1) that

for all . From (3.4) and (2.2), we have

for all .

If we take and in (3.5), we get

If we take and in (3.6), we get

From (3.7) and (3.8), we also have

Consequently, by induction, we have (3.3) holds for all . This implies

and

If there exists an integer number such that

then we have

which implies and . Therefore, and and so is a coupled fixed point of F.

Now, we assume that for all . Since and for all , by (3.1) and (3.2), we have

So, we have the sequence defined by is a decreasing sequence. Since P is regular, there exists such that as .

Next, we prove that . Suppose that . From (3.12), letting , we have

Since , we have and and then

which contradictions with . Consequently, we must get

By the property (), we have

Now, we show that and are Cauchy sequences in cone ball-metric space . Suppose on the contrary that at least one of and are not a Cauchy sequence in . Then there exists such that and sequences of positive integers and such that for all positive integers k,

and

Further, for the integer , we can choose is the smallest integer for which (3.15) holds. Then we have

Using (3.15) and (3.16) and the rectangle inequality, we have

Letting , we have

By the rectangle inequality, we get

Taking and using (3.13), (3.14), and (3.17), we get

Since , we have and , that is,

which contradictions with (3.17). Therefore, and are Cauchy sequences in . Since X complete, we get and as for some .

Finally, we prove that is a coupled fixed point of F. Since F is a continuous, taking in (3.2), we get

and

Therefore, and , that is, F has a coupled fixed point. This completes the proof. □

Corollary 3.2Letbe a partially ordered set andbe a cone ball-metric induced by the cone metricdonXwith a regular conePsuch thatis a complete cone ball-metric space. Letbe a continuous mapping having the mixed monotone property onX. Suppose that there existssuch that

for allfor whichand. If there existssuch that

thenFhas a coupled fixed point, that is, there existsuch that

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

Theorem 3.3Letbe a partially ordered set andbe a cone ball-metric induced by the cone metricdonXwith a regular conePsuch thatis a complete cone ball-metric space. Letbe a mapping having the mixed monotone property onX. Suppose that there existssuch that

for allfor whichand. If there existssuch that

andXhas the following property:

(i) if a nondecreasing sequenceconverges tox, thenfor all,

(ii) if a nonincreasing sequenceconverges toy, thenfor all,

thenFhas a coupled fixed point, that is, there existsuch that

Proof By the similar the proof as in Theorem 3.1, we have the nondecreasing sequence converges to x and the nonincreasing sequence converges to y for some . By (i), (ii), we get and for all . Thus, by the rectangle inequality of , we get

Taking the limit as , we have , and thus and . Therefore, F has a coupled fixed point in . This completes the proof. □

Corollary 3.4Letbe a partially ordered set andbe a cone ball-metric induced by the cone metricdonXwith a regular conePsuch thatis a complete cone ball-metric space. Letbe a mapping having the mixed monotone property onX. Suppose that there existssuch that

for allfor whichand. If there existssuch that

andXhas the following property:

(i) if a nondecreasing sequenceconverges tox, thenfor all,

(ii) if a nonincreasing sequenceconverges toy, thenfor all,

thenFhas a coupled fixed point, that is, there existsuch that

Theorem 3.5In addition to the hypotheses in Theorem 3.1, suppose that, are comparable then, that is, .

Proof From Theorem 3.1, we have two sequences , in X such that, for all ,

and also and as . Now, we assume that . Since F has the mixed monotone property, we have for all . From (3.1) and property of cone-ball metric , we have

This implies

So we have is a decreasing sequence. Similar to the prove in Theorem 3.1, we get as .

By the rectangular inequality and (3.21), we have

From above inequality, taking , we obtain that and then . This completes the proof. □

Theorem 3.6In addition to the hypotheses in Theorem 3.3, suppose that, are comparable then, that is, .

Proof By the similar method as in the prove of Theorem 3.5 and by applying Theorem 3.3, we can get the conclusion. □

Theorem 3.7Letbe a partially ordered set andbe a cone ball-metric induced by the cone metricdonXwith a regular conePsuch thatis a complete cone ball-metric space. Letbe a mapping having the mixed monotone property onX. Suppose that there existssuch that

for allfor whichand. If there existssuch that

and either

(a) Fis continuous or

(b) Xhas the following property:

(i) if a nondecreasing sequenceconverges tox, thenfor all,

(ii) if a nonincreasing sequenceconverges toy, thenfor all,

thenFhas a coupled fixed point.

Proof For any with and , it follows from (3.22) that

and

From (3.23) and (3.24), we have

for all with and , where

for all . It is easy to verify that . If we apply Theorems 3.1 and 3.3, we know that F has a coupled fixed point. □

Corollary 3.8Letbe a partially ordered set andbe a cone ball-metric induced by the cone metricdonXwith a regular conePsuch thatis a complete cone ball-metric space. Letbe a mapping having the mixed monotone property onX. Suppose that there existssuch that

for allfor whichand. If there existssuch that

and either

(a) Fis continuous or

(b) Xhas the following property:

(i) if a nondecreasing sequenceconverges tox, thenfor all,

(ii) if a nonincreasing sequenceconverges toy, thenfor all,

thenFhas a coupled fixed point.

Let Ξ denote the class of functions which satisfies the following condition:

For any sequence in ,

Theorem 3.9Letbe a partially ordered set andbe a cone ball-metric induced by the cone metricdonXwith a regular conePsuch thatis a complete cone ball-metric space. Letbe a mapping having the mixed monotone property onX. Suppose that there existssuch that

for allfor whichandIf there existssuch that

and either

(a) Fis continuous or

(b) Xhas the following property:

(i) if a nondecreasing sequenceconverges tox, thenfor all,

(ii) if a nonincreasing sequenceconverges toy, thenfor all,

thenFhas a coupled fixed point.

Proof If we taking for all in Theorem 3.1, then, from (a), we get the conclusion. Also, if we take for all in Theorem 3.3, then, from (b), we obtain the conclusion. □

Theorem 3.10Letbe a partially ordered set andbe a cone ball-metric induced by the cone metricdonXwith a regular conePsuch thatis a complete cone ball-metric space. Letbe a mapping having the mixed monotone property onXand such that, whenever . Suppose that there existssuch that

for allfor which. If there existssuch that

and either

(a) Fis continuous or

(b) Xhas the following property:

(i) if a nondecreasing sequenceconverges tox, thenfor all,

(ii) if a nonincreasing sequenceconverges toy, thenfor all,

thenFhas a coupled fixed point.

Proof From the assumption, there exist such that and . Now, we define such that and . Further, the fact , we get . Thus, we now have

Let and . From the fact that and the mixed monotone property, we have

Continuing this procedure, we have two sequences and such that

and

for all . If there exists a nonnegative integer k such that (say), then we have

that is, . Therefore, is a coupled fixed point of F.

Therefore, we assume that

for all . In view of (3.29), for all , the inequality (3.28) holds with

The rest of the proof can be completed by repeating the same steps given in Theorem 3.1 and Theorem 3.3. This completes the proof. □

Example 3.11 Let be a usual norm space with a regular cone and be a cone metric space with a metric for all . Then is a complete cone metric space. Therefore, we get a cone ball metric such that

for all and so is a complete cone ball-metric space.

Let a partial order ⪯ on X be defined as follows: For ,

Let be defined by

Let hold, then we have . Therefore, we have

and

So the left side of (3.28) is

and then (3.28) is satisfied for all . Thus, Theorem 3.10 is applicable to this example with and . Therefore, F has a coupled fixed points that is a point and .

Remark 3.12 Example 3.11 is not applied by Theorems 3.1, 3.3, and 3.9. This is evident by the fact that the inequality (3.1), (3.19), and (3.27) are not satisfied when , and . Moreover, the coupled fixed point is not unique.

In this section, we study the necessary condition for the uniqueness of a coupled fixed point. If is a partially ordered set, then we endow the product of with the following partial order relation: for any ,

Theorem 4.1In addition to the hypotheses in Theorem 3.1, suppose that, for any, there exists a pointwhich is comparable toand. ThenFhas a unique coupled fixed point.

Proof By Theorem 3.1, we get F has a coupled fixed point , that is,

We may assume that are another coupled fixed points of F and so

Next, we prove that and . By assumption, there exists which is comparable to and . We put and and construct two sequences and by

for all . Since is comparable with , we may assume that . It easy to see that for all . From (3.1), we have

This implies that is a decreasing sequence and so

for some .

Now, we show that . We may assume that . By the similar method as in the proof of Theorem 3.1, we can conclude that

Since , we get and . Therefore, we have

which is a contradiction. Thus, we have as . Similarly, one can prove as .

Finally, we have

Taking in above inequalities, we have , that is, and .

For the case when is similar. This completes the proof. □

Theorem 4.2In addition to the hypotheses in Theorem 3.3, suppose that, for any, there exists a pointwhich is comparable toand. ThenFhas a unique coupled fixed point.

Proof By the similar method given in the prove of Theorem 4.1 and by applying Theorem 3.3, we can get the conclusion. □

Open problems:

• In our theorems, can the mixed monotone property be replaced by a more general property (see the work of Sintunavarat et al.[36])?

• In our theorems, can the mixed monotone property be replaced by another property (see the work of Ðorić et al.[16])?

• Can the coupled fixed-point theorems in this paper be extended to coupled best proximity point theorems (see the work of Sintunavarat et al.[37])?

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

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

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.

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

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