Coupled coincidence point and common coupled fixed point theorems lacking the mixed monotone property

In this paper, we prove the coupled coincidence point theorems for a -compatible mapping in partially ordered cone metric spaces over a solid cone without the mixed g-monotone property. In the case of a totally ordered space, these results are automatically obvious under the assumption given. Therefore, these results can be applied in a much wider class of problems. We also prove the uniqueness of a common coupled fixed point in this setup and give some example which is not applied to the existence of a common coupled fixed point by using the mixed g-monotone property but can be applied to our results.

MSC: 47H10, 54H25.

The famous Banach contraction principle states that if is a complete metric space and is a contraction mapping (i.e., for all , where α is a non-negative number such that ), then T has a unique fixed point. This principle is one of the cornerstones in the development of nonlinear analysis. Fixed point theorems have applications not only in the various branches of mathematics, but also in economics, chemistry, biology, computer science, engineering, and others. Due to the importance, generalizations of Banach’s contraction principle have been investigated heavily by several authors.

Following this trend, the problem of existence and uniqueness of fixed points in partially ordered sets has been studied thoroughly because of its interesting nature. In 1986, Turinici [1] presented the first result in this direction. Afterward, Ran and Reurings [2] gave some applications of Turinici’s theorem to matrix equations. The results of Ran and Reurings were further extended to ordered cone metric spaces in [3-5]. In 2005, Nieto and Rodríguez-López [6] extended Ran and Reurings’s theorems for nondecreasing mappings and obtained a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

The notion of coupled fixed points was introduced by Guo and Lakshmikantham [7]. Since then, the concept has been of interest to many researchers in metrical fixed point theory. In 2006, Bhaskar and Lakshmikantham [8] introduced the concept of a mixed monotone property (see further Definition 2.4). They proved classical coupled fixed point theorems for mappings satisfying the mixed monotone property and also discussed an application of their result by investigating the existence and uniqueness of a solution of the periodic boundary value problem. Following this result, Harjani et al.[9] (see also [10,11]) studied the existence and uniqueness of solutions of a nonlinear integral equation as an application of coupled fixed points. Very recently, motivated by the work of Caballero et al.[12], Jleli and Samet [13] discussed the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem

where such that , is the Riemann-Liouville fractional derivative and is continuous, (f is singular at ) for all , is nondecreasing with respect to the first component and decreasing with respect to its second and third components.

Since their important role in the study of the existence and uniqueness of a solution of the periodic boundary value problem, a nonlinear integral equation, and the existence and uniqueness of a positive solution for the singular nonlinear fractional differential equation boundary value problem, a wide discussion on coupled fixed point theorems aimed the interest of many scientists.

In 2009, Lakshmikantham and Ćirić [14] extended the concept of a mixed monotone property to a mixed g-monotone mapping and proved coupled coincidence point and common coupled fixed point theorems which are more general than the result of Bhaskar and Lakshmikantham in [8]. A number of articles on coupled fixed point, coupled coincidence point, and common coupled fixed point theorems have been dedicated to the improvement; see [15-30] and the references therein.

On the other hand, in 2007, Huang and Zhang [31] have re-introduced the concept of a cone metric space which is replacing the set of real numbers by an ordered Banach space E. They went further and defined the convergence via interior points of the cone by which the order in E is defined. This approach allows the investigation of cone spaces in the case when the cone is not necessarily normal. They also continued with results concerned with the normal cones only. One of the main results from [31] is the Banach contraction principle in the setting of normal cone spaces. Afterward, many authors generalized their fixed point theorems in cone spaces with normal cones. In other words, the fixed point problem in the setting of cone metric spaces is appropriate only in the case when the underlying cone is non-normal but just has interior that is nonempty. In this case only, proper generalizations of results from the ordinary metric spaces can be obtained. In 2011, Janković et al.[32] gave some examples showing that theorems from ordinary metric spaces cannot be applied in the setting of cone metric spaces, when the cone is non-normal.

Recently, Nashine et al.[33] established common coupled fixed point theorems for mixed g-monotone and -compatible mappings satisfying more general contractive conditions in ordered cone metric spaces over a cone that is only solid (i.e., has a nonempty interior) which improve works of Karapınar [34] and Shatanawi [35]. This result is an ordered version extension of the results of Abbas et al.[36].

In this work, we show that the mixed g-monotone property in common coupled fixed point theorems in ordered cone metric spaces can be replaced by another property due to Ðorić et al.[37]. This property is automatically satisfied in the case of a totally ordered space. Therefore, these results can be applied in a much wider class of problems. Our results generalize and extend many well-known comparable results in the literature. An illustrative example is presented in this work when our results can be used in proving the existence of a common coupled fixed point, while the results of Nashine et al.[33] cannot.

In this section, we give some notations and a property that are useful for our main results. Let E be a real Banach space with respect to a given norm and let be a zero vector of E. A nonempty subset P of E is called a cone if the following conditions hold:

1. P is closed and ;

2. , , ;

3. , .

Given a cone , a partial ordering with respect to P is naturally defined by if and only if for . We will write to indicate that but , while will stand for , where denotes the interior of P.

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

The least positive number K satisfying the above statement is called a normal constant of P. In 2008, Rezapour and Hamlbarani [38] showed that there are no normal cones with a normal constant .

In what follows, we always suppose that E is a real Banach space with cone P satisfying (such cones are called solid).

Definition 2.1 ([31])

Let X be a nonempty set and satisfy

1. for all and if and only if ;

2. for all ;

3. for all .

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

Definition 2.2 ([31])

Let be a cone metric space, be a sequence in X, and .

1. If for every with , there is such that for all , then is said to converge to x. This limit is denoted by or as .

2. If for every with , there is such that for all , then is called a Cauchy sequence in X.

3. If every Cauchy sequence in X is convergent in X, then is called a complete cone metric space.

Let be a cone metric space. Then the following properties are often used (particularly when dealing with cone metric spaces in which the cone need not be normal):

() if , where and , then ;

() if for each , then ;

() if , and , then ;

() if , and , then there exists such that for all , we have .

Definition 2.3 Let X be a nonempty set. Then is called an ordered cone metric space if

(i) is a cone metric space,

(ii) is a partially ordered set.

Let be a partially ordered set. By , we mean for . Elements are called comparable if or holds. A mapping f is said to be g-nondecreasing (resp., g-nonincreasing) if, for all , implies (resp., ). If g is the identity mapping, then f is said to be nondecreasing (resp., nonincreasing).

Definition 2.4 ([8,14])

Let be a partially ordered set and let and . The mapping F is said to have a mixedg-monotone property if F is monotone g-nondecreasing in its first argument and monotone g-nonincreasing in its second argument, that is, for any ,

and

hold. If in the previous relations g is the identity mapping, then it is said that F has a mixed monotone property.

Definition 2.5 ([8,14])

Let X be a nonempty set and , . An element is called

(C₁) a coupled fixed point of F if and ;

(C₂) a coupled coincidence point of mappings g and F if

and in this case is called a coupled point of coincidence;

(C₃) a common coupled fixed point of mappings g and F if

Definition 2.6 ([36])

Let X be a nonempty set. Mappings and are called

() w-compatible if whenever and ;

() -compatible if whenever .

It is easy to see that w-compatible implies -compatible. The following example shows that the converse of the above argument is not true.

Example 2.7 Let and and be defined by

It is easy to see that and , but . Hence, F and g are not w-compatible.

However, is possible only if and for all points in this case, we get . Therefore, F and g are -compatible.

For elements x, y of a partially ordered set , we will write whenever x and y are comparable (i.e., or holds).

Next, we give a new property due to Ðorić et al.[37].

Let X be a nonempty set and let and . We will consider the following condition:

In particular, when , it reduces to

Remark 2.8 We obtain that the conditions (2.3) and (2.4) are trivially satisfied if is the totally ordered.

The following examples show that the condition (2.3) ((2.4), resp.) may be satisfied when F does not have the mixed g-monotone property (monotone property, resp.).

Example 2.9 Let , ,

for all . Since but for all , the mapping F does not have the mixed g-monotone property. But it has property (2.3) since

(1) For each , we get and for all .

(2) For each , we get and for all .

Example 2.10 Let , ,

for all . Since but for all , the mapping F does not have the mixed monotone property. But it has property (2.4) since

(1) For each , we get and for all .

(2) For each , we get and for all .

(3) The other two cases are trivial.

In this section, we give the existence of coupled coincidence point theorems in ordered cone metric spaces lacking the mixed g-monotone property. Our first main result is the following theorem.

Theorem 3.1Letbe an ordered cone metric space over a solid conePand letand. Suppose that the following hold:

(i) andis a complete subspace ofX;

(ii) gandFsatisfy property (2.3);

(iii) there existsuch thatand;

(iv) there existsforandsuch that for allsatisfyingand,

holds;

(v) ifwheninX, thenfornsufficiently large.

Then there existsuch that

that is, Fandghave a coupled coincidence point.

Proof Starting from , (condition (iii)) and using the fact that (condition (i)), we can construct sequences and in X such that

for all . By (iii), we get , and the condition (ii) implies that

Proceeding by induction, we get that and, similarly, for all . Therefore, we can apply the condition (3.1) to obtain

which implies that

Similarly, starting with and using and for all , we get

Combining (3.3) and (3.4), we obtain that

Now, starting from and using and for all , we get that

Similarly, starting from and using and for all , we get that

Again adding up, we obtain that

Finally, adding up (3.5) and (3.6), it follows that

with

since .

From the relation (3.7), we have

If , then is a coupled coincidence point of F and g. So, let .

For any , repeated use of the triangle inequality gives

Since as , we get as .

From (), we have for and large n,

By (), we get

Since

and

then by (), we get and for n large enough. Therefore, we get and are Cauchy sequences in . By completeness of , there exist such that and as .

By (v), we have and for all . Now, we prove that and .

If and for some , from (3.1) we have

which further implies that

Since , then for , there exists such that

for all . Therefore,

Now, according to (), it follows that and . Similarly, we can prove that . Hence, is a coupled coincidence point of the mappings F and g.

So, we suppose that for all . Using (3.1), we get

which further implies that

Since and , then for , there exists such that , , and for all . Thus,

Now, according to (), it follows that and . Similarly, . Hence, is a coupled coincidence point of the mappings F and g. □

Remark 3.2 In Theorem 3.1, the condition (ii) is a substitution for the mixed g-monotone property that has been used in most of the coupled coincidence point theorems so far. Therefore, Theorem 3.1 improves the results of Nashine et al.[33]. Moreover, it is an ordered version extension of the results of Abbas et al.[36].

Corollary 3.3Letbe an ordered cone metric space over a solid conePand letand. Suppose that the following hold:

(i) andis a complete subspace ofX;

(ii) gandFsatisfy property (2.3);

(iii) there existsuch thatand;

(iv) there existandsuch that for allsatisfyingand,

holds;

(v) ifwheninX, thenfornsufficiently large.

Then there existsuch that

that is, Fandghave a coupled coincidence point.

Putting , where is the identity mapping from X into X in Theorem 3.1, we get the following corollary.

Corollary 3.4Letbe an ordered cone metric space over a solid conePand let. Suppose that the following hold:

(i) Xis complete;

(ii) gandFsatisfy property (2.4);

(iii) there existsuch thatand;

(iv) there existsforandsuch that for allsatisfyingand,

holds;

(v) ifwheninX, thenfornsufficiently large.

Then there existsuch that

that is, Fhas a coupled fixed point.

Our second main result is the following.

Theorem 3.5Letbe an ordered cone metric space over a solid coneP. Letandbe mappings. Suppose that the following hold:

(i) andis a complete subspace ofX;

(ii) gandFsatisfy property (2.3);

(iii) there existsuch thatand;

(iv) there is somesuch that for allsatisfyingand, there exists

such that

(v) ifwheninX, thenfornsufficiently large.

Then there existsuch that

that is, Fandghave a coupled coincidence point.

Proof Since (condition (i)), we can start from , (condition (iii)) and construct sequences and in X such that

for all . From (iii), we get and the condition (ii) implies that

By repeating this process, we have . Similarly, we can prove that for all .

Since and for all , from (iv), we have that there exist and

such that

Similarly, one can show that there exists

such that

Now, denote . Since the cases and are trivial, we have to consider the following four possibilities.

Case 1. and . Adding up, we get that

Case 2. and . Then

Case 3. and . This case is treated analogously to Case 1.

Case 4. and . This case is treated analogously to Case 2.

Thus, in all cases, we get for all , where . Therefore,

and by the same argument as in Theorem 3.1, it is proved that and are Cauchy sequences in . By the completeness of , there exist such that and .

From (v), we get and for all . Now, we prove that and .

If and for some , from (iv) we have

where . Let be fixed. If or , then for n sufficiently large, we have that

By property (), it follows that . If , then we get that

Now, it follows that for n sufficiently large,

Therefore, again by property (), we get that . Similarly, we can prove that . Hence, is a coupled point of coincidence of F and g.

Then, we suppose that for all . For this, consider

where . Let be fixed. If or , then for n sufficiently large, we have that

By property (), it follows that . If , then we get that

Now, it follows that for n sufficiently large,

Thus, again by property (), we get that .

Similarly, is obtained. Hence, is a coupled point of coincidence of the mappings F and g. □

Remark 3.6 It would be interesting to relate our Theorem 3.5 with Theorem 2.1 of Long et al.[39].

Putting , where is the identity mapping from X into X in Theorem 3.5, we get the following corollary.

Corollary 3.7Letbe an ordered cone metric space over a solid coneP. Letbe mappings. Suppose that the following hold:

(i) Xis complete;

(ii) Fsatisfies property (2.4);

(iii) there existsuch thatand;

(iv) there is somesuch that for allsatisfyingand, there exists

such that

(v) ifwheninX, thenfornsufficiently large.

Then there existsuch that

that is, Fhas a coupled fixed point.

Some questions arise naturally from Theorems 3.1 and 3.5. For example, one may ask if there are necessary conditions for the existence and uniqueness of a common coupled fixed point of F and g?

The next theorem provides a positive answer to this question with additional hypotheses to Theorems 3.1 and 3.5.

For the given partial order ⪯ on the set X, we will denote also by ⪯ the order on given by

Theorem 4.1In addition to the hypotheses of Theorem 3.1, suppose that for every, there existssuch that

and

IfFandgare-compatible, thenFandghave a unique common coupled fixed point, that is, there exists a uniquesuch that

Proof From Theorem 3.1, the set of coupled coincidence points of F and g is nonempty. Suppose and are coupled coincidence points of F, that is, , , and . We will prove that

By assumption, there exists such that

and

Put , and choose so that and . Then, similarly as in the proof of Theorem 3.1, we can inductively define sequences , with

for all n. Further, set , , , and, in a similar way, define the sequences , and , . Then it is easy to show that

and

as .

Since

we have and . It is easy to show that, similarly,

for all , that is, and for all . Thus, from (3.1), we have

that is,

In the same way, starting from , we can show that

Thus,

In a similar way, starting from , resp. , and adding up the obtained inequalities, one gets that

Finally, adding up (4.3) and (4.4), we obtain that

where λ is determined as in (3.8), and hence .

By inequality (4.5) n time, we have

It follows from as that

for all and large n. Since

it follows by () that for large n, and so when . Similarly, when . By the same procedure, one can show that and as . By the uniqueness of the limit, we get and , i.e., (4.2) is proved. Therefore, is the unique coupled point of coincidence of F and g.

Note that if is a coupled point of coincidence of F and g, then is also a coupled point of coincidence of F and g. Then and therefore is the unique coupled point of coincidence of F and g.

Next, we show that F and g have a common coupled fixed point. Let . Then we have . Since F and g are -compatible, we have

Thus, is a coupled point of coincidence of F and g. By the uniqueness of a coupled point of coincidence of F and g, we get . Therefore, , that is, is a common coupled fixed point of F and g.

Finally, we show the uniqueness of a common coupled fixed point of F and g. Let be another common coupled fixed point of F and g. So,

Then and are two common coupled points of coincidence of F and g and, as was previously proved, it must be , and so . This completes the proof. □

Next, we give some illustrative example which supports Theorem 4.1, while the results of Nashine et al.[33] do not.

Example 4.2 Let be ordered by the following relation:

Let with for all and

It is well known (see, e.g., [40]) that the cone P is not normal. Let

for all , for a fixed (e.g., for ). Then is a complete ordered cone metric space over a non-normal solid cone.

Let and be defined by

Consider and , we have for , we get , but

So, the mapping F does not satisfy the mixed g-monotone property. Therefore, Theorems 3.1 and 3.2 of Nashine et al.[33] cannot be used to reach this conclusion.

Now, we show that Theorem 4.1 can be used for this case.

Take and . We will check that the condition (3.1) in Theorem 3.1 holds.

For satisfying and , we have

Next, we show that F and g are -compatible. We note that if , then we get only one case, that is, , and hence

Therefore, F and g are -compatible.

Moreover, other conditions in Theorem 4.1 are also satisfied. Now, we can apply Theorem 4.1 to conclude the existence of a unique common coupled fixed point of F and g that is a point .

The following uniqueness result corresponding to Theorem 3.5 can be proved in the same way as Theorem 4.1.

Theorem 4.3In addition to the hypotheses of Theorem 3.5, suppose that for every, there existssuch that

and

IfFandgare-compatible, thenFandghave a unique coupled common fixed point, that is, there exists a uniquesuch that

The authors declare that they have no competing interests.

All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

The authors would like to thank Professor Hichem Ben-El-Mechaiekh and the referee for valuable comments. The second author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST) and the third author would like to thank the Commission on Higher Education, the Thailand Research Fund and KMUTT under Grant No. MRG5580213 for financial support during the preparation of this manuscript.

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