Open Access

# Coupled fixed point theorems for nonlinear contractions without mixed monotone property

Fixed Point Theory and Applications20122012:170

DOI: 10.1186/1687-1812-2012-170

Accepted: 19 September 2012

Published: 3 October 2012

## Abstract

In this paper, we show the existence of a coupled fixed point theorem of nonlinear contraction mappings in complete metric spaces without the mixed monotone property and give some examples of a nonlinear contraction mapping, which is not applied to the existence of coupled fixed point by using the mixed monotone property. We also study the necessary condition for the uniqueness of a coupled fixed point of the given mapping. Further, we apply our results to the existence of a coupled fixed point of the given mapping in partially ordered metric spaces. Moreover, some applications to integral equations are presented.

MSC:47H10, 54H25.

### Keywords

coupled fixed point F-invariant set transitive property mixed monotone property partially ordered set

## 1 Introduction

Let X be an arbitrary nonempty set. A fixed point for a self mapping is a point such that . The applications of fixed point theorems are very important in diverse disciplines of mathematics, statistics, chemistry, biology, computer science, engineering and economics in dealing with problems arising in approximation theory, potential theory, game theory, mathematical economics, theory of differential equations, theory of integral equations, theory of matrix equations etc. (see, e.g., [16]). For example, fixed point theorems are incredibly useful when it comes to prove the existence of various types of Nash equilibria (see, e.g., [1]) in economics. Fixed point theorems are also helpful for proving the existence of weak periodic solutions for a model describing the electrical heating of a conductor taking into account the Joule-Thomson effect (see, e.g., [7]).

One of the very popular tools of a fixed point theory is the Banach contraction principle which first appeared in 1922. It states that if is a complete metric space and is a contraction mapping (i.e., for all , where k is a non-negative number such that ), then T has a unique fixed point. Several mathematicians have been dedicated to improvement and generalization of this principle (see [814]).

Especially, in 2004, Ran and Reurings [15] showed the existence of fixed points of nonlinear contraction mappings in metric spaces endowed with a partial ordering and presented applications of their results to matrix equations. Since 2004 some authors have studied fixed point theorems in partially ordered metric spaces (see [1619] and references therein). Subsequently, Nieto and Rodríguez-López [18] extended the results in [15] for non-decreasing mappings and obtained a unique solution for a first-order ordinary differential equation with periodic boundary conditions (see also [19]).

One of the interesting and crucial concepts, a coupled fixed point theorem, was introduced by Guo and Lakshmikantham [20]. In 2006 Bhaskar and Lakshmikantham [21] introduced the notion of the mixed monotone property of a given mapping. Furthermore, they proved some coupled fixed point theorems for mappings which satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. They also established the classical coupled fixed point theorems and gave some of their applications. The main results of Bhaskar and Lakshmikantham are as follows.

Theorem 1.1 (Bhaskar and Lakshmikantham [21])

Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on X. Assume that there exists a with
(1.1)
for all for which and . If there exists such that

then there exists such that and .

Theorem 1.2 (Bhaskar and Lakshmikantham [21])

Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Suppose that X has the following property:
1. (i)

if is a non-decreasing sequence with , then for all ,

2. (ii)

if is a non-increasing sequence with , then for all .

Let be a mapping having the mixed monotone property on X. Assume that there exists with
(1.2)
for all for which and . If there exists such that

then there exists such that and .

Because of the important role of Theorems 1.1 and 1.2 in nonlinear differential equations, nonlinear integral equations and differential inclusions, many authors have studied the existence of coupled fixed points of the given mappings in several spaces and applications (see [2231] and references therein).

In this paper, we establish the existence of a coupled fixed point of the given mapping in complete metric spaces without the mixed monotone property. We also give some illustrative examples to illustrate our main theorems. Furthermore, we find the necessary condition to guarantee the uniqueness of the coupled fixed point. Our results improve and extend some coupled fixed point theorems of Bhaskar and Lakshmikantham [21] and others. As an application, we apply the main results to the setting of partially ordered metric spaces and also present some applications to integral equations.

## 2 Preliminaries

In this section, we give some definitions, examples and remarks which are useful for main results in this paper.

Throughout this paper, denotes a collection of subsets of X, and denotes a partially ordered set with the partial order . By , we mean . A mapping is said to be non-decreasing (resp., non-increasing) if for all , implies (resp., ).

Definition 2.1 (Bhaskar and Lakshmikantham [21])

Let be a partially ordered set and . The mapping F is said to have the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any ,
(2.1)
and
(2.2)

Definition 2.2 (Bhaskar and Lakshmikantham [21])

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

Example 2.3 Let and be defined by

for all . It is easy to see that F has a unique coupled fixed point .

Example 2.4 Let and be defined by

for all . We can see that a coupled fixed point of F is , where and are disjoint sets.

Next, we give the notion of an F-invariant set which is due to Samet and Vetro [32].

Definition 2.5 (Samet and Vetro [32])

Let be a metric space and be a given mapping. Let M be a nonempty subset of . We say that M is an F-invariant subset of if and only if, for all ,
1. (i)

;

2. (ii)

.

Here, we introduce the new property which is useful for our main results.

Definition 2.6 Let be a metric space and M be a subset of . We say that M satisfies the transitive property if and only if, for all ,

Remark 2.7 We can easily check that the set is trivially F-invariant, which satisfies the transitive property.

Example 2.8 Let endowed with the usual metric and be defined by

It easy to see that is F-invariant, which satisfies the transitive property.

Example 2.9 Let endowed with the usual metric and be defined by

It easy to see that is F-invariant, which satisfies the transitive property.

The following example plays a key role in the proof of our main results in a partially ordered set.

Example 2.10 Let be a metric space endowed with a partial order . Let be a mapping satisfying the mixed monotone property, that is, for all , we have
(2.3)
and
(2.4)
Define a subset by

Then M is an F-invariant subset of , which satisfies the transitive property.

## 3 Coupled fixed point theorems without the mixed monotone property

Theorem 3.1 Let be a complete metric space and M be a nonempty subset of . Assume that there is a function with and for each , and also suppose that is a mapping such that
(3.1)
for all . Suppose that either
1. (a)

F is continuous or

2. (b)
if for any two sequences , with ,

for all , then for all .

If there exists such that and M is an F-invariant set which satisfies the transitive property, then there exists such that and , that is, F has a coupled fixed point.

Proof From , we can construct two sequences and in X such that
(3.2)
for all . If there exists such that and , then

Thus, is a coupled fixed point of F. This finishes the proof. Therefore, we may assume that or for all .

Since and M is an F-invariant set, we get
Again, using the fact that M is an F-invariant set, we have
By repeating this argument, we get

for all . Denote for all .

Now, we show that
for all . Since for all , from (3.1), it follows that
(3.3)
Since M is an F-invariant set and for all , we get for all . From (3.1) and for all , we get
(3.4)
Adding (3.3) and (3.4), we get
(3.5)
for all . From (3.5) and for all , we have

for all , that is, is a monotone decreasing sequence. Therefore, for some .

Now, we show that . Suppose that . Taking of both sides of (3.5), from for all , it follows that
which is a contradiction. Thus, and
(3.6)
Next, we prove that and are Cauchy sequences. Suppose that at least one, or , is not a Cauchy sequence. Then there exists and two subsequences of integers and with such that
(3.7)
for all . Further, corresponding to , we can choose in such a way that it is the smallest integer with satisfying (3.7). Then we have
(3.8)
Using (3.7), (3.8) and the triangle inequality, we have
(3.9)

Letting and using (3.6), we have .

Since and M satisfies the transitive property, we get
(3.10)
From (3.1) and (3.10), we get
(3.11)
and
(3.12)
Adding (3.11) and (3.12), we get
(3.13)
for all . Taking of both sides of (3.13), from for all , it follows that
which is a contradiction. Therefore, and are Cauchy sequences. Since X is complete, there exists such that
(3.14)
Finally, we show that and . If the assumption (a) holds, then we have
(3.15)
and
(3.16)

Therefore, and , that is, F has a coupled fixed point.

Suppose that (b) holds. We obtain that a sequence converges to x and a sequence converges to y for some . By the assumption, we have for all . Since for all , by the triangle inequality and (3.1), we get
(3.17)

Taking , we have , and so . Similarly, we can conclude that . Therefore, F has a coupled fixed point. This completes the proof. □

Now, we give an example to validate Theorem 3.1.

Example 3.2 Let endowed with the usual metric for all and endowed with the usual partial order defined by . Define a continuous mapping by

for all . Let and . Then we have , but , and so the mapping F does not satisfy the mixed monotone property.

Now, let be a function defined by for all . Then we obtain and for any . By simple calculation, we see that for all ,

Therefore, if we apply Theorem 3.1 with , we know that F has a unique coupled fixed point, that is, a point is a unique coupled fixed point.

Remark 3.3 Although the mixed monotone property is an essential tool in the partially ordered metric spaces to show the existence of coupled fixed points, the mappings do not have the mixed monotone property in a general case as in the above example. Therefore, Theorem 3.1 is interesting, as a new auxiliary tool, in showing the existence of a coupled fixed point.

If we take the mapping for some in Theorem 3.1, then we get the following:

Corollary 3.4 Let be a complete metric space and M be a nonempty subset of . Suppose that is a mapping such that there exists such that
(3.18)
for all . Suppose that either
1. (a)

F is continuous or

2. (b)
for any two sequences , with , if

for all , then for all .

If there exists such that and M is an F-invariant set which satisfies the transitive property, then there exists such that and , that is, F has a coupled fixed point.

Now, from Theorem 3.1, we have the following question:

(Q1) Is it possible to guarantee the uniqueness of the coupled fixed point of F?

Now, we give positive answers to this question.

Theorem 3.5 In addition to the hypotheses of Theorem  3.1, suppose that for all , there exists such that and . Then F has a unique coupled fixed point.

Proof From Theorem 3.1, we know that F has a coupled fixed point. Suppose that and are coupled fixed points of F, that is, , , and .

Now, we show that and . By the hypothesis, there exists such that and . We put and and construct two sequences and by

for all .

Since M is F-invariant and , we have
that is,
From , if we use again the property of F-invariant, then it follows that
and so
By repeating this process, we get
(3.19)
for all . From (3.1) and (3.19), we have
(3.20)
Since M is F-invariant and for all , we have
(3.21)
for all . From (3.1) and (3.21), we get
(3.22)
Thus, from (3.20) and (3.22), we have
(3.23)
for all . By repeating this process, we get
(3.24)
for all . From and , it follows that for each . Therefore, from (3.24), we have
(3.25)
Similarly, we can prove that
(3.26)
By the triangle inequality, for all , we have
(3.27)

Taking in (3.27) and using (3.25) and (3.26), we have , and so and . Therefore, F has a unique coupled fixed point. This completes the proof. □

Next, we give a simple application of our results to coupled fixed point theorems in partially ordered metric spaces.

Corollary 3.6 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Assume that there is a function with and for each and also suppose that is a mapping such that F has the mixed monotone property and
(3.28)
for all for which and . Suppose that either
1. (a)

F is continuous or

2. (b)

X has the following property:

3. (i)

if is a non-decreasing sequence with , then for all ,

4. (ii)

if is a non-increasing sequence with , then for all .

If there exists such that

then there exists such that and , that is, F has a coupled fixed point.

Proof First, we define a subset by
From Example 2.10, we can conclude that M is an F-invariant set which satisfies the transitive property. By (3.28), we have
(3.29)
for all with . Since such that
we get
For the assumption (b), for any two sequences , such that is a non-decreasing sequence in X with and is a non-increasing sequence in X with , we have
and

for all . Therefore, we have for all , and so the assumption (b) of Theorem 3.1 holds.

Now, since all the hypotheses of Theorem 3.1 hold, F has a coupled fixed point. This completes the proof. □

Corollary 3.7 In addition to the hypotheses of Corollary  3.6, suppose that for all , there exists such that , and , . Then F has a unique coupled fixed point.

Proof First, we define a subset by

From Example 2.10, we can conclude that M is an F-invariant set which satisfies the transitive property. Thus, the proof of the existence of a coupled fixed point is straightforward by following the same lines as in the proof of Corollary 3.6.

Next, we show the uniqueness of a coupled fixed point of F. Since for all , there exists such that , and , , we can conclude that and . Therefore, since all the hypotheses of Theorem 3.5 hold, F has a unique coupled fixed point. This completes the proof. □

Corollary 3.8 (Bhaskar and Lakshmikantham [21])

Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on X. Assume that there exists with
(3.30)
for all for which and . If there exists such that

then there exists such that and .

Proof Taking for some in Corollary 3.6(a), we can get the conclusion. □

Corollary 3.9 (Bhaskar and Lakshmikantham [21])

Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Suppose that X has the following property:
1. (i)

if is a non-decreasing sequence with , then for all ,

2. (ii)

if is a non-increasing sequence with , then for all .

Let be a continuous mapping having the mixed monotone property on X. Assume that there exists with
(3.31)
for all for which and . If there exists such that

then there exists such that and .

Proof Taking for some in Corollary 3.6(b), we can get the conclusion. □

## 4 Applications

In this section, we apply our theorem to the existence theorem for a solution of the following nonlinear integral equations:
(4.1)

where T is a real number such that and .

Let denote the space of -valued continuous functions on the interval . We endowed X with the metric defined by

It is clear that is a complete metric space.

Now, we consider the following assumptions:

Definition 4.1 An element is called a coupled lower and upper solution of the integral equation (4.1) if and
and

for all .

(1) is continuous;

(2) for all and for all for which and , we have

where is continuous, non-decreasing and satisfies and for each .

Next, we give the existence theorem for a unique solution of the integral equations (4.1).

Theorem 4.2 Suppose that and hold. Then the integral equations (4.1) have the unique solution if there exists a coupled lower and upper solution for (4.1).

Proof Define the mapping by

Let . It is obvious that M is an F-invariant subset of which satisfies the transitive property. It is easy to see that (b) given in Theorem 3.1 is satisfied.

Next, we prove that F has a coupled fixed point .

Now, let . Using , for all , we have
which implies that
(4.2)
Therefore, we get

for all . This implies that the condition (3.1) of Theorem 3.1 is satisfied. Moreover, it is easy to see that there exists such that and all conditions in Theorem 3.1 are satisfied. Therefore, we apply Theorem 3.1 and then we get the solution . □

## Declarations

### Acknowledgements

This project was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No.55000613). The first author would like to thank the Research Professional Development Project under the Science Achievement Scholarship of Thailand (SAST), the third 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 Number: 2011-0021821).

## Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT)
(2)
Department of Mathematics Education and the RINS, Gyeongsang National University

## References

1. Border KC: Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge University Press, New York; 1985.View Article
2. Cataldo, A, Lee, EA, Liu, X, Matsikoudis, ED, Zheng, H: A constructive Fixed point theorem and the feedback semantics of timed systems. Technical Report UCB/EECS-2006–4, EECS Dept., University of California, Berkeley (2006)
3. Guo Y: A generalization of Banach’s contraction principle for some non-obviously contractive operators in a cone metric space. Turk. J. Math. 2012, 36: 297–304.
4. Hyvärinen A: Fast and robust fixed-point algorithms for independent component analysis. IEEE Trans. Neural Netw. 1999, 10(3):626–634. 10.1109/72.761722View Article
5. Noumsi A, Derrien S, Quinton P: Acceleration of a content based image retrieval application on the RDISK cluster. IEEE International Parallel and Distributed Processing Symposium 2006.
6. Yantir A, Gulsan Topal S: Positive solutions of nonlinear m-point BVP on time scales. Int. J. Differ. Equ. 2008, 3(1):179–194. 0973–6069
7. Badii M: Existence of periodic solutions for the thermistor problem with the Joule-Thomson effect. Ann. Univ. Ferrara, Sez. 7: Sci. Mat. 2008, 54: 1–10. 10.1007/s11565-008-0041-5
8. Arvanitakis AD: A proof of the generalized Banach contraction conjecture. Proc. Am. Math. Soc. 2003, 131: 3647–3656. 10.1090/S0002-9939-03-06937-5
9. Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9
10. Mongkolkeha C, Sintunavarat W, Kumam P: Fixed point theorems for contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 93
11. Sintunavarat W, Kumam P:Weak condition for generalized multi-valued -weak contraction mappings. Appl. Math. Lett. 2011, 24: 460–465. 10.1016/j.aml.2010.10.042
12. Sintunavarat W, Kumam P: Gregus type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type. J. Inequal. Appl. 2011., 2011: Article ID 3
13. Sintunavarat W, Kumam P: Common fixed point theorems for hybrid generalized multi-valued contraction mappings. Appl. Math. Lett. 2012, 25: 52–57. 10.1016/j.aml.2011.05.047
14. Sintunavarat W, Kumam P:Common fixed point theorems for generalized v-operator classes and invariant approximations. J. Inequal. Appl. 2011., 2011: Article ID 67
15. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4
16. Cho YJ, Saadati R, Wang S: Common fixed point theorems on generalized distance in order cone metric spaces. Comput. Math. Appl. 2011, 61: 1254–1260. 10.1016/j.camwa.2011.01.004
17. Graily E, Vaezpour SM, Saadati R, Cho YJ: Generalization of fixed point theorems in ordered metric spaces concerning generalized distance. Fixed Point Theory Appl. 2011., 2011: Article ID 30
18. Nieto JJ, Lopez RR: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23: 2205–2212. 10.1007/s10114-005-0769-0
19. Sintunavarat W, Cho YJ, Kumam P: Common fixed point theorems for c -distance in ordered cone metric spaces. Comput. Math. Appl. 2011, 62: 1969–1978. 10.1016/j.camwa.2011.06.040
20. Guo D, Lakshmikantham V: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal., Theory Methods Appl. 1987, 11: 623–632. 10.1016/0362-546X(87)90077-0
21. Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
22. Abbas M, Sintunavarat W, Kumam P: Coupled fixed point in partially ordered G -metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 31
23. Cho YJ, He G, Huang NJ: The existence results of coupled quasi-solutions for a class of operator equations. Bull. Korean Math. Soc. 2010, 47: 455–465.
24. Cho YJ, Shah MH, Hussain N: Coupled fixed points of weakly F -contractive mappings in topological spaces. Appl. Math. Lett. 2011, 24: 1185–1190. 10.1016/j.aml.2011.02.004
25. Cho YJ, Rhoades BE, Saadati R, Samet B, Shantawi W: Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type. Fixed Point Theory Appl. 2012., 2012: Article ID 8
26. Gordji ME, Cho YJ, Baghani H: Coupled fixed point theorems for contractions in intuitionistic fuzzy normed spaces. Math. Comput. Model. 2011, 54: 1897–1906. 10.1016/j.mcm.2011.04.014View Article
27. Sintunavarat W, Cho YJ, Kumam P: Coupled coincidence point theorems for contractions without commutative condition in intuitionistic fuzzy normed spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81
28. Sintunavarat W, Cho YJ, Kumam P: Coupled fixed point theorems for weak contraction mapping under F -invariant set. Abstr. Appl. Anal. 2012., 2012: Article ID 324874
29. Sintunavarat W, Kumam P: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 93
30. Sintunavarat W, Cho YJ, Kumam P: Coupled fixed-point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 128
31. Sintunavarat W, Petruşel A, Kumam P:Common coupled fixed point theorems for -compatible mappings without mixed monotone property. Rend. Circ. Mat. Palermo 2012. doi:10.1007/s12215–012–0096–0
32. Samet B, Vetro C: Coupled fixed point F -invariant set and fixed point of N -order. Ann. Funct. Anal. 2010, 1: 46–56.