1. Introduction

The fixed point theory centers on the process of solving the equation of the form . One of the most widely used theory is Banach fixed point theory and its several extensions in generalized metric spaces. Therefore, fixed point theory on partially ordered sets has been studied recently in ^{[1, 3, 8, 10, 11, 13]}. For example, fixed point theorems for nonlinear and semi-linear operators on order intervals ^[1], coupled fixed point theorems ^[9] and extended the theoretical results to fixed points in partially sets ^[10] etc. On the other hand, given non-empty subsets and of the partially ordered set and a non-self mapping from to , one can perceive that the equation is improbable to have a solution. Naturally, best proximity point theorems on partially ordered set are also be studied in ^{[4, 5, 6, 7]}.

It is well-known that those abstract results can be applied to obtain an abundance of concrete results for some special problems, for instance, (a) differential and difference equation; (b) integral equation; (c) periodic boundary value problems. The purpose of this paper is to obtain the existence of solution of the integral equation for mixed monotone, contractions in the setting of partially ordered sets endowed with metrics. It is remarked that the unique solution of integral equations in this paper are established in the setting of ordered metric spaces whereas the fixed point theorems in ^{[1, 3, 9, 12]} are elicited in the framework of fixed point theorems on partially ordered metric space.

2. Fixed Point Theorems in Partially Ordered Metric Spaces

Definition 2.1 ^[2] Let be a partially ordered set, be a mapping. If , then is said to have the monotone increasing property.

Let be a partially ordered set and suppose is a complete metric space. Let be an increasing and continuous mapping. The following Theorems establish the fact that the contractive nature of is not restricted to the entire set but only restricted to the comparable elements of .

Definition 2.2 ^[14] Let be a set and let be a given real number. A functional is said to be a -metric if the following conditions are satisfied:

1. if and only if ;

2. ,for all;

3. .

A pair is called a -metric space.

Theorem 2.1 ^[10] If there exists with , whenever and there exists , with , then has a fixed point.

Theorem 2.2 ^[11] Assume that there exist upper and lower bounds of the pair for any If there exists with , whenever and there exists , with or , then has a unique fixed point . Moreover, for any , the orbit converges to the fixed point .

Let , , we define the following order relation in , for , then is a partially ordered set. Define the metric on as the follow:

then is a complete metric space.

Next, we consider the existence of solutions for the following integral equation for an unknown function (see ^[3]):

(1)

where are given continuous functions.

Let be the set of real continuous functions on . It is easy to check that is a complete metric space. Define a mapping by

(2)

Then is a solution of if and only if it is a fixed point of .

Theorem 2.3 Consider the integral equation under the following assumptions:

;

for all, if, then ;

;

or

Then has a unique solution . Moreover, for any , the orbit converges to the solution .

Proof. Let , then

It implies that . So is an increasing and continuous mapping.

Obviously, there exist upper and lower bounds of the pair for any . Hence, all conditions of Theorem 2.2 are fulfilled. This means that has a unique solution .

Moreover, for any , the orbit converges to the solution .

Next we present an example as follows.

Example 2.1 In the integral equation , let ,,,,,. Then become

(3)

Let , then . Now, all conditions of Theorem 2.3 are satisfied. On the other hand, we can easy to solve the integral equation and the unique solution is .

3. Coupled Fixed Point Theorems in Partially Ordered Metric Spaces

Now, we endow the product space with the partial order as the following:

Definition 3.1 ^[2] Let be a partially ordered set, be a mapping. If is monotone increasing in and is monotone decreasing in , that is, for any , if and and if and imply

Thus we say that has the mixed monotone property.

Definition 3.2 ^[2] We call an element a coupled fixed point of the mapping , if .

Theorem 3.1 ^[2] Let be a continuous mapping satisfy the mixed monotone property on . Assume that there exists with

If there exists such that and .

Then, there exist such that and .

Theorem 3.2 ^[2] In addition to the hypothesis of Theorem 3.1, suppose that ever pair of elements of has an upper bound or a lower bound in , then .

We assume that and are related by the relation .

Next, we will study the existence of a unique solution to the integral equation , as an application to the fixed pointed Theorem 3.2.

Let , then is a partially ordered set if we define the following order relation in : and , for , and for all .

Consider the integral equation under the following assumptions:

;

there exists , for all ,

if , then

Let

then and

Define by

(4)

Now, we will show that has the mixed monotone property. Indeed, for , that is , for all , we have

Hence, for , that is, Similarly, if , that is , for all , we have

Hence, for , that is, .

Thus is monotone increasing in and is monotone decreasing in.

Now, for , that is, for all , we have

Assume that

there exists such that and .

Theorem 3.3 Suppose the integral equation satisfy , then has a unique solution .

Proof. From the above analysis and Theorem 3.2, we can immediately obtain the result.

Example 3.1 In integral equation , let

then become

(5)

Let , thensatisfy.

Then

Let , then and .

So, all conditions of Theorem 3.3 are fulfilled. This means that has a unique solution .

Next, we will study the existence of a unique solution to the following system of integral equation as another application of the fixed pointed Theorem 3.2.

(6)

where .

A solution of the above system is a pair satisfying the above relations for all .

We consider endowed with the partial order relation:

We will also consider the following metric on :

Notice that is a metric and can be represented by using the supermum type norm

Then we have the following existence and uniqueness result.

Theorem 3.4 Consider the integral system under the following assumptions:

(1) and are continuous andis integrable with respect to the first variable.

(2) has the generalized mixed monotone property with respect to the last two variables for all .

(3) There exist in such that for each with and (or reversely), we have

for each .

(4) ,

(5) There exist such that

or

for all .

Then there exists a unique solution of the system .

Proof. We can prove that all the assumptions of Theorem 3.2 are satisfied. We define by

for each .

Then system can be written as a couple fixed point problem for :

First, we will show that has the mixed monotone property. Indeed, for , that is , for all , we have

Hence, for all , that is , .

Similarly, if , that is , for all , we have for all , that is, .

Then, for all and or ( and ), we have

We see that all the assumptions of Theorem 3.2 are satisfied and the conclusion follows.

Next, we conclude our work by an example.

Example 3.2 In integral equations system , let ,

Then (6) become

(7)

Then

Let then and

Hence, all conditions of Theorem 3.4 are fulfilled. This means that has a unique solution.

4. Coupled Fixed Point Theorems in b-metric Spaces

Theorem 4.1 ^[14] Let be a complete -metric spaces with and be a continuous mapping with the mixed monotone property on . Suppose that the following conditions are satisfied:

there exists such that

there exists such that and

Then there exists such that and .

In this section, we present an existence Theorem for such a nonlinear coupled system

(8)

where with , , and are given mapping.

Next, we consider the following -metric on

It is note that is a complete-metric space with .

Theorem 4.2 Consider the nonlinear coupled system . Suppose that the following conditions hold:

is continuous;

has the generalized mixed monotone property with respect to the last two variables for all ;

There exist continuous mappings

for each , , , with and ( or reversely), we have

; , where .

There exists , such that

for all.

Then, there exists a pair coupled solution for system .

Proof. We can prove that all the assumptions of Theorem 4.1 are satisfied. Define : by

for each . Then system can be regarded as a couple fixed point question of :

In the first place, we will prove that has the mixed monotone property. For , we have

Thus, for every element . Similarly, we can know that

Then, for all and or (and), we have

where

From the above proof, we find that all the assumptions of Theorem 4.1 are satisfied.

Example 4.1 For the integral equation , let ,, ,

Then become

Define

Obviously, , Let , we have , and . Then, all the conditions of Theorem 4.1 are satisfied. It indicates that has a pair coupled solution

Acknowledgements

The author Youhua Qian gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through grant No. 11572288, and the financial support of China Scholarship Council (CSC) through grant No. 201408330049. We are also grateful to the Professor Jinlu Li for his constructive comments and suggestions.

References

[1]	W. Y. Feng and G. Zhang, New fixed point theorems on order intervals and their applications, Fixed Point Theory Appl. (2015), 2015: 218.
	In article		View Article
[2]	M. A. Khamsi, Generalized metric spaces: A survey, J. Fixed Point Theory Appl. 17. (2015), 455-475.
	In article		View Article
[3]	S. Radenovi´ c, T. Doˇ senovi´ c,T.A. Lampert and Z. Golubov´ ı´ c, A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equa-tions, Applied Mathematics and Computation . 273 (2016), 155-164.
	In article		View Article
[4]	M. Jleli and B. Samet, Best proximity point results for MK-proximal contractions on ordered sets, J. Fixed Point Theory Appl. 17 (2015), 439-452.
	In article		View Article
[5]	S. Sadiq Basha, Best proximity point theorems on partially ordered sets, Optim Lett. (2013), 7:1035-1043.
	In article		View Article
[6]	S. Sadiq Basha, Common best proximity points: global minimal solution, Top (2013), 21: 182-188.
	In article		View Article
[7]	S. Sadiq Basha, Common best proximity points: global minimization of multi-objective func-tions, J Glob Optim. (2012), 54: 367-373.
	In article		View Article
[8]	B. Samet, Coupled point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Analysis. 72 (2010), 4508-4517.
	In article		View Article
[9]	T. Gnana Bhaskar, V.Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Analysis 65 (2006), 1379-1393.
	In article		View Article
[10]	J. J. Nieto and R. Rodr´ ıguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 22 (2005), 223-239.
	In article		View Article
[11]	A. C. Ran and M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Amer. Math. Soc. 132 (2004), 1435-1443.
	In article		View Article
[12]	T. G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis. 65 (2006), 1379-1393.
	In article		View Article
[13]	Umit Aksoy, Erdal Karapinar and ` Inci M. Erhan Fixed points of generalized admissible contractions on b-metric spaces with an application to boundary value problems, (2016), 17(6), 1095-1108.
	In article
[14]	M. F. Bota, A. Petruşel, G. Petruşel, B. Samet. Coupled fixed point theorems for single-valued operators in b-metric spaces. Fixed Point Theory & Applications, (1)(2015):1-15.
	In article		View Article
[15]	W. Sintunavarat, Nonlinear integral equations with new admissibility types in b-metric spaces, Journal of Fixed Point Theory and Applications, (2015):1-20.
	In article