# Some Fixed Point Theorems on Ordered Metric Spaces and Application

- Ishak Altun
^{1}Email author and - Hakan Simsek
^{1}

**2010**:621469

**DOI: **10.1155/2010/621469

© I. Altun and H. Simsek. 2010

**Received: **2 July 2009

**Accepted: **13 January 2010

**Published: **21 January 2010

## Abstract

We present some fixed point results for nondecreasing and weakly increasing operators in a partially ordered metric space using implicit relations. Also we give an existence theorem for common solution of two integral equations.

## 1. Introduction

Existence of fixed points in partially ordered sets has been considered recently in [1], and some generalizations of the result of [1] are given in [2–6]. Also, in [1] some applications to matrix equations are presented, in [3, 4] some applications to periodic boundary value problem and to some particular problems are, respectively, given. Later, in [6] O'Regan and Petruşel gave some existence results for Fredholm and Volterra type integral equations. In some of the above works, the fixed point results are given for nondecreasing mappings.

We can order the purposes of the paper as follows.

First, we give a slight generalization of some of the results of the above papers using an implicit relation in the following way.

In [1, 3], the authors used the following contractive condition in their result, there exists such that

Afterwards, in [2], the authors used the nonlinear contractive condition, that is,

where is anondecreasing function with for , instead of (1.1). Also in [2], the authors proved a fixed point theorem using generalized nonlinear contractive condition, that is,

for , where is as above. In the Section 3, we generalized the above contractive conditions using the implicit relation technique in such a way that

for , where is a function as given in Section 2. We can obtain various contractive conditions from (1.4). For example, if we choose

in (1.4), then, we have (1.3). Similarly we can have the contractive conditions in [7–9] from (1.4).

In some of the above mentioned theorems, the fixed point results are given for nondecreasing mappings. Also in these theorems the following condition is used:

In Section 4, we give some examples such that two weakly increasing mappings need not be nondecreasing. Therefore, we give a common fixed point theorem for two weakly increasing operators in partially ordered metric spaces using implicit relation technique. Also we did not use the condition (1.6) in this theorem. At the end, to see the applicability of our result, we give an existence theorem for common solution of two integral equations using a result of the Section 4.

## 2. Implicit Relation

Implicit relations on metric spaces have been used in many articles. See for examples, [10–15].

Let denote the nonnegative real numbers, and let be the set of all continuous functions satisfying the following conditions:

is nonincreasing in variables ;

implies ;

, for all .

Example 2.1.

where , , .

Let and . If then which implies , a contradiction. Thus and . Similarly, let and then If , then Thus is satisfied with . Also , for all . Therefore, .

Example 2.2.

, where .

Let and . If then which is a contradiction. Thus and Similarly, let and then we have If then Thus is satisfied with Also for all Therefore, .

Example 2.3.

where is right continuous and , for

Let and If then which is a contradiction. Thus and Similarly, let and then we have If , then Thus is satisfied with Also , for all Therefore, .

Example 2.4.

, where , , and

Let and Then Similarly, let and then we have If then Thus is satisfied with . Also , for all . Therefore, .

## 3. Fixed Point Theorem for Nondecreasing Mappings

We need the following lemma for the proof of our theorems.

Lemma 3.1 (See [16]).

Let be a right continuous function such that for every , then , where denotes the -times repeated composition of with itself.

Theorem 3.2.

hold. If there exists an with , then has a fixed point.

Proof.

From , we have . Therefore, letting in (3.16), we have This is a contradiction since for Thus is a Cauchy sequence in so there exists an with .

which is a contradiction to . Thus and so .

Remark 3.3.

Note that if we take that

implies ,

Instead of in Theorem 3.2, again we can have the same result.

If we combine Theorem 3.2 with Example 2.1, we obtain the following result.

Corollary 3.4.

hold. If there exists an with , then has a fixed point.

Remark 3.5.

Theorem of [2] follows from Example 2.3, Remark 3.3, and Theorem 3.2.

Remark 3.6.

We can have some new results from other examples and Theorem 3.2.

Remark 3.7.

and hence has a unique fixed point. If condition (3.27) fails, it is possible to find examples of functions with more than one fixed point. There exist some examples to illustrate this fact in [3].

## 4. Fixed Point Theorem for Weakly Increasing Mappings

Now we give a fixed point theorem for two weakly increasing mappings in ordered metric spaces using an implicit relation. Before this, we will define an implicit relation for the contractive condition of the theorem.

Let be the set of all continuous functions satisfying and the following conditions:

implies ;

and , for all .

We can easily show that, all functions in the Examples in Section 2 are in .

Definition 4.1 (See [17, 18]).

Let be a partially ordered set. Two mappings are said to be weakly increasing if and for all .

Note that, two weakly increasing mappings need not be nondecreasing.

Example 4.2.

then it is obvious that and for all . Thus and are weakly increasing mappings. Note that both and are not nondecreasing.

Example 4.3.

Let be endowed with the coordinate ordering, that is, and . Let be defined by and , then and . Thus and are weakly increasing mappings.

Example 4.4.

Thus and are weakly increasing mappings. Note that but , then is not nondecreasing. Similarly, is not nondecreasing.

Theorem 4.5.

hold, then and have a common fixed point.

Remark 4.6.

Note that, in this theorem we remove the condition "there exists an with '' of Theorem 3.2. Again we can consider the result of Remark 3.7 for this theorem.

Proof of Theorem 4.5.

From , we have . Therefore, letting in (4.32), we have . This is a contradiction since for . Thus is a Cauchy sequence in , so is a Cauchy sequence. Therefore, there exists an with .

which is a contradiction to . Thus and so .

Remark 4.7.

We can have some new results from Theorem 4.5 with some examples for .

For example, we can have the following corollary.

Corollary 4.8.

hold, then and have a common fixed point.

Proof.

Let , then it is obvious that . Therefore, the proof is complete from Theorem 4.5.

## 5. Application

Consider the integral equations

The purpose of this section is to give an existence theorem for common solution of (5.1) using Corollary 4.8. This section is related to those [19–22].

Let be a partial order relation on .

Theorem 5.1.

Consider the integral equations (5.1).

(i) and are continuous;

for each and comparable ;

(iv) .

Then the integral equations (5.1) have a unique common solution in .

Proof.

Then is a partially ordered set. Also is a complete metric space. Moreover, for any increasing sequence in converging to , we have for any . Also for every , there exists which is comparable to and [6].

Hence for each comparable . Therefore, all conditions of Corollary 4.8 are satisfied. Thus the conclusion follows.

## Declarations

### Acknowledgment

The authors thank the referees for their appreciation, valuable comments, and suggestions.

## Authors’ Affiliations

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