- Research Article
- Open Access

# Contractive-Like Mapping Principles in Ordered Metric Spaces and Application to Ordinary Differential Equations

- J Caballero
^{1}, - J Harjani
^{1}and - K Sadarangani
^{1}Email author

**2010**:916064

https://doi.org/10.1155/2010/916064

© J. Caballero et al. 2010

**Received:**25 November 2009**Accepted:**30 March 2010**Published:**10 May 2010

## Abstract

The purpose of this paper is to present a fixed point theorem for generalized contractions in partially ordered complete metric spaces. We also present an application to first-order ordinary differential equations.

## Keywords

- Ordinary Differential Equation
- Partial Order
- Point Theorem
- Lower Solution
- Nondecreasing Mapping

## 1. Introduction

Existence of fixed point in partially ordered sets has been considered recently in [1–17]. Tarski's theorem is used in [9] to show the existence of solutions for fuzzy equations and in [11] to prove existence theorems for fuzzy differential equations. In [2, 6, 7, 10, 13] some applications to ordinary differential equations and to matrix equations are presented. In [3–5, 17] some fixed point theorems are proved for a mixed monotone mapping in a metric space endowed with partial order and the authors apply their results to problems of existence and uniqueness of solutions for some boundary value problems.

In the context of ordered metric spaces, the usual contraction is weakened but at the expense that the operator is monotone. The main tool in the proof of the results in this context combines the ideas in the contraction principle with those in the monotone iterative technique [18].

In [19] the following generalization of Banach's contraction principle appears.

Theorem 1.1.

where . Then has a unique fixed point and converges to for each .

Recently, in [2] the authors prove a version of Theorem 1.1 in the context of ordered complete metric spaces. More precisely, they prove the following result.

Theorem 1.2.

Besides, suppose that for each there exists which is comparable to and . If there exists with , then has a unique fixed point.

The purpose of this paper is to generalize Theorem 1.2 with the help of the altering functions.

We recall the definition of such functions.

Definition 1.3.

An altering function is a function which satisfies the following.

(a) is continuous and nondecreasing.

(b) if and only if .

Altering functions have been used in metric fixed point theory in recent papers [20–22].

In [7] the authors use these functions and they prove some fixed point theorems in ordered metric spaces.

## 2. Fixed Point Theorems

Definition 2.1.

This definition coincides with the notion of a nondecreasing function in the case and represents the usual total order in .

In the sequel, we prove the main result of the paper.

Theorem 2.2.

where is an altering function and .

If there exist with , then has a fixed point.

Proof.

Assume that .

In what follows, we will show that is a Cauchy sequence.

This fact and (2.15) give us which is a contradiction.

This shows that is a Cauchy sequence.

and this proves that is a fixed point.

Theorem 2.3.

where is an altering function and . If there exists with , then has a fixed point.

Proof.

As is an altering function, this gives us and, thus,

Now, we present an example where it can be appreciated that the hypotheses in Theorems 2.2 and 2.3 do not guarantee uniqueness of the fixed point. This example appears in [10].

is a partially ordered set whose different elements are not comparable. Besides, is a complete metric space considering as the Euclidean distance. The identity map is trivially continuous and nondecreasing and condition (2.2) of Theorem 2.2 is satisfied since elements in are only comparable to themselves. Moreover, and has two fixed points in .

Theorem 2.4.

Adding condition (2.28) to the hypotheses of Theorem 2.2 (resp., Theorem 2.3), we obtain uniqueness of the fixed point of .

Proof.

Suppose that there exist which are fixed points of and . We distinguish two cases.

Case 1.

which is a contradiction.

Case 2.

Thus,

Assume that .

and this implies that .

and, consequently, , which is a contradiction.

Hence, .

and taking limit, we obtain .

This finishes the proof.

Remark 2.5.

Under the assumptions of Theorem 2.4, it can be proved that for every , , where is the fixed point (i.e., the operator is Picard).

In fact, for and comparable to then using the same argument that is in Case 1 of Theorem 2.4 can prove that and, hence, .

and taking limit as , we obtain or, equivalently, = .

Remark 2.6.

Notice that if is totally ordered, condition (2.28) is obviously satisfied.

Remark 2.7.

Considering the identity mapping in Theorem 2.4, we obtain Theorem 1.2, being the main result of [2].

## 3. Application to Ordinary Differential Equations

In this section we present an example where our results can be applied.

This example is inspired by [10].

where and is a continuous function.

Clearly, satisfies condition (2.28), since for the function .

Moreover, in [10] it is proved that with the above-mentioned metric satisfies condition (2.21).

Now, let denote the class of functions satisfying the following.

(i) is nondecreasing.

(ii) , .

(iii) ,

where is the class of functions defined in Section 1.

Examples of such functions are , with , , and .

Recall now the following definition

Definition 3.1.

Now, we present the following theorem about the existence of solution for problem (3.1) in presence of a lower solution.

Theorem 3.2.

where . Then the existence of a lower solution for (3.1) provides the existence of a unique solution of (3.1).

Proof.

Notice that if is a fixed point of , then is a solution of (3.1).

In the sequel, we check that hypotheses in Theorem 2.4 are satisfied.

This proves that the operator satisfies condition (2.2) of Theorem 2.2.

Finally, letting be a lower solution for (3.1), we claim that

Finally, Theorem 2.4 gives that has a unique fixed point.

Remark 3.3.

Notice that if , then . In fact, as , then is nondecreasing and, consequently, is also nondecreasing.

Moreover, as , then , and, thus, .

Finally, as , and as , then it is easily seen that .

Example 3.4.

It is easily seen that . Taking into account Remark 3.3, .

where is the function above mentioned.

This example can be treated by our Theorem 3.2 but it cannot be covered by the results of [6] because is not increasing.

## Declarations

### Acknowledgments

This research was partially supported by "Ministerio de Educación y Ciencia", Project MTM 2007/65706. This work is dedicated to Professor W. Takahashi on the occasion of his retirement.

## Authors’ Affiliations

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This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.