Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain

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.

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].

Let

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

Theorem 1.1.

Let

where

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.

Let

where

Besides, suppose that for each

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

(a)

(b)

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.

If

This definition coincides with the notion of a nondecreasing function in the case

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

Theorem 2.2.

Let

where

If there exist

Proof.

If

Put

Using the fact that

If there exists

Assume that

Then, from (2.4), we have

Letting

and, consequently,

In what follows, we will show that

Suppose that

Further, corresponding to

Using (2.10), (2.11), and the triangular inequality, we have

Letting

Again, the triangular inequality gives us

Letting

As

Taking into account (2.13) and (2.15) and the fact that

As

Since

This fact and (2.15) give us

This shows that

Since

and this proves that

In what follows, we prove that Theorem 2.2 is still valid for

Theorem 2.3.

Let

where

Proof.

Following the proof of Theorem 2.2, we only have to check that

Letting

or, equivalently,

As

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].

Let

In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 2.2 and 2.3. This condition appears in [16] and says that

In [10] it is proved that condition (2.27) is equivalent to

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

Case 1.

If

which is a contradiction.

Case 2.

Using condition (2.28), there exists

Since

Thus,

Assume that

Taking into account that

and this implies that

Since

and, consequently,

Hence,

Analogously, it can be proved that

Finally, as

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

In fact, for

If

Finally,

and taking limit as

Remark 2.6.

Notice that if

Remark 2.7.

Considering

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].

We study the existence of solution for the following first-order periodic problem

where

Previously, we considered the space

is a complete metric space.

Clearly,

Moreover, in [10] it is proved that

Now, let

(i)

(ii)

(iii)

where

Examples of such functions are

Recall now the following definition

Definition 3.1.

A lower solution for (3.1) is a function

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

Theorem 3.2.

Consider problem (3.1) with

such that for

where

Proof.

Problem (3.1) can be written as

This problem is equivalent to the integral equation

where

Define

Notice that if

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

The mapping

and this implies, taking into account that

Besides, for

Using the Cauchy-Schwarz inequality in the last integral, we get

The first integral gives us

As

Taking into account (3.14), (3.15), and (3.16), from (3.13) we get

Since

or, equivalently,

This implies that

Putting

This proves that the operator

Finally, letting

In fact,

Multiplying by

and this gives us

As

and so

This and (3.24) give us

and, consequently,

Finally, Theorem 2.4 gives that

Remark 3.3.

Notice that if

Moreover, as

Finally, as

Example 3.4.

Consider

It is easily seen that

Now, we consider problem (3.1) with

such that for

where

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

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.