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

Abstract

The purpose of this paper is to present some fixed point theorems for Meir-Keeler contractions in a complete metric space endowed with a partial order.

**MSC: **47H10.

1 Introduction and preliminaries

The Banach contraction mapping principle is one of the pivotal results of analysis. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics.

Generalization of the above principle has been a heavily investigated branch of research. In particular, Meir and Keeler

**Theorem 1.1**.

^{n}

The purpose of this article is to present a version of Theorem 1.1 in the context of ordered metric spaces.

Existence of fixed point in partially ordered sets has been recently studied in

In the context of ordered metric spaces, the usual contraction is weakened but at the expense that the operator is monotone.

2 Fixed point results: nondecreasing case

Our starting point is the following definition.

**Definition 2.1**.

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

**Remark 2.2**.

In what follows, we present the following theorem which is a version of Theorem 1.1 in the context of ordered metric spaces when the operator is nondecreasing.

**Theorem 2.3**.

_{0 }∈ _{0 }≤ _{0}

**Remark 2.4**.

**Remark 2.5**.

_{0 }= _{0}, then the proof is finished.

Suppose that _{0 }< _{0 }and

Put _{n+1 }= ^{n}_{0}. Obviously, (_{n}) is a nondecreasing sequence.

For better readability, we divide the proof into several steps.

**Step 1**: lim_{n → ∞ }_{n}, _{n + 1}) = 0.

In fact, if the sequence (_{n}) is not strictly nondecreasing, then we can find _{0 }∈ ℕ such that _{n}) is strictly nondecreasing, then by Remark 2.5, (_{n}, _{n+1})) is strictly decreasing, and hence, it is convergent. Put _{n → ∞ }_{n}, _{n+1}) (notice that _{n}, _{n+1}):

Now, we will prove that

Suppose that

Applying condition (??) to

Since _{n→∞ }_{n}, _{n+1}), there exists _{0 }∈ ℕ such that

This is a contradiction because _{n}, _{n+1}):

Therefore,

**Step 2**: (_{n}) is a Cauchy sequence.

In fact, fix

By condition (??), there exists

On the other hand, by (??), there exists _{0 }∈ ℕ such that

Fix _{0 }and in order to prove that {_{n}} is a Cauchy sequence it is sufficient to show that

We will use mathematical induction.

For

Now, we assume that (??) holds for some fixed

Then, using (??) and the inductive hypothesis, we get

Now, we consider two cases.

**Case 1: **_{n-1}_{n+p}) ≥

In this case, taking into account (??),

and, since _{n-1 }< _{n+p}, by (??)

This proves that (??) is satisfied by

**Case 2: **_{n-1}, _{n+p}) <

As _{n-1}_{n+p}) > 0 (because {_{n}} is a nondecreasing sequence and _{n}_{n+1}) > 0 for any _{0 }= _{n-1}_{n+p}) we can get

(notice that _{n-1 }< _{n+p}) and this proves that (??) is satisfied by

Therefore, {_{n}} is a Cauchy sequence.

Since _{n→∞ }_{n }=

and, therefore,

This finishes the proof.

In what follows we prove that Theorem 2.3 is still valid for

**Theorem 2.6**.

As (_{n}) is a nondecreasing sequence in _{n }→ _{n(k)}) such that _{n(k) }≤

If there exists _{0 }∈ ℕ such that _{n}) gives us that

Particularly,

Suppose that for any _{n(k) }<

Applying condition (??) to _{n(k)}

As _{n(k) }→ _{n(k)+1 }→

As (_{n(k)+1}) is a subsequence of (_{n}) and _{n }→ _{n(k)+1 }→

Now, the uniqueness of the limit in complete metric spaces gives us

This finishes the proof.

Now, we present an example where it can be appreciated that assumptions in Theorems 2.3 and 2.6 do not guarantee uniqueness of the fixed point.

Let ^{2 }and consider the usual order

Then, (_{2}) is a complete metric space considering _{2 }the Euclidean distance. The identity map

In what follows, we present a sufficient condition for the uniqueness of the fixed point in Theorems 2.3 and 2.6. This condition appears in

**Theorem 2.7**.

We consider two cases.

**Case 1: **Suppose that

Without loss of generality, we suppose

Putting

which is a contradiction.

**Case 2: **Suppose that

By condition (??), there exists

Suppose

Monotonicity of ^{n }^{n}

We consider two possibilities:

(a) Suppose that there exists _{0 }∈ ℕ such that ^{n}_{0}, and, consequently, ^{n}

(b) Suppose that ^{n}^{n}^{n}^{n}

Thus, {^{n}

Suppose

Applying condition (??) of Theorem 2.3 for

As lim_{n→∞ }^{n}^{n}_{0 }∈ ℕ such that

and, since

which contradicts to ^{n}

Therefore,

This finishes the proof.

3 Fixed point results: nonincreasing case

We start this section with the following definition.

**Definition 3.1**.

The main result of this section is the following theorem.

**Theorem 3.2**.

(_{0 }∈ _{0 }≤ _{0 }_{0 }≥ _{0},

(

_{0 }= _{0}, then it is obvious that inf{_{0 }< _{0 }(the same argument serves for _{0 }< _{0}).

In virtue that ^{n}_{0}) are comparable.

Suppose that there exists _{0 }∈ ℕ such that

In this case,

Now, we suppose that ^{n}_{0 }≠ ^{n+1}_{0 }for any

Since ^{n}_{0 }and ^{n+1}_{0 }are comparable applying the contractive condition we obtain

and this inequality is satisfied by any

Thus, {^{n}x_{0}, ^{n+1}_{0})} is a decreasing sequence of positive real numbers and, consequently, lim_{n→∞ }^{n}x_{0}, ^{n+1}_{0}) =

Using a similar argument that in Theorem 2.3, we prove that

Finally, the fact lim_{n→∞ }^{n}x_{0}, ^{n+1}_{0}) = 0 implies that inf{

This finishes the proof of (a).

(b) Suppose that

Taking into account that the mapping

is continuous and the fact that

By (

The uniqueness of the fixed point is proved as in Theorem 2.7.

**Remark 3.3**. _{n}) _{n-1 }_{n + p }

4 Examples

In this section, we present some examples which illustrate our results.

**Example 4.1**. ^{2 }_{2}. (_{2})

**Example 4.2**. _{2 }

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The three authors have contributed equally in this paper. They read and approval the final manuscript.

Acknowledgements

Partially supported by Ministerio de Ciencia y Tecnología, project MTM 2007-65706.