- Research Article
- Open Access

# Fixed Point Theorems for Generalized Weakly Contractive Condition in Ordered Metric Spaces

- HemantKumar Nashine
^{1}and - Ishak Altun
^{2}Email author

**2011**:132367

https://doi.org/10.1155/2011/132367

© H. K. Nashine and I. Altun. 2011

**Received:**30 September 2010**Accepted:**11 February 2011**Published:**10 March 2011

## Abstract

Fixed point results with the concept of generalized weakly contractive conditions in complete ordered metric spaces are derived. These results generalize the existing fixed point results in the literature.

## Keywords

- Fixed Point Theorem
- Contractive Condition
- Lower Semicontinuity
- Cauchy Sequence
- Common Fixed Point

## 1. Introduction and Preliminaries

There are a lot of generalizations of the Banach contraction mapping principle in the literature. One of the most interesting of them is the result of Khan et al. [1]. They addressed a new category of fixed point problems for a single self-map with the help of a control function which they called an altering distance function.

A function is called an altering distance function if is continuous, nondecreasing, and holds.

Khan et al. [1] given the following result.

Theorem 1.1.

for all and for some . Then has a unique fixed point.

In fact, Khan et al. [1] proved a more general theorem of which the above result is a corollary. Another generalization of the contraction principle was suggested by Alber and Guerre-Delabriere [2] in Hilbert Spaces by introducing the concept of weakly contractive mappings.

where is positive on and .

Rhoades [3] showed that most results of [2] are still valid for any Banach space. Also, Rhoades [3] proved the following very interesting fixed point theorem which contains contractions as special case .

Theorem 1.2.

Let be a complete metric space. If is a weakly contractive mapping, and in addition, is continuous and nondecreasing function, then has a unique fixed point.

In fact, Alber and Guerre-Delabriere [2] assumed an additional condition on which is . But Rhoades [3] obtained the result noted in Theorem 1.2 without using this particular assumption. Also, the weak contractions are closely related to maps of Boyd and Wong [4] and Reich type [5]. Namely, if is a lower semicontinuous function from the right, then is an upper semicontinuous function from the right, and moreover, (1.2) turns into . Therefore, the weak contraction is of Boyd and Wong type. And if we define for and , then (1.2) is replaced by . Therefore, the weak contraction becomes a Reich-type one.

Recently, the following generalized result was given by Dutta and Choudhury [6] combining Theorem 1.1 and Theorem 1.2.

Theorem 1.3.

for all , where , are both continuous and nondecreasing functions with if and only if . Then, has a unique fixed point.

Also, Zhang and Song [7] given the following generalized version of Theorem 1.2.

Theorem 1.4.

Then, there exists a unique point such that .

Very recently, Abbas and Doric [8] and Abbas and Ali Khan [9] have obtained common fixed points of four and two mappings, respectively, which satisfy generalized weak contractive condition.

In recent years, many results appeared related to fixed point theorem in complete metric spaces endowed with a partial ordering in the literature [10–25]. Most of them are a hybrid of two fundamental principle: Banach contraction theorem and the weakly contractive condition. Indeed, they deal with a monotone (either order-preserving or order-reversing) mapping satisfying, with some restriction, a classical contractive condition, and such that for some , either or , where is a self-map on metric space. The first result in this direction was given by Ran and Reurings [22, Theorem 2.1] who presented its applications to matrix equation. Subsequently, Nieto and Rodŕiguez-López [18] extended the result of Ran and Reurings [22] for nondecreasing mappings and applied to obtain a unique solution for a first-order ordinary differential equation with periodic boundary conditions.

Further, Harjani and Sadarangani [26] proved the ordered version of Theorem 1.2, Amini-Harandi and Emami [12] proved the ordered version of Rich type fixed point theorem, and Harjani and Sadarangani [27] proved ordered version of Theorem 1.3.

The aim of this paper is to give a generalized ordered version of Theorem 1.4. We will do this using the concept of weakly increasing mapping mentioned by Altun and Simsek [11] (also see [28, 29]).

## 2. Main Results

We will begin with a single map. The following theorem is a generalized version of Theorems 2.1 and 2.2 of Harjani and Sadarangani [27].

Theorem 2.1.

holds. Then, has a fixed point.

Proof.

for all .

First, we will prove that .

and so there exist and a subsequence of such that .

that is, . Thus, by the property of , we have , which is a contradiction. Therefore, we have .

Next, we show that is Cauchy.

By the property of , we have . Thus, the proof is complete.

The following corollary is a generalized version of Theorems 1.2 and 1.3 of Harjani and Sadarangani [26].

Corollary 2.2.

holds. Then, has a fixed point.

Remark 2.3.

and hence, has a unique fixed point. If condition (2.32) fails, it is possible to find examples of functions with more than one fixed point. There exist some examples to illustrate this fact in [18].

Example 2.4.

Also, it is easy to see that the other conditions of Theorem 2.1 are satisfied, and so has a fixed point in . Also, note that the weak contractive condition of Theorem 1.3 of this paper and Corollary 2.2 of [7] is not satisfied.

Now, we will give a common fixed point theorem for two maps. For this, we need the following definition, which is given in [28].

Definition 2.5.

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. There exist some examples to illustrate this fact in [11].

Theorem 2.6.

holds. Then, and have a common fixed point.

Remark 2.7.

Note that in this theorem, we remove the condition "there exists an with " of Theorem 2.1. Again, we can consider the result of Remark 2.3 for this theorem.

Proof of Theorem 2.6.

This implies , and so there exist and a subsequence of such that .

This is a contradiction. Thus, is a Cauchy sequence in , so is a Cauchy sequence. Therefore, there exists a with .

which is a contradiction. Thus, , and so .

Corollary 2.8.

holds. Then, and have a common fixed point.

Corollary 2.9.

holds. Then, and have a common fixed point.

## 3. Some Applications

In this section, we present some applications of previous sections, first we obtain some fixed point theorems for single mapping and pair of mappings satisfying a general contractive condition of integral type in complete partially ordered metric spaces. Second, we give an existence theorem for common solution of two integral equations.

Set is a Lebesgue integrable mapping which is summable and nonnegative and satisfies , for each .

Theorem 3.1.

holds. Then, has a fixed point.

Proof.

where and . Hence by Theorem 2.1 has unique fixed fixed point.

Theorem 3.2.

holds. Then, and have a common fixed point.

Proof.

where and . Hence, Theorem 2.6 has unique fixed fixed point.

Let be a partial order relation on .

Theorem 3.3.

Consider the integral equations (3.14).

(i) and are continuous,

(ii)for each ,

(iii)there exist a continuous function such that

for each and comparable ,

(iv) .

Then, the integral equations (3.14) 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 [21].

for each comparable . Therefore, all conditions of Corollary 2.8 are satisfied. Thus, the conclusion follows.

## Declarations

### Acknowledgments

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

## Authors’ Affiliations

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